diff --git a/data_all_eng_slimpj/shuffled/split2/finalzakt b/data_all_eng_slimpj/shuffled/split2/finalzakt new file mode 100644 index 0000000000000000000000000000000000000000..da50fa75d3e81dcf520552f9b3ca7d3b39ce076f --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzakt @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nMulti-view data contain information relevant for the identification of patterns or clusters that allow us to specify groups of subjects or objects. Our focus is on patients for which we have bio-medical and\/or clinical observations describing patient characteristics obtained from various diagnostic procedures or produced by different molecular technologies~\\cite{fu2020overview}. The different types of subject characteristics constitute views related to these patients. Integrative clustering of these views facilitates the detection of patient groups, with the consequence of improved clinical diagnostic and treatment schemes.\n\nSimple integration of single view clustering results is not appropriate for the diversity and complexity of available medical observations. Even state-of-the-art multi-view approaches have their limitations. Ensemble clustering has the potential to overcome some of them \\cite{ronan2018openensembles}\\cite{ alqurashi2019clustering}. For instance, while spectral clustering might be the optimal method for a specific image-based analysis, agglomerative clustering might be more appropriate for tabular data. This can be the case, where patient data reflect some hierarchical structure in a disease of interest and its subtypes \\cite{ciriello2013emerging}. Moreover, in real-world applications the data views originate from highly heterogeneous input sources. Thus, each view needs to be clustered with the best possible and most adequate strategy. Multi-view clustering methods are widely applied within the bio-medical domain. Molecular data from different biological layers are retrieved for the same set of patients. The clusters inferred from these multi-omics observations facilitate the stratification of cancer patients into sub-groups, paving the way towards precision medicine. \n\nHere, we present Parea, a generic and flexible methodology to build clustering ensembles of arbitrary complexity. To be precise, we introduce Parea$_{\\textit{hc}}$, an ensemble method which performs hierarchical clustering and data fusion for disease subtype discovery. The name of our method is derived from the Greek word {\\it Parea}, meaning a group of friends who gather to share experiences, values, and ideas.\n\nThe manuscript is structured as follows: Section~\\ref{sec:Parea_general} formally describes the ensemble structures we have developed. Section~\\ref{sec:approach} presents our multi-view hierarchical ensemble clustering approach for disease subtype detection. We discuss related work in Section~\\ref{sec:other_methods} and introduce the methods we used for benchmark comparisons. In Section~\\ref{sec:results} the results are presented and discussed. A brief introduction to the \\textit{Pyrea} Python package is given in Section~\\ref{sec:pyrea}. We conclude with Section~\\ref{sec:conclusion}.\n\n\\section{General ensemble architecture}\n\\label{sec:Parea_general}\n\nThe following concept for multi-view ensemble clustering is proposed. Each view $V \\in \\mathbb{R}^{n\\times p}$ is associated with a specific clustering method $c$, where $n$ is the number of samples and $p$ is the number of predictors, and in total we have $N$ data views. An ensemble, called $\\mathcal{E}$, can be modelled using a set of views $\\mathcal{V}$ and an associated fusion algorithm $f$. \n\n\\begin{equation}\n \\mathcal{V} \\mapsfrom \\{(V \\in \\mathbb{R}^{n\\times p}, c)\\} \n\\end{equation}\n\n\\begin{equation}\n \\mathcal{E}(\\mathcal{V}, f) \\mapsto \\widetilde{V}\\in \\mathbb{R}^{p\\times p}\n\\end{equation}\n\n\\begin{equation}\n \\mathcal{V} \\mapsfrom \\{(\\widetilde{V}\\in \\mathbb{R}^{p\\times p}, c)\\} \n\\end{equation}\n\nFrom the above equations we can see that a specified ensemble $\\mathcal{E}$ creates a view $\\widetilde{V} \\in \\mathbb{R}^{p\\times p}$ which again can be used to specify $\\mathcal{V}$, including an associated clustering algorithm $c$. With this concept it is possible to \\textit{layer-wise} stack views and ensembles into arbitrarily complex ensemble architectures. It should be noted, however, that the resulting view of a specified ensemble $\\mathcal{E}$ forms an affinity matrix of dimension $p \\times p$, and thus only those clustering methods which are congruent with an affinity or a distance matrix for input are applicable. \n\n\\section{Proposed Ensemble Approach}\n\\label{sec:approach}\n\nThe Parea$_{\\textit{hc}}$ ensemble approach comprises two different strategies: Parea$_{\\textit{hc}}^{1}$ is limited to the application of two selected hierarchical clustering methods. Parea$_{\\textit{hc}}^{2}$ allows for the hierarchical clustering methods in the data fusion process to be varied. \n\nThe two hierarchical clustering methods of Parea$_{\\textit{hc}}^{1}$ for multiple data views are $hc_{1}$ and $hc_{2}$. The resulting fused matrices $\\widetilde{V}$ are clustered again with the same methods and the results are combined to form a final consensus (see Figure \\ref{fig:Parea_arch}, panel (a)). A formal description of Parea$_{\\textit{hc}}^{1}$ is give by:\n\n\\begin{equation}\n\\mathcal{V}_{1} \\mapsfrom \\{(V_{1},hc_{1}),(V_{2},hc_{1}),\\ldots, (V_{N},hc_{1})\\}, \n\\quad\n\\mathcal{V}_{2} \\mapsfrom \\{(V_{1},hc_{2}),(V_{2},hc_{2}),\\ldots, (V_{N},hc_{2})\\}\n\\end{equation}\n\n\\begin{equation}\n\\mathcal{E}_{1}(\\mathcal{V}_{1}, f) \\mapsto \\widetilde{V}_{1},\n\\quad\n\\mathcal{E}_{2}(\\mathcal{V}_{2}, f) \\mapsto \\widetilde{V}_{2}\n\\end{equation}\n\n\n\\begin{equation}\n \\mathcal{V}_{3} \\mapsfrom \\{(\\widetilde{V}_{1},hc_{1}),(\\widetilde{V}_{2},hc_{2})\\} \n\\end{equation}\n\n\n\\begin{equation}\n\\mathcal{E}_{3}(\\mathcal{V}_{3}, f) \\mapsto \\widetilde{V}_{3}.\n\\end{equation}\nThe affinity matrix $\\widetilde{V}_{3}$ is then clustered with $hc_{1}$ and $hc_{2}$ from the first layer, and the consensus of the obtained clustering solutions constitute the final cluster assignments: \n\n\\begin{equation}\n\\mathcal{V}_{4} \\mapsfrom \\{(\\widetilde{V_{3}},hc_{1}),(\\widetilde{V_{3}},hc_{2})\\}\n\\end{equation}\n\n\n\\begin{equation}\n\\text{cons}(\\mathcal{V}_{4}) \n\\end{equation}\nGiven the proposed ensemble architecture, a genetic algorithm infers the optimal combination of $hc_{1}$ and $hc_{2}$ using the silhouette coefficient \\cite{rousseeuw1987silhouettes} as a fitness function. For the data fusion algorithm $f$, we utilize a method introduced in \\cite{pfeifer2021hierarchical}. See Figure \\ref{fig:Parea_arch} for a graphical illustration of the described ensemble architecture. \n\nIn the Parea$_{\\textit{hc}}^{2}$ approach the views are clustered with up to $N$ different hierarchical clustering methods $hc_{1}, hc_{2}, \\ldots, hc_{N}$, where $N$ is the number of data views.\nA formal description of the Parea$_{\\textit{hc}}^{2}$ is given by:\n\n\n\\begin{equation}\n\\mathcal{V}_{1} \\mapsfrom \\{(V_{1},hc_{1}),(V_{2},hc_{2}),\\ldots, (V_{N},hc_{N})\\}\n\\end{equation}\n\n\n\\begin{equation}\n\\mathcal{E}_{1}(\\mathcal{V}_{1}, f) \\mapsto \\widetilde{V}_{1}\n\\end{equation}\nThe affinity matrix $\\widetilde{V}_{1}$ is then clustered with $hc_{1}$, $hc_{2}$, and $hc_{3}$. The consensus of the obtained clustering results are the final cluster assignments: \n\n\n\\begin{equation}\n \\mathcal{V}_{2} \\mapsfrom \\{(\\widetilde{V}_{1},hc_{1}),(\\widetilde{V}_{1},hc_{2}), (\\widetilde{V}_{1},hc_{3})\\} \n\\end{equation}\n\n\n\\begin{equation}\n\\text{cons}(\\mathcal{V}_{2}) \n\\end{equation}\nThe best combination of clustering methods ($hc_{1}$, $hc_{2}$, and $hc_{3}$) is again inferred by a genetic algorithm, where the silhouette coefficient \\cite{rousseeuw1987silhouettes} is deployed as a fitness function. We consider eight different hierarchical clustering methods, namely single-linkage and complete-linkage clustering \\cite{murtagh2012algorithms}, an unweighted pair-group method using arithmetic averages (UPGMA) \\cite{sokal1958statistical}, a weighted pair-group method using arithmetic averages (WPGMA) \\cite{sokal1958statistical} clustering, a weighted pair-group method using centroids (WPGMC) \\cite{gower1967comparison}, an unweighted pair-group method using centroids (UPGMC) \\cite{sokal1958statistical} clustering, and clustering based on Ward's minimum variance \\cite{ward1963hierarchical}\\cite{murtagh2014ward}.\n\n\\begin{figure*}[h!]\n\\begin{center}\n\\includegraphics[width=17.3cm]{Figures\/Parea_general_crop.pdf\n\\end{center}\n\\caption{\\textbf{(a)} The Parea$_{\\textit{hc}}^{1}$ ensemble architecture. The views are organised in two ensembles. Each ensemble is associated with a specific clustering method. \\textbf{(b)} The Parea$_{\\textit{hc}}^{2}$ ensemble architecture. The views can be clustered using different hierarchical clustering techniques. }\\label{fig:Parea_arch}\n\\end{figure*} \n\n\\section{Alternative approaches}\n\\label{sec:other_methods}\n\nA comparison with alternative approaches was conducted using a set of state-of-the-art multi-view clustering methods implemented within the Python package mvlearn \\cite{perry2021mvlearn}. Part of this set is a multi-view spectral clustering approach with and without the use of a co-training framework \\cite{kumar2011co}. An additional method is based on multi-view $k$-means \\cite{chao2017survey} clustering, plus an implementation of multi-view spherical $k$-means using the co-EM framework as described in \\cite{bickel2004multi}.\n\nFor disease subtype detection we compared Parea$_{\\textit{hc}}$ with NEMO \\cite{rappoport2019nemo}, SNF (Similar Network Fusion) \\cite{wang2014similarity}, HCfused \\cite{pfeifer2021hierarchical}, and PINSplus \\cite{nguyen2019pinsplus}. SNF models the similarity between subjects or objects as a network and then fuses these networks via an interchanging diffusion process. Spectral clustering is applied to the fused network to infer the final cluster assignments. NEMO builds upon SNF, but provides solutions to partial data and implements a novel \\textit{eigen-gap} method \\cite{von2007tutorial} to infer the optimal number of clusters.\nThe method implemented within PINSplus systematically adds noise to the data and infers the best number of clusters based on the stability against this noise. When the best $k$ (number of clusters) is detected, binary matrices are formulated reflecting the cluster solutions for each single-view contribution. A final agreement matrix is derived by counting the number of times two subjects or objects appear in the same cluster. This agreement matrix is then used for a standard clustering method, such as $k$-means. \n\nHCfused is a hierarchical clustering and data fusion algorithm. It is based on \\textit{bottom-up} agglomerative clustering to create a fused affinity matrix. At each step two clusters are merged within the view that provides the minimal distance between these clusters. The number of times two samples appear in the same cluster is reflected by a co-association matrix. The final cluster assignments are obtained from hierarchical clustering based on Ward's minimum variance \\cite{ward1963hierarchical}\\cite{murtagh2014ward}.\n\n\\section{Results and discussion}\n\\label{sec:results}\n\n\\subsection*{Evaluation on machine learning benchmark data sets}\n\nWe tested Parea$_{\\textit{hc}}$ on the IRIS data set available from the UCI Machine Learning Repository\\footnote{See \\url{https:\/\/archive.ics.uci.edu\/ml\/datasets\/iris}}. It contains three classes of 50 instances each, where each class refers to a type of iris plant.\nWe compared the results of the ensemble with each of the possible ensemble members, and also compared the outcomes with a consensus approach, where the cluster solutions of all single algorithms were combined. The data pool was sampled 50 times and the clustering methods were applied on each iteration. The number of clusters $k$ was set to three corresponding to the ground truth.\nThese sanity checks revealed superior results for the Parea$_{\\textit{hc}}^{1}$ ensemble method compared to a simple consensus approach, as judged by the Adjusted Rand Index (ARI) (see Figure \\ref{fig:IRIS}). We could further observe that the underlying genetic algorithm infers ward.D and ward.D2 as the best performing method combination (Figure \\ref{fig:IRIS}, panel (b)). \n\nIn an additional investigation, we evaluated the accuracy of the discussed methods on the nutrimouse data set \\cite{martin2007novel}. The data set originates from a nutrigenomic study of the mouse in which the effects of five regimens with contrasted fatty acid compositions on liver lipids and hepatic gene expression in mice were considered. Two views were acquired from forty mice. First, gene expressions of 120 genes were measured in liver cells, as potentially relevant for the nutrition study. Second, lipid concentrations of 21 hepatic fatty acids were measured by gas chromatography.\n\nFor the nutrimouse data set Parea$_{\\textit{hc}}^{2}$ performs better than Parea$_{\\textit{hc}}^{1}$ (see Figure \\ref{fig:100leaves}, panel (a)). This observation suggests that higher accuracy can be achieved when the views are analysed with disjoint clustering strategies. Parea$_{\\textit{hc}}^{2}$ performs best when the median NMI (\\textit{Normalised Mutual Information}) is used as a metric. However, at the same time we observed higher variance for the Parea ensembles compared to multi-view spectral clustering. It is worth noting that the alternative spectral-based approaches from the mvlearn Python package performed as well as Parea$_{\\textit{hc}}^{1}$. \n\nLast, we further studied Parea's performance on the one-hundred plant species leaves multi-view data set, available from the UCI Machine Learning Repository\\footnote{See \\url{https:\/\/archive.ics.uci.edu\/ml\/datasets\/One-hundred+plant+species+leaves+data+set}}. For each feature, a 64 element vector is observed per leaf sample. These vectors form contiguous descriptors for shape as well as texture and margin. In contrast to the IRIS data set it is composed of multiple views. As can be seen in Figure \\ref{fig:100leaves}, Parea$_{\\textit{hc}}^{1}$ and Parea$_{\\textit{hc}}^{2}$ compete well with the spectral-based approaches. Multi-view $k$-means clustering does not capture the ground truth class distribution (see Figure \\ref{fig:100leaves}, panel (b)).\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/Parea_vs_Single.pdf}\n \\caption{}\n \n \\end{subfigure}\n \n \\begin{subfigure}[b]{0.49\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/Heatmap.pdf}\n \\caption{}\n \n \\end{subfigure}\n \\caption{(a) Parea$_{\\textit{hc}}$ versus each of the available ensemble methods executed on the \\textit{single-view} IRIS data set. (b) The pairwise Parea$_{\\textit{hc}}$ ensembles inferred by the genetic algorithm for clustering the IRIS data set.}\n \\label{fig:IRIS}\n\\end{figure}\n\n\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/Mouse_diet.pdf}\n \\caption{}\n \n \\end{subfigure}\n \n \\begin{subfigure}[b]{0.50\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/100leaves.pdf}\n \\caption{}\n \n \\end{subfigure}\n \\caption{\n Parea$_{\\textit{hc}}$ versus a set of state-of-the-art multi-view methods implemented within the mvlearn package \\cite{perry2021mvlearn}.\n (a) The \\textit{multi-view} nutrimouse data set. (b) The \\textit{multi-view} 100 leaves data set.\n }\n \\label{fig:100leaves}\n\\end{figure}\n\n\n\\subsection*{Multi-omics clustering for disease subtype discovery}\n\nWe applied the aforementioned ensemble methods to patient data of seven different cancer types, namely glioblastoma multiforme (GBM), kidney renal clear cell carcinoma (KIRC), liver hepatocellular carcinoma (LIHC), skin cutaneous melanoma (SKCM), ovarian serous cystadenocarcinoma (OV), sarcoma (SARC), and acute myeloid leukemia (AML), aiming at the externally known survival outcome. The Parea$_{\\textit{hc}}$ ensemble approach was studied on multi-omics data, including gene expression data (mRNA), DNA methylation, and micro-RNAs. The obtained results were compared with the alternative approaches SNF, NEMO, PINSplus, and HCfused. All data were retrieved from the ACGT lab at Tel Aviv University\\footnote{See \\url{http:\/\/acgt.cs.tau.ac.il\/multi_omic_benchmark\/download.html}}, which makes available a data repository recently proposed as a convenient benchmark for multi-omics clustering approaches \\cite{rappoport2018multi}. The data were pre-processed as follows: patients and features with more than $20\\%$ missing values were removed and the remaining missing values were imputed with $k$-nearest neighbor imputation. In the methylation data, we selected those 5000 features with maximal variance in each data set. All features were then normalised to have mean zero and standard deviation one. \n\nThe resulting multi-omics views were then clustered with the discussed multi-view clustering approaches. It is important to mention that the cancer patients were exclusively analysed based on their genomic footprints. The survival statuses and times of all patients were retrieved to validate the quality of the inferred patient clusters. Here, a well performing clustering method is determined by its capability to separate patients into groups with similar event statuses in terms of survival. To this end we randomly sampled 100 patients 30 times from the data pool and performed the Cox log-rank test, which is an inferential procedure for the comparison of event time distributions among independent (i.e. clustered) patient groups.\n\nIn the case of Parea$_{\\textit{hc}}$, the optimal number of clusters was determined by the silhouette coefficient. We set the number of HCfused \\cite{pfeifer2021hierarchical} fusion iterations to 30. The maximum possible number of clusters was fixed to 10. The obtained Cox log-rank $p$-values ($\\alpha=0.05$ significance level) are displayed in Figure \\ref{fig:TCGA} and Table \\ref{tab:table1}. Our Parea$_{\\textit{hc}}$ ensembles outperformed the alternative approaches in six out of seven cases. SKCM is the only cancer type for which HCfused achieved better results. Notably, spectral-based clustering (see NEMO \\cite{rappoport2019nemo} and SNF \\cite{wang2014similarity}) does not perform as well as with the benchmark machine learning data sets. We further learned that Parea$_{\\textit{hc}}^{1}$ is more accurate than Parea$_{\\textit{hc}}^{2}$ (Figure \\ref{fig:TCGA}, Table \\ref{tab:table1}). \n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=14cm]{Figures\/TCGA_median.pdf\n\\end{center}\n\\caption{Parea$_{\\textit{hc}}$ in comparison with the alternative approaches SNF, NEMO, PINSplus, and HCfused. Colored bars represent the method-specific median Cox log-rank $p$-values for the seven different cancer types. The vertical line refers to $\\alpha=0.05$ significance level.}\n\\label{fig:TCGA}\n\\end{figure}\n\n \n \n\n\n\\begin{table}[h!]\n \\begin{center}\n \\caption{Survival analysis of TCGA cancer data}\n \\label{tab:table1}\n \\begin{tabular}{l l l l l l l l}\n \\textbf{Cancer type} & \\textbf{Sample size} & SNF & PINSplus & NEMO & HCfused &Parea$_{\\textit{hc}}^{1}$ & Parea$_{\\textit{hc}}^{2}$ \\\\\n \\hline\n GBM & 538 & 0.1304 & 0.2223 & \\textbf{0.0347} & 0.0997 & \\textbf{0.0447}& \\textbf{0.0347}\\\\\n KIRC & 606 & 0.3962 & 0.4005 & 0.3464 & 0.0561 & \\textbf{0.0137} & \\textbf{0.0400}\\\\\n \n LIHC & 423 & 0.5357 & 0.6731 & 0.4354 & 0.2062 & \\textbf{0.0334} & \\textbf{0.0436}\\\\\n SKCM & 473 & 0.5153 & 0.3956 & 0.4565 & 0.0699 & 0.1677 & 0.1629\\\\\n OV & 307 & 0.4042 & 0.5300 & 0.3593 & 0.2594 & 0.1685 & 0.2870 \\\\\n SARC & 265 & 0.1622 & 0.2024 & 0.0979 & \\textbf{0.0408} & \\textbf{0.0076} & \\textbf{0.0109} \\\\\n AML & 173 & 0.0604 & 0.1973 & \\textbf{0.0440} & 0.1148 & \\textbf{0.0167} & 0.0502\\\\ \\hline\n \n \\end{tabular}\n \\end{center}\n \\centering\nDisplay of the median Cox log-rank p-values. Significant results ($\\alpha=0.05$) are highlighted in bold. \n\\end{table}\n\n\\section{Pyrea Software Library}\n\\label{sec:pyrea}\nPyrea is a software framework which allows for flexible ensembles to be built, layer-wise, using a variety of clustering algorithms and fusion techniques. It can be used to implement the ensemble methods discussed throughout this paper and any other custom ensemble structure that users may wish to create. It is made available as a Python package, which can be installed via the \\textit{pip} package manager. See the project's GitHub repository under \\url{https:\/\/github.com\/mdbloice\/Pyrea} for source code and usage examples, such as how to implement some of the ensembles mentioned above. Comprehensive documentation is available under \\url{https:\/\/pyrea.readthedocs.io}. Pyrea is MIT licensed, and available for Windows, macOS, and Linux.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe have presented Parea$_{\\textit{hc}}$, an ensemble approach for multi-view hierarchical clustering for cancer subtype discovery. We could show that Parea$_{\\textit{hc}}$ competes well with current state-of-the-art multi-view clustering techniques, on classical machine learning data sets as well as for real-world multi-omics cancer patient data. The proposed methodology for building ensembles is highly versatile and allows for ensembles to be stacked layer-wise. Additionally, the Parea ensemble strategy is not limited to a specific clustering technique. Within our developed Python package \\textit{Pyrea}, we enable flexible ensemble building, while providing a wide-range of clustering algorithms, data fusion techniques, and metrics to infer the best number of clusters. \n\n\n\n\n\n \n\n\n\n \n \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\nThe problem of group synchronization is critical for various tasks in data science, including structure from motion (SfM), simultaneous localization and mapping (SLAM), Cryo-electron microscopy imaging, sensor network localization, multi-object matching and community detection. Rotation synchronization, also known as rotation averaging, is the most common group synchronization problems in 3D reconstruction. It asks to recover camera rotations from \nmeasured\nrelative rotations \nbetween\npairs of cameras. Permutation synchronization, which has applications in multi-object matching, asks to obtain globally consistent matches of objects from possibly erroneous measurements of matches between some pairs of objects. The simplest example of group synchronization is $\\mathbb Z_2$ synchronization, which appears in community detection. \n\nThe general problem of group synchronization can be mathematically formulated as follows.\nAssume a graph $G([n],E)$ with $n$ vertices indexed by $[n]=\\{1,\\ldots,n\\}$, a group $\\mathcal{G}$, and a set of group elements $\\{g_i^*\\}_{i=1}^n \\subseteq \\mathcal{G}$. \nThe \nproblem asks to recover $\\{g_i^*\\}_{i=1}^n$ from noisy and corrupted measurements $\\{g_{ij}\\}_{ij \\in E}$ of the group ratios $\\{g_i^*g_j^{*-1}\\}_{ij \\in E}$.\nWe note that one can only recover, or approximate, the original group elements $\\{g_i^*\\}_{i\\in [n]}$ up to a right group action.\nIndeed, for any $g_0\\in \\mathcal G$, $g_{ij}^*$ can also be written as $g_i^*g_0(g_j^*g_0)^{-1}$ and thus $\\{g_i^*g_0\\}_{i\\in [n]}$ is also a solution. \nThe above mentioned synchronization problems (rotation, permutation, and $\\mathbb Z_2$ synchronization) correspond to the groups $SO(3)$, $S_N$, and $\\mathbb Z_2$, respectively.\n\nThe most challenging issue for group synchronization is the practical scenario of highly corrupted and noisy measurements. Traditional least squares solvers often fail to produce accurate results in such a scenario. \nMoreover, some basic estimators that seem to be robust to corruption\noften do not tolerate in practice high level of noise. \nWe aim to propose a general method for group synchronization that may tolerate high levels and different kinds of corruption and noise. While our basic ideas are formally general, in order to carefully refine and test them, we focus on the special problem of rotation synchronization, which is also known as rotation averaging \\cite{RotationAveraging13}.\nWe choose this problem as it is the most common, and possibly most difficult, synchronization problem in 3D computer vision.\n\n\\subsection{Related Works}\n\\label{sec:related}\nMost previous group synchronization solvers minimize an energy function. \nFor the discrete groups $\\mathbb Z_2$ and $S_N$, least squares energy minimization is commonly used.\nRelevant robustness results, under special corruption and noise models, are discussed in \\citet{Z2Afonso2, Z2abbe, Z2Afonso, chen_partial, Huang13, PPM_vahan, deepti}.\n\nFor Lie groups, such as $SO(D)$, that is, the group of $D \\times D$ orthogonal matrices with determinant 1, where $D\\geq 2$, least squares minimization was proposed to handle Gaussian noise \\citep{rotationNP,StrongDuality18,Govindu04_Lie}. However, when the measurements are also adversarially corrupted, this framework does not work well and other corruption-robust energy functions need to be used \\cite{ChatterjeeG13_rotation, L12, HartleyAT11_rotation, SO2ML, wang2013exact}. \nThe most common corruption-robust energy function uses least absolute deviations.\n\\citet{wang2013exact} prove that under a very special probabilistic setting with $\\mathcal G = SO(D)$, the pure minimizer of this energy function can exactly recover the underlying group elements with high probability. However, their assumptions are strong and they use convex relaxation, which changes the original problem and is expensive to compute. \n \\citet{SO2ML} apply a trimmed averaging procedure for robustly solving $SO(2)$ synchronization. They are able to recover the ground truth group elements under a special deterministic condition on the topology of the corrupted subgraph. However, the verification of this condition and its extension to $SO(D)$, where $D>2$, are nontrivial.\n\\citet{HartleyAT11_rotation} used the Weiszfeld algorithm\n to minimize the least-absolute-deviations energy function\nwith $\\mathcal G = SO(3)$. Their method iteratively computes geodesic medians. However, they update only one rotation matrix per iteration, which results in slow empirical convergence and may increase the possibility of getting stuck at local minima. \\citet{L12} proposed a robust Lie-algebraic averaging method over $\\mathcal G = SO(3)$. They apply an iteratively reweighted least squares (IRLS) procedure in the tangent space of $SO(3)$ to optimize different robust energy functions, including the one that uses least absolute deviations. They claim that the use of the $\\ell_{1\/2}$ norm for deviations results in highest empirical accuracy. The empirical robustness of the two latter works is not theoretically guaranteed, even in simple settings. A recent deep learning method \\cite{NeuroRA} solves a supervised version of rotation synchronization,\nbut it does not apply to the above unsupervised formulation of the problem.\n\n\\citet{truncatedLS} use least absolute deviations minimization for solving 1D translation synchronization, where $\\mathcal G=\\mathbb R$ with addition. \nThey propose a special version of IRLS and provide a deterministic exact recovery guarantee that depends on properties of the graph and its Laplacian. They do not explain their general result in an adversarial setting, but in a very special noisy setting.\n\n \nRobustness results were established for least absolute deviations minimization in camera location estimation, which is somewhat similar to group synchronization \\cite{HandLV15,LUDrecovery}. These results assume special probabilistic setting, however, they have relatively weak assumptions on the corruption model.\n\n\n\n Several energy minimization solutions have been proposed to $SE(3)$ synchronization \\cite{SE3_MCMC, SE3_SDP_jesus, SE3_Rosen, SE3_sync, SE3_RPCA}. This problem asks to jointly estimate camera rotations and locations from relative measurements of both. Neither of these solutions successfully address highly corrupted scenarios.\n\n\nOther works on group synchronization, which do not minimize energy functions but aim to robustly recover corrupted solutions,\nscreen corrupted edges using cycle consistency information. For a group $\\mathcal{G}$ with group identity denoted by $e$, any $m \\geq 3$, any cycle\n$L=\\{i_1i_2, i_2i_3\\dots i_m i_1\\}$ of length $m$ and any corresponding product of ground-truth group ratios along $L$, $g^*_L=g_{i_1i_2}^*g_{i_2i_3}^*\\cdots g_{i_mi_1}^*$, the cycle-consistency constraint is\n$g^*_L= e$.\nIn practice, one is given the product of measurements, that is, $g_L=g_{i_1i_2}g_{i_2i_3}\\cdots g_{i_mi_1}$, and in order to ``approximately satisfy the cycle-consistency constraint'' one tries to enforce $g_L$ to be sufficiently close to $e$.\n\\citet{Zach2010} uses the cycle-consistency constraint to detect corrupted relative rotations in $SO(3)$. It seeks to maximize a log likelihood function, which is based on the cycle-consistency constraint, using either belief propagation or convex relaxation. However, no theoretical guarantees are provided for the accuracy of outlier detection. Moreover, the log likelihood function implies very strong assumptions on the joint densities of the given relative rotations.\n\\citet{shen2016} classify the relative rotations as uncorrupted if they belong to any cycle that approximately satisfies the cycle-consistency constraint. However, this work only exploits local information and cannot handle the adversarial corruption case, where corrupted cycles can be approximately consistent. \n\nAn iterative reweighting strategy, IR-AAB \\cite{AAB}, was proposed to detect and remove corrupted pairwise directions for the different problem of camera location estimation. It utilizes another notion of cycle-consistency to infer the corruption level of each edge. \\citet{cemp} extend the latter idea, and interpret it as a message passing procedure, to solve group synchronization with any compact group. They refer to their new procedure as cycle-edge message passing (CEMP). While We follow ideas of \\citet{cemp,AAB}, we directly solve for group elements, instead of estimating corruption levels, using them to initial cleaning of edges and solving the cleaner problem with another method.\n\nTo the best of our knowledge, the unified frameworks for group synchronization are \\citet{ICMLphase,cemp,AMP_compact}. However, \\citet{ICMLphase} and \\citet{AMP_compact} assume special probabilistic models that do not address adversarial corruption. Furthermore, \\citet{ICMLphase} only applies to Lie groups and the different setting of multi-frequencies.\n\n\n\\subsection{Contribution of This Work}\nCurrent group synchronization solvers often do not perform well with highly corrupted and noisy group ratios. In order to address this issue, we propose a rotation synchronization solver that can address in practice high levels of noise and corruption. Our main ideas seem to generalize to group synchronization with any compact group, but more careful developments and testing are needed for other groups. We emphasize the following specific contributions of this work:\n\\begin{itemize}\n\\item\nWe propose the message passing least squares (MPLS) framework as an alternative paradigm to IRLS for group synchronization, and in particular, rotation synchronization. It uses the theoretically guaranteed CEMP algorithm\nfor estimating the underlying corruption levels.\nThese estimates are then used for learning better weights for the weighted least squares problem.\n\\item \nWe explain in Section \\ref{sec:issue} why the common IRLS solver may not be accurate enough and in Section \\ref{sec:mpls} why MPLS can overcome these obstacles. \n\\item While MPLS can be formally applied to any compact group, we refine and test it for the group $\\mathcal G = SO(3)$. We demonstrate state-of-the-art results for rotation synchronization with both synthetic data having nontrivial scenarios and real SfM data.\n\n\n\\end{itemize}\n\n\n\n\n \n\n\n\n\n\n\\section{Setting for Robust Group Synchronization}\\label{sec:adversarial}\nSome previous robustness theories for group synchronization typically assume a very special and often unrealistic corruption probabilistic model for very special groups \\cite{deepti,wang2013exact}. In general, simplistic probabilistic models for corruption,\nsuch as generating potentially corrupted group ratios according to the Haar measure on $\\mathcal G$ \\cite{wang2013exact},\nmay not generate some nontrivial scenarios that often occur in practice. \nFor example, in the application of rotation synchronization that arise in SfM, the corrupted camera relative rotations can be self-consistent due to the ambiguous scene structures \\citep{1dsfm14}. However, in common probabilistic models, such as the one in \\citet{wang2013exact}, cycles with corrupted edges are self-consistent with probability zero. A more realistic model is the adversarial corruption model for the different problem of camera location \\cite{LUDrecovery, HandLV15}. However, it also assumes very special probabilistic models for the graph and camera locations, which are not realistic. A more general model of adversarial corruption with noise is due to \\citet{cemp} and we review it here.\n\nWe assume a graph $G([n],E)$ \nand a compact group $\\mathcal G$\nwith a bi-invariant metric $d$, that is, for any $g_1$, $g_2$, $g_3\\in \\mathcal G$, \n $d(g_1,g_2)=$ $d(g_3g_1,g_3g_2)=$ $d(g_1g_3,g_2g_3)$. \n For $\\mathcal G = SO(3)$, or any Lie group, $d$ is commonly chosen to be the geodesic distance.\n\n Since $\\mathcal G$ is compact, we can scale $d$ and assume \n\n that $d(\\cdot)\\leq 1$.\n \n \n \nWe partition $E$ into $E_g$ and $E_b$, which represent sets of good (uncorrupted) and bad (corrupted) edges, respectively. \nWe will need a topological assumption on $E_b$, or equivalently, $E_g$. A necessary assumption is that $G([n],E_g)$ is connected, though further restrictions on $E_b$ may be needed for \nestablishing theoretical guarantees\n\\cite{cemp}.\n\nIn the noiseless case, the adversarial corruption model generates group ratios in the following way.\n\\begin{align}\\label{eq:model}\ng_{ij}=\\begin{cases}\ng^*_{ij}:=g_i^*g_j^{*-1}, & ij \\in E_g;\\\\\n\\tilde g_{ij} \\neq g^*_{ij}, & ij \\in E_b.\n\\end{cases}\n\\end{align}\nThat is, for edges $ij\\in E_b$, the corrupted group ratio $\\tilde g_{ij}\\neq g_{ij}^*$ can be arbitrarily chosen from $\\mathcal G$.\nThe corruption is called adversarial since \none can maliciously corrupt the group ratios for $ij\\in E_b$ and also maliciously choose $E_b$ as long as the needed assumptions on $E_b$ are satisfied. One can even form cycle-consistent corrupted edges, so that they can be confused with the good edges.\n\nIn the noisy case, we assume a noise model for $d(g_{ij},g_{ij}^*)$, where $ij\\in E_g$. In theory, one may need to restrict this model \\cite{cemp}, but in practice we test highly noisy scenarios.\n\nFor $ij\\in E$ we define the corruption level of $ij$ as \\[s_{ij}^* = d(g_{ij},g_{ij}^*).\\]\nWe use ideas of \\citet{cemp} to estimate $\\{s_{ij}^*\\}_{ij \\in E}$, but then we propose new ideas to estimate $\\{g_{i}^*\\}_{i \\in [n]}$. While exact estimation of both quantities is equivalent in the noiseless case \\cite{cemp}, this property is not valid when adding noise.\n\n \n\\section{Issues with the Common IRLS}\\label{sec:issue}\nWe first review the least squares minimization, least absolute and unsquared deviations minimization and IRLS for group synchronization. We then explain why IRLS may not form a good solution for the group synchronization problem, and in particular for Lie algebraic groups, such as the rotation group.\n\nThe least squares minimization \ncan be formulated as follows:\n\\begin{align}\n\\label{eq:l2}\n \\min_{\\{g_i\\}_{i=1}^n \\subseteq \\mathcal G}\\sum_{ij\\in E}d^2(g_{ij},g_ig_j^{-1}),\n\\end{align}\nwhere one often relaxes this formulation.\nThis formulation is generally sensitive to outliers and thus more robust energy functions are commonly used when considering corrupted group ratios. More specifically, one may choose a special function $\\rho(x) \\neq x^2$ and solve the following least unsquared deviation formulation\n\\begin{align}\\label{eq:lp}\n \\min_{\\{g_i\\}_{i=1}^n \\subseteq \\mathcal G}\\sum_{ij\\in E}\\rho\\left(d(g_{ij},g_ig_j^{-1})\\right).\n\\end{align}\nThe special case of $\\rho(x)=x$ \\cite{HandLV15,HartleyAT11_rotation,ozyesil2015robust, wang2013exact} is referred to as least absolute deviations. Some other common choices are $\\rho(x)=x^2\/(x^2+\\sigma^2)$ \\cite{ChatterjeeG13_rotation} and $\\rho(x)=\\sqrt x$ \\cite{L12}. \n\n\nThe least unsquared formulation is typically solved using IRLS, where at iteration $t$ one solves the weighted least squares problem:\n\\begin{align} \\label{eq:irls}\n \\{g_{i,t}\\}_{i\\in [n]}&= \\operatorname*{arg\\,min}_{\\{g_i\\}_{i=1}^n \\subseteq \\mathcal G}\\sum_{ij\\in E} w_{ij,t-1}d^2(g_{ij},g_ig_j^{-1}).\n\\end{align}\nIn the first iteration the weights can be initialized in a certain way, but in the next iterations the weights are updated using the residuals of this solution. Specifically, for $ij \\in E$ and iteration $t$, the residual is $r_{ij,t}=d(g_{ij},g_{i,t}g_{j,t}^{-1})$ and the weight $w_{ij,t}$ is \n\\begin{align}\nw_{ij,t}&=\nF(r_{ij,t}),\n\\label{eq:weight_irls}\n\\end{align}\nwhere the function $F$ depends on the choice of $\\rho$. For $\\rho(x)=x^p$, where $0 \\tau\\}}$ with a parameter $\\tau>0$. We decrease $\\tau$ as the iteration number increases in order to avoid overweighing outliers. By doing this we aim to address the third drawback of IRLS mentioned in Section \\ref{sec:issue}. We remark that the truncated function is $F(x)\\mathbf{1}_{\\{x\\leq \\tau\\}}$ and the additional term $10^{-8}\\mathbf{1}_{\\{x> \\tau\\}}$ is needed to ensure that the graph with weights resulting from $F_{\\tau}$ is connected.\n\n\n\n\n\\begin{algorithm}[h]\n\\caption{Message Passing Least Squares (MPLS)}\\label{alg:MPLS}\n\\begin{algorithmic}\n\\REQUIRE $\\{g_{ij}\\}_{ij\\in E}$, $\\{d_{ij,k}\\}_{k\\in C_{ij}}$, nonincreasing $\\{\\tau_t\\}_{t\\geq 0}$, increasing $\\{\\beta_t\\}_{t=0}^T$, decreasing $\\{\\alpha_t\\}_{t\\geq 1}$\n\\STATE \\textbf{Steps:}\n\\STATE Compute $\\{s_{ij,T}\\}_{ij\\in E}$ by CEMP\n\\STATE $w_{ij,0}=F_{\\tau_0}(s_{ij,T})$\n\\hspace*{\\fill} $ij\\in E$\n\\STATE $t=0$\n\\WHILE {not convergent}\n\\STATE $t=t+1$\n\\STATE $\\{g_{i,t}\\}_{i\\in [n]}=\\operatorname*{arg\\,min}\\limits_{g_i\\in\\mathcal G}\\sum\\limits_{ij\\in E} w_{ij,t-1}d^2(g_{ij},g_ig_j^{-1})$\n\n\\STATE $r_{ij,t}=d(g_{ij}, g_{i,t}g_{j,t}^{-1})$\n\\hspace*{\\fill} $ij\\in E$\n\n\\STATE $q_{ij,k}^{t} = \\exp(-\\beta_T(r_{ik,t}+r_{jk,t}))$\n\\hspace*{\\fill} $k\\in C_{ij},\\, ij\\in E$\n\n\\STATE $h_{ij,t}\n =\\frac{\\sum_{k\\in C_{ij}}q_{ij,k}^t d_ {ij,k}}{\\sum_{k\\in C_{ij}}q_{ij,k}^t}$\n\\hspace*{\\fill} $ij\\in E$\n\\STATE $w_{ij,t}=F_{\\tau_t}(\\alpha_t h_{ij,t}+(1-\\alpha_t)r_{ij,t})$\n\\hspace*{\\fill} $ij\\in E$\n\\ENDWHILE\n\\ENSURE $\\left\\{g_{i,t}\\right\\}_{i\\in [n]}$\n\\end{algorithmic}\n\\end{algorithm}\nThe initial step of the algorithm estimates the corruption levels $\\{s_{ij}^*\\}_{ij \\in E}$ by CEMP. The initial weights for the IRLS procedure \nfollow \\eqref{eq:weight_irls} with additional truncation. \nAt each iteration, the group ratios $\\{g_{i,t}\\}_{i\\in [n]}$ are estimated from the weighted least squares procedure in \\eqref{eq:irls}. However, the weights $w_{ij,t}$ are updated in a very different way. First of all, for each $ij \\in E$ the corruption level $s_{ij}^*$ is re-estimated in two different ways and a convex combination of the two estimates is taken. The first estimate is a residual $r_{ij,t}$ computed with the newly updated estimates $\\{g_{i,t}\\}_{i\\in [n]}$. This is the error of approximating the given measurement $g_{ij}$ by the newly estimated group ratio. The other estimate practically applies CEMP to re-estimate the corruption levels. For edge $ij \\in E$, the latter estimate of $s_{ij}^*$ is denoted by $h_{ij,t}$. \nFor interpretation, we can replace \\eqref{eq:pr_f} with \n$\\Pr(s_{ij}^*|r_{ij,t})=\\exp(-\\beta_T x)$ and use it to derive analogs of \\eqref{eq:pijk}\nand \n\\eqref{eq:sijt}.\nUnlike CEMP, we use the single parameter, $\\beta_T$, as we assume that CEMP provides a sufficiently good initialization.\nAt last, a similar weight as in \\eqref{eq:weight_irls}, but truncated, is applied to the combined estimate $\\alpha_t h_{ij,t}+(1-\\alpha_t)r_{ij,t}$. \n\n\nWe remark that utilizing the estimate $h_{ij,t}$ for the corruption level addresses the first drawback of IRLS discussed in Section \\ref{sec:issue}. Indeed, assume the case where $ij\\in E_b$ and $r_{ij,t}$ is close to 0. Here, $w_{ij,t}$ computed by IRLS is relatively large; however, since $ij\\in E_b$, $w_{ij,t}$ needs to be small. Unlike $r_{ij,t}$ in IRLS, we expect that $h_{ij,t}$ in MPLS should not be too small as long as for some $k\\in C_{ij}$, $d_{ij,k}$ are sufficiently large. This happens as long as there exists some $k\\in C_{ij}$ for which the cycle $ijk$ is good. Indeed, in this case $s_{ij}^*$ is sufficiently large and for good cycles $d_{ij,k}=s_{ij}^*$.\n\n\nWe further remark that $h_{ij,t}$ is a good approximation of $s_{ij}^*$ under certain conditions. For example, if for all $k\\in C_{ij}$, $r_{ik,t}\\approx s_{ik}^*$ and $r_{jk,t}\\approx s_{jk}^*$, then plugging in the definition of $p_{ij,k}^t$ to the expression of $h_{ij,t}$, using the fact that $\\beta_T$ is sufficiently large and at last applying Proposition \\ref{prop:good cycle}, we obtain that\n\\begin{align}\n h_{ij,t}\n &= \\sum_{k\\in C_{ij}}\\frac{\\exp(-\\beta_T(r_{ik,t}+r_{jk,t}))}{\\sum_{k\\in C_{ij}}\\exp(-\\beta_T(r_{ik,t}+r_{jk,t}))}d_ {ij,k} \\nonumber\\\\\n &\\approx \\sum_{k\\in C_{ij}}\\frac{\\exp(-\\beta_T(s_{ik}^*+s_{jk}^*)) }{\\sum_{k\\in C_{ij}}\\exp(-\\beta_T(s_{ik}^*+s_{jk}^*))} d_ {ij,k} \\\\\n &\\approx \\sum_{k\\in C_{ij}}\\frac{\\mathbf{1}_{\\{ijk \\text{ is a good cycle}\\}}}{\\sum_{k\\in C_{ij}}\\mathbf{1}_{\\{ijk \\text{ is a good cycle}\\}}} d_ {ij,k}=s_{ij}^*.\\nonumber\n\\end{align}\nThis intuitive argument for a restricted case conveys the idea that ``local good information'' can be used to estimate $s_{ij}^*$. \nThe theory of CEMP \\cite{cemp} shows that under weaker conditions such information can propagate through the whole graph within a few iterations, but we cannot extend it to MPLS.\n\n\nIf the graph $G([n], E)$ is dense with sufficiently many good cycles, then we expect that this good information can propagate in few iterations and that $h_{ij,t}$ will have a significant advantage over $r_{ij,t}$. However, in real scenarios of rotation synchronization in SfM, one may encounter sparse graphs, which may not have enough cycles and, in particular, not enough good cycles. In this case, utilizing $h_{ij,t}$ is mainly useful in the early iterations of the algorithm. On the other hand, when $\\{g_{i,t}\\}_{i \\in [n]}$ are close to $\\{g_i^*\\}_{i \\in [n]}$, $\\{r_{ij,t}\\}_{i \\in [n]}$ will be sufficiently close to $\\{s_{ij}^*\\}_{i \\in [n]}$. Aiming to address rotation synchronization, we decrease $\\alpha_t$, the weight of $h_{ij,t}$, with $t$. In other applications, different choices of $\\alpha_t$ can be used \\cite{robust_multi_object2020}.\n\n\nThe second drawback of IRLS, discussed in Section \\ref{sec:issue}, is the possible difficulty of implementing the weighted least squares step of \\eqref{eq:irls}. This issue is application-dependent, and since in this work we focus on rotation synchronization (equivalently, $SO(3)$ synchronization), we show in the next subsection how MPLS can deal with the above issue in this specific problem. \nNevertheless, we claim that our framework can also be applied to other compact group synchronization problems and we demonstrate this claim in a follow up work \\cite{robust_multi_object2020}.\n\n\\subsection{MPLS for $SO(3)$ synchronization} \n\\label{sec:so3}\nRotation synchronization, or $SO(3)$ synchronization, aims to solve 3D rotations $\\{\\boldsymbol{R}_i^*\\}_{i\\in [n]} \\in SO(3)$ from measurements $\\{\\boldsymbol{R}_{ij}\\}_{ij\\in E} \\in SO(3)$ of the 3D relative rotations $\\{\\boldsymbol{R}_i^*\\boldsymbol{R}_j^{*-1}\\}_{ij\\in E} \\in SO(3)$. \nThroughout the rest of the paper, we use the following normalized geodesic distance for $\\boldsymbol{R}_1,\\boldsymbol{R}_2 \\in SO(3)$:\n\\begin{equation}\n\\label{eq:geo_distance}\n d(\\boldsymbol{R}_1,\\boldsymbol{R}_2)=\\|\\log(\\boldsymbol{R}_1\\boldsymbol{R}_2^{-1})\\|_F\/(\\sqrt2\\pi),\n\\end{equation}\nwhere $\\log$ is the matrix logarithm and the normalization factor ensures that the diameter of $SO(3)$ is $1$. \nWe provide some relevant preliminaries of the Riemannian geometry of $SO(3)$ in Section \\ref{sec:prelim_riem} and then describe the implementation of MPLS for $SO(3)$, which we refer to as MPLS-$SO(3)$, in Section \\ref{sec:mpls_so3_details}.\n\n\\subsubsection{Preliminaries: $SO(3)$ and $\\mathfrak{so}(3)$} \\label{sec:prelim_riem}\nWe note that $SO(3)$ is a Lie group, and its corresponding Lie algebra, $\\mathfrak{so}(3)$, is the space of all skew symmetric matrices, which is isomorphic to $\\mathbb R^3$. For each $\\boldsymbol{R}\\in SO(3)$, its corresponding element in $\\mathfrak{so}(3)$ is $\\boldsymbol\\Omega=\\log(\\boldsymbol{R})$, where $\\log$ denotes matrix logarithm. Each $\\boldsymbol\\Omega\\in \\mathfrak{so}(3)$ can be represented as $[\\boldsymbol{\\omega}]_{\\times}$ for some $\\boldsymbol\\omega =(\\omega_1, \\omega_2, \\omega_3)^T \\in \\mathbb R^3$ in the following way:\n\\begin{align*}\n [\\boldsymbol{\\omega}]_{\\times} :=\n \\left(\\begin{array}{ccc}\n 0 & - \\omega_3 & \\omega_2 \\\\\n \\omega_3 & 0& -\\omega_1 \\\\\n -\\omega_2 & \\omega_1 & 0\n \\end{array}\\right).\n\\end{align*}\nIn other words, we can map any $\\boldsymbol\\omega\\in \\mathbb R^3$ to \n$\\boldsymbol\\Omega=[\\boldsymbol{\\omega}]_{\\times} \\in \\mathfrak{so}(3)$ and $\\boldsymbol{R}=\\exp([\\boldsymbol{\\omega}]_{\\times}) \\in SO(3)$, where $\\exp$ denotes the matrix exponential function. We remark that geometrically $\\boldsymbol\\omega$ is the tangent vector at $\\boldsymbol I$ of the geodesic path from $\\boldsymbol I$ to $\\boldsymbol{R}$.\n\n\n\\subsubsection{Details of MPLS-$SO(3)$}\n\\label{sec:mpls_so3_details}\nWe note that in order to adapt MPLS to the group $SO(3)$, we only need a specific algorithm to solve the following formulation of the weighted least squares problem at iteration $t$\n\\begin{align}\n \\nonumber &\\min\\limits_{\\boldsymbol{R}_{i,t}\\in \\mathcal SO(3)}\\sum\\limits_{ij\\in E} w_{ij,t}d^2(\\boldsymbol{R}_{ij},\\boldsymbol{R}_{i,t}\\boldsymbol{R}_{j,t}^{-1})\\\\\n =& \\min\\limits_{\\boldsymbol{R}_{i,t}\\in \\mathcal SO(3)}\\sum\\limits_{ij\\in E} w_{ij,t}d^2(\\boldsymbol{I}, \\boldsymbol{R}_{i,t}^{-1}\\boldsymbol{R}_{ij}\\boldsymbol{R}_{j,t}),\\label{eq:wlsSO3}\n\\end{align}\nwhere the last equality follows from the bi-invariance of $d$.\nThe constraints on orthogonality and determinant of $\\boldsymbol{R}_i$ are non-convex. If one relaxes those constraints, with an appropriate choice of the metric $d$, then the solution of the least squares problem in the relaxed Euclidean space often lies away from the embedding of $SO(3)$ into that space. \nFor this reason, we follow the common choice of $d$ according to \\eqref{eq:geo_distance} and implement the Lie-algebraic Averaging (LAA) procedure \\cite{Govindu04_Lie, ChatterjeeG13_rotation, L12,consensusSO3}. \nWe review LAA, explain why it may be problematic and why our overall implementation may overcome its problems. \nLAA aims to move from $\\boldsymbol{R}_{i,t}$ to $\\boldsymbol{R}_{i,t+1}$ along the manifold using the right group action $\\boldsymbol{R}_{i,t}=\\boldsymbol{R}_{i,t-1}\\Delta \\boldsymbol{R}_{i,t}$, where \n$\\Delta \\boldsymbol{R}_{i,t}\\in SO(3)$. \nFor this purpose, it defines $\\Delta\\boldsymbol{R}_{ij,t}=\\boldsymbol{R}_{i,t-1}^{-1}\\boldsymbol{R}_{ij}\\boldsymbol{R}_{j,t-1}$\nso that \n\\begin{multline*}\n(\\Delta \\boldsymbol{R}_{i,t})^{-1}\\Delta \\boldsymbol{R}_{ij,t} \\Delta \\boldsymbol{R}_{j,t} = \\\\(\\Delta \\boldsymbol{R}_{i,t})^{-1}\\boldsymbol{R}_{i,t-1}^{-1}\\boldsymbol{R}_{ij}\\boldsymbol{R}_{j,t-1} \\Delta \\boldsymbol{R}_{j,t}=\\boldsymbol{R}_{i,t}^{-1}\\boldsymbol{R}_{ij}\\boldsymbol{R}_{j,t} \n\\end{multline*}\nand \\eqref{eq:wlsSO3} \ncan be transformed to the still hard to solve equation \n\\begin{equation}\\label{eq:deltaR}\n \\min\\limits_{\\Delta \\boldsymbol{R}_{i,t}\\in\\mathcal SO(3)}\\sum\\limits_{ij\\in E} w_{ij,t}d^2(\\boldsymbol I,(\\Delta \\boldsymbol{R}_{i,t})^{-1}\\Delta \\boldsymbol{R}_{ij,t} \\Delta \\boldsymbol{R}_{j,t}).\n\\end{equation}\nLAA then maps $\\{\\Delta \\boldsymbol{R}_{i,t}\\}_{i\\in [n]}$ and $\\{\\Delta \\boldsymbol{R}_{ij,t}\\}_{ij\\in E}$ to the tangent space of $\\boldsymbol{I}$ by $\\Delta \\boldsymbol\\Omega_{i,t}=\\log \\Delta \\boldsymbol{R}_{i,t}$\nand $\\Delta \\boldsymbol\\Omega_{ij,t}=\\log \\Delta \\boldsymbol{R}_{ij,t}$.\nApplying \\eqref{eq:geo_distance} and the fact that the Riemannian logarithmic map, which is represented by $\\log$, preserves the geodesic distance and using a ``naive approximation'':\n$d(\\boldsymbol I,(\\Delta \\boldsymbol{R}_{i,t})^{-1}\\Delta \\boldsymbol{R}_{ij,t} \\Delta \\boldsymbol{R}_{j,t}) $. Therefore, \nLAA uses the following approximation \n\\begin{multline}\n\\label{eq:app_laa}\n d(\\boldsymbol{I},(\\Delta \\boldsymbol{R}_{i,t})^{-1}\\Delta \\boldsymbol{R}_{ij,t}\\Delta \\boldsymbol{R}_{j,t}) \n = \\\\\n \\|\\log((\\Delta \\boldsymbol{R}_{i,t})^{-1}\\Delta \\boldsymbol{R}_{ij,t} \\Delta \\boldsymbol{R}_{j,t})\\|_F\/(\\sqrt2\\pi)\n \\approx \\\\\n\\|-\\log(\\Delta \\boldsymbol{R}_{i,t})+\\log(\n\\Delta \\boldsymbol{R}_{ij,t})+\\log( \\Delta \\boldsymbol{R}_{j,t}))\\|_F\/(\\sqrt2\\pi) = \\\\\n\\|\\Delta \\boldsymbol\\Omega_{i,t}-\\Delta \\boldsymbol\\Omega_{j,t}-\\Delta \\boldsymbol\\Omega_{ij,t}\\|_F\/(\\sqrt2\\pi).\n\\end{multline}\nConsequently, LAA transforms \\eqref{eq:deltaR} as follows:\n\\begin{equation}\\label{eq:delta_omega}\n \\min\\limits_{\\Delta \\boldsymbol\\Omega_{i,t}\\in\\mathfrak{so}(3)}\\sum\\limits_{ij\\in E} w_{ij,t}\\|\\Delta \\boldsymbol\\Omega_{i,t}-\\Delta \\boldsymbol\\Omega_{j,t}-\\Delta \\boldsymbol\\Omega_{ij,t}\\|^2_F.\n\\end{equation}\nHowever, the approximation in \\eqref{eq:app_laa}\nis only valid when $\\Delta \\boldsymbol{R}_{ij,t}$, $\\Delta \\boldsymbol{R}_{i,t}$, $\\Delta \\boldsymbol{R}_{j,t}$ $\\approx \\boldsymbol{I}$, \nwhich is unrealistic. \n\nOne can check that the following conditions: $\\boldsymbol{R}_{ij}\\approx \\boldsymbol{R}_i^*\\boldsymbol{R}_j^{*-1}$ ($s_{ij}^*\\approx 0$), $\\boldsymbol{R}_{i,t} \\approx \\boldsymbol{R}_i^*$ and $\\boldsymbol{R}_{j,t} \\approx \\boldsymbol{R}_j^*$ for $t \\geq 0$ imply that $\\Delta \\boldsymbol{R}_{ij,t}$, $\\Delta \\boldsymbol{R}_{i,t}$, $\\Delta \\boldsymbol{R}_{j,t}$ $\\approx \\boldsymbol{I}$ and thus \nimply \\eqref{eq:app_laa}. Therefore, to make LAA work we need to give large weights to edges $ij$ with small $s_{ij}^*$ and provide a good initialization $\\{\\boldsymbol{R}_{i,0}\\}_{i\\in [n]}$ that is reasonably close to $\\{\\boldsymbol{R}_{i}^*\\}_{i\\in [n]}$ and so that $\\{\\boldsymbol{R}_{i,t}\\}_{i\\in [n]}$ for all $t\\geq 1$ are still close to the ground truth. \nOur heuristic argument is that good approximation by CEMP, followed by MPLS, addresses these requirements. Indeed, to address the first requirement, we note that good initialization by CEMP can result in $s_{ij,T} \\approx s_{ij}^*$ and by the nature of $F$, $w_{ij,0}$ is large when $s_{ij,T}$ is small. \nAs for the second requirement, \nwe assign the weights $s_{ij,T}$, obtained by CEMP, to each $ij\\in E$ and find the minimum spanning tree (MST) for the weighted graph by Prim's algorithm. We initialize the rotations by fixing $\\boldsymbol{R}_{1,0}=\\boldsymbol{I}$, multiplying relative rotations along the computed MST and consequently obtaining $\\boldsymbol{R}_{i,0}$ for any node $i$. We summarize our MPLS version of rotation averaging in Algorithm~\\ref{alg:SO3}. \n\\begin{algorithm}[h]\n\\caption{MPLS-$SO(3)$}\\label{alg:SO3}\n\\begin{algorithmic}\n\\REQUIRE $\\{\\boldsymbol{R}_{ij}\\}_{ij\\in E}$, $\\{d_{ij,k}\\}_{k\\in C_{ij}}$, $\\{\\tau_t\\}_{t\\geq 0}$, $\\{\\beta_t\\}_{t=0}^T$, $\\{\\alpha_t\\}_{t\\geq 1}$\n\\STATE \\textbf{Steps:}\n\n\\STATE Compute $\\{s_{ij,T}\\}_{ij\\in E}$ by CEMP\n\\STATE Form an $n \\times n$ weight matrix $\\boldsymbol{W}$, where $W_{ij}=W_{ji}= s_{ij,T}$ for $ij\\in E$, and $W_{ij}=W_{ji}=0$ otherwise\n\\STATE $G([n],E_{ST})=$ minimum spanning tree of $G([n],W)$\n\\STATE $\\boldsymbol{R}_{1,0}=\\boldsymbol{I}$\n\\STATE find $\\{\\boldsymbol{R}_{i,0}\\}_{i>1}$ by $\\boldsymbol{R}_i=\\boldsymbol{R}_{ij}\\boldsymbol{R}_j$ for $ij\\in E_{ST}$ \n\\STATE $t=0$\n\\STATE $w_{ij,0}=F_{\\tau_0}(s_{ij,T})$ \n\\WHILE {not convergent}\n\\STATE $t=t+1$\n\\STATE $\\Delta \\boldsymbol\\Omega_{ij,t}=\\log(\\boldsymbol{R}_{i,t-1}^{-1}\\boldsymbol{R}_{ij}\\boldsymbol{R}_{j,t-1})$ \\hspace*{\\fill} $ij\\in E$\n\n\\STATE $\\{\\Delta \\boldsymbol\\Omega_{i,t}\\}_{i\\in [n]}=$\n\\STATE \\quad \\quad $\\operatorname*{arg\\,min}\\limits_{\\Delta \\boldsymbol\\Omega_{i,t}\\in\\mathfrak{so}(3)}\\sum\\limits_{ij\\in E} w_{ij,t}\\|\\Delta \\boldsymbol\\Omega_{i,t}-\\Delta \\boldsymbol\\Omega_{j,t}-\\Delta \\boldsymbol\\Omega_{ij,t}\\|^2_F$\n\\STATE $\\boldsymbol{R}_{i,t}=\\boldsymbol{R}_{i,t-1}\\exp(\\Delta \\boldsymbol\\Omega_{i,t})$ \\hspace*{\\fill} $i\\in [n]$\n\\STATE $r_{ij,t}=\\|\\Delta \\boldsymbol\\Omega_{i,t}-\\Delta \\boldsymbol\\Omega_{j,t}-\\Delta \\boldsymbol\\Omega_{ij,t}\\|_F\/(\\sqrt2\\pi)$ \\hspace*{\\fill} $ij\\in E$\n\n\\STATE $q_{ij,k}^{t} =\\exp(-\\beta_T(r_{ik,t}+r_{jk,t}))$\n\\hspace*{\\fill} $k\\in C_{ij},\\, ij\\in E$\n\\STATE $h_{ij,t}\n =\\frac{\\sum_{k\\in C_{ij}}q_{ij,k}^t d_ {ij,k}}{\\sum_{k\\in C_{ij}}q_{ij,k}^t}$ \\hspace*{\\fill} $ij\\in E$\n\\STATE $w_{ij,t}=F_{\\tau_t}(\\alpha_t h_{ij,t}+(1-\\alpha_t)r_{ij,t})$\n\\hspace*{\\fill} $ij\\in E$\n\\ENDWHILE\n\\ENSURE $\\left\\{\\boldsymbol{R}_{i,t}\\right\\}_{i\\in [n]}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsection{Computational Complexity}\n\\label{sec:complexity}\nCEMP requires the computation \nof $d_{ij,k}$ for $ij \\in E$ and $k\\in C_{ij}$. Its computational complexity per iteration is thus of order $O(|E|)$ as we use $|C_{ij}|=50$ for all $ij \\in E$. Since we advocate few iterations ($T=5$) of CEMP, or due to its fast convergence under special settings \\cite{cemp}, we can assume that its total complexity is $O(|E|)$.\nThe computational complexity of MPLS depends on the complexity of solving the weighted least squares problem, which depends on the group. For MPLS-$SO(3)$, the most expensive part is solving the weighted least squares problem in the tangent space, whose complexity is at most $O(n^3)$. This is thus also the complexity of MPLS-$SO(3)$ per iteration. Unlike CEMP, we have no convergence guarantees yet for MPLS.\n\n\n\n\\section{Numerical Experiments}\n\\label{sec:numerics}\nWe test the proposed MPLS algorithm on rotation synchronization, while comparing with state-of-the-art methods. We also try simpler ideas than MPLS that are based on the basic strategy of CEMP.\nAll computational tasks were implemented on a machine with 2.5GHz Intel i5 quad core processors and 8GB memory. \n\n\\subsection{Implementation}\n\\label{sec:implementation}\nWe use the following default parameters for Algorithm \\ref{alg:cemp}:\n$|C_{ij}|=50$ for $ij\\in E$; $T=5$; $\\beta_t=2^{t}$ and $t=0, \\ldots, 5$. If an edge is not contained in any 3-cycle, we set its corruption level as 1.\nFor MPLS-$SO(3)$, which we refer to in this section as MPLS, we use the above parameters of Algorithm \\ref{alg:cemp} and the following ones for $t\\geq 1$: \n $$\\alpha_t=1\/(t+1) \\ \\text{ and } \\ \\tau_{t}=\\inf_x\\left\\{\\hat P_t(x)> \\max\\{1-0.05t\\,,0.8\\}\\right\\}.$$\nHere, $\\hat P_t$ denotes the empirical distribution of \n$\\{\\alpha_t h_{ij,t}+$ $(1-\\alpha_t)r_{ij,t}\\}_{ij\\in E}$.\nThat is, for $t=0$, 1, 2, 3, we ignore $0\\%$, $5\\%$, $10\\%$, $15\\%$ of edges that have highest $\\alpha_t h_{ij,t}+$ $(1-\\alpha_t)r_{ij,t}$, and for $t \\geq 4$ we ignore $20\\%$ of such edges. \n$F(x)$ for MPLS is chosen as $x^{-3\/2}$ and it corresponds to $\\rho(x)=\\sqrt x$. For simplicity and consistency, we use these choices of parameters for all of our experiments. We remark that our choice of $\\beta_t$ in Algorithm \\ref{alg:cemp} is supported by the theory of \\citet{cemp}. \nWe found that MPLS is not so sensitive to its parameters. One can choose other values of $\\{\\beta_t\\}_{t\\geq 0}$, for example any geometric sequence with ratio 2 or less, and stop after several iterations. Similarly, one may replace 0.8 and 0.05\nin the definition of $\\tau_t$ with $0.7-0.9$ and $0.01-0.1$, respectively, and perform similarly on average.\n\nWe test two previous state-of-the-art IRLS methods: IRLS-GM \\cite{ChatterjeeG13_rotation} with $\\rho(x) = x^2\/(x^2+25)$, $F(x)=25\/(x^2+25)^2$\nand IRLS-$\\ell_{1\/2}$ \\cite{L12} with $\\rho(x)=\\sqrt x$, $F(x)=x^{-3\/2}$.\nWe use their implementation by \\citet{L12}. \n\nWe have also separately implemented the part of initializing the rotations \nof MPLS in Algorithm~\\ref{alg:SO3} and refer to it by CEMP+MST. Recall that it solves rotations by direct propagation along the minimum weighted spanning tree of the graph with weights obtained by \nAlgorithm \\ref{alg:cemp} (CEMP).\nWe also test the application of this initialization to \nthe main algorithms in \\citet{ChatterjeeG13_rotation} and \\citet{L12} and refer to the resulting methods by CEMP+IRLS-GM\nand CEMP+IRLS-$\\ell_{1\/2}$, respectively. We remark that the original algorithms initialize by a careful least absolute deviations minimization.\nWe use the convergence criterion\n$\\sum_{i\\in[n]}\\|\\Delta \\boldsymbol\\Omega_{i,t}\\|_F\/(\\sqrt2n)< 0.001$\nof \\citet{L12} for all the above algorithms.\n\nBecause the solution is determined up to a right group action, we align our estimated rotations $\\{\\hat\\boldsymbol{R}_i\\}$ with the ground truth ones $\\{ \\boldsymbol{R}_i^*\\}$. That is, we find a rotation matrix $\\boldsymbol{R}_\\text{align}$ so that $\\sum_{i\\in [n]}\\|\\hat\\boldsymbol{R}_i\\boldsymbol{R}_\\text{align}-\\boldsymbol{R}_i^*\\|_F^2$ is minimized. For synthetic data, we report the following mean estimation error in degrees: $180\\cdot\\sum_{i\\in [n]} d(\\hat \\boldsymbol{R}_i\\boldsymbol{R}_\\text{align}\\,, \\boldsymbol{R}_i^*)\/n$. For real data, we also report the median of $\\{180\\cdot d(\\hat \\boldsymbol{R}_i\\boldsymbol{R}_\\text{align}\\,, \\boldsymbol{R}_i^*)\\}_{i\\in [n]}$.\n\n\n\n\n\n\n\\subsection{Synthetic Settings}\nWe test the methods in the following two types of artificial scenarios. In both scenarios, the graph is generated by the Erd\\H{o}s-R\\'{e}nyi model $G(n,p)$ with $n=200$ and $p=0.5$.\n\\subsubsection{Uniform Corruption}\nWe consider the following random model for generating $\\boldsymbol{R}_{ij}$\n\\begin{equation}\n \\boldsymbol{R}_{ij}=\\begin{cases}\n \\text{Proj}(\\boldsymbol{R}_{ij}^*+\\sigma \\boldsymbol{W}_{ij}),&\\text{w.p. } 1-q;\\\\\n \\tilde \\boldsymbol{R}_{ij}\\sim \\text{Haar}($SO(3)$),& \\text{w.p. } q,\n \\end{cases}\n\\end{equation}\nwhere Proj denotes the projection onto $SO(3)$; $\\boldsymbol{W}_{ij}$ is a $3 \\times 3$ Wigner matrix whose elements follow i.i.d.~standard normal distribution; $\\sigma\\geq 0$ is a fixed noise level; $q$ is the probability that an edge is corrupted and Haar$(SO(3))$ is the Haar probability measure on $SO(3)$. We clarify that for any $3 \\times 3$ matrix $\\boldsymbol{A}$, $\\text{Proj}(\\boldsymbol{A}) = \\operatorname*{arg\\,min}_{\\boldsymbol{R}\\in SO(3)} \\|\\boldsymbol{R}-\\boldsymbol{A}\\|_F$.\n\nWe test the algorithms with four values of $\\sigma:$ $0$, $0.1$, $0.5$, and $1$. We average the mean error over 10 random samples from the uniform model and report it as a function of $q$ in Figure \\ref{fig:s1}.\n\n We note that MPLS consistently outperforms the other methods for all tested values of $q$ and $\\sigma$.\n In the noiseless case, MPLS exactly recovers the group ratios even when $70\\%$ of the edges are corrupted.\n It also nearly recovers with $80\\%$ corrupted edges, where the estimation errors for IRLS-GM and IRLS-$\\ell_{1\/2}$ are higher than 30 degrees. MPLS is also shown to be stable under high level of noise. Since all algorithms produce poor solutions when $q=0.9$, we only show results for $0\\leq q\\leq 0.8$.\n \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8cm]{uniform.pdf}\n \\caption{Performance under uniform corruption. The mean error (in degrees) is plotted against the corruption probability $q$ for 4 values of $\\sigma$. }\n \\label{fig:s1}\n\\end{figure}\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8cm]{adv.pdf}\n \\caption{\n Performance under self-consistent corruption. The mean error is plotted against the corruption probability $q$ for 4 values of $\\sigma$.}\n \\label{fig:adv}\n\\end{figure}\n\n\n\n\n\n\n\n\\begin{table*}[!htbp]\n\\centering\n\\resizebox{2\\columnwidth}{!}{\n\\renewcommand{\\arraystretch}{1.3}\n\\tabcolsep=0.1cm\n\\begin{tabular}{|l||c|c||c|c|c|c||c|c|c|c||c|c|c|c||c|c|c|c|}\n\\hline\nAlgorithms & \\multicolumn{2}{c||}{}& \\multicolumn{4}{c||}{IRLS-GM} &\n\\multicolumn{4}{c||}{IRLS-$\\ell_{\\frac12}$} &\n \\multicolumn{4}{c||}{CEMP+MST}&\n \\multicolumn{4}{c|}{MPLS} \\\\\n\\text{Dataset}& $n$ & $m$ & {\\large$\\tilde{e}$} & {\\large $\\hat{e}$} & runtime & iter $\\#$ \n& {\\large$\\tilde{e}$} & {\\large $\\hat{e}$} & runtime & iter $\\#$\n& {\\large$\\tilde{e}$} & {\\large $\\hat{e}$} & runtime & iter $\\#$ &{\\large$\\tilde{e}$} & {\\large $\\hat{e}$} & runtime & iter $\\#$\\\\\\hline\nAlamo& 564 & 71237 &\n3.64 & 1.30 & 14.2 & 10+8 & \n3.67 & 1.32 & 15.5 & 10+9 & \n4.05 & 1.62 & \\textbf{10.38} & 6 &\n\\textbf{3.44} & \\textbf{1.16} & 20.6 & 6+8\n\\\\\\hline\n\nEllis Island& 223 & 17309 &\n3.04 & 1.06 & 3.2 & 10+9 &\n2.71 & 0.93 & 2.8 & 10+13 &\n2.94 & 1.11 & \\textbf{2.4} & 6 &\n\\textbf{2.61} & \\textbf{0.88} & 4.0 & 6+11\n\\\\\\hline\n\nGendarmenmarkt& 655 & 32815&\n\\textbf{39.24} & \\textbf{7.07} & 6.5 & 10+14 &\n39.41 & 7.12 & 7.3 & 10+19 &\n45.33 & 8.62 & \\textbf{4.7} & 6 &\n44.94 & 9.87 & 17.8 & 6+25\n\\\\\\hline\n\nMadrid Metropolis & 315 & 14903 &\n5.30 & 1.78 & 3.8 & 10+30 &\n4.88 & 1.88 & 2.7 & 10+12 &\n5.10 & 1.66 & \\textbf{2.1} & 6 &\n\\textbf{4.65} & \\textbf{1.26} & 5.2 & 6+23\n\\\\\\hline\nMontreal N.D.& 442 & 44501 &\n\n1.25 & 0.58 & 6.5 & 10+6 &\n1.22 & 0.57 & 7.3 & 10+8 &\n1.33 & 0.79 & \\textbf{6.3} & 6 &\n\\textbf{1.04} & \\textbf{0.51} & 9.3 & 6+7\n\\\\\\hline\nNotre Dame & 547 & 88577 &\n\n2.63 & 0.78 & 17.2 & 10+7 &\n2.26 & 0.71 & 22.5 & 10+10 &\n2.35 & 0.94 & \\textbf{13.2} & 6 &\n\\textbf{2.06} & \\textbf{0.67} & 31.5 & 6+8\n\\\\\\hline\nNYC Library& 307 & 13814 &\n\n2.71 & 1.37 & 2.5 & 10+14 &\n2.66 & 1.30 & 2.6 & 10+15 &\n3.00 & 1.41 & \\textbf{1.9} & 6 &\n\\textbf{2.63} & \\textbf{1.24} & 4.5 & 6+14\n\\\\\\hline\nPiazza Del Popolo & 306 & 18915 &\n\n4.10 & 2.17 & 2.8 & 10+9 &\n3.99 & 2.09 & 3.1 & 10+13 &\n\\textbf{3.44} & \\textbf{1.57} & \\textbf{2.6} & 6 &\n3.73 & 1.93 & 3.5 & 6+3\n\\\\\\hline\nPiccadilly& 2031 & 186458 &\n 5.12 & 2.02 & 153.5 & 10+16 &\n 5.19 & 2.34 & 170.2 & 10+19 &\n 4.66 & 1.98 & \\textbf{45.8} & 6 &\n \\textbf{3.93} & \\textbf{1.81} & 191.9 & 6+21\n \n\\\\\\hline\nRoman Forum& 989 & 41836 &\n\n2.66 & 1.58 & 8.6 & 10+9 &\n2.69 & 1.57 & 11.4 & 10+17 &\n2.80 & 1.45 & \\textbf{6.1} & 6 &\n\\textbf{2.62} & \\textbf{1.37} & 8.8 & 6+8\n\n\\\\\\hline\nTower of London& 440 & 15918 &\n\n3.42 & 2.52 & 2.6 & 10+8 &\n3.41 & 2.50 & 2.4 & 10+12 &\n\\textbf{2.84} & \\textbf{1.57} & \\textbf{2.2} & 6 &\n3.16 & 2.20 & 2.7 & 6+7\n\n\\\\\\hline\nUnion Square& 680 & 17528 &\n\n6.77 & 3.66 & 5.0 & 10+32 &\n6.77 & 3.85 & 5.6 & 10+47 &\n7.47 & 3.64 & \\textbf{2.5} & 6 &\n\\textbf{6.54} & \\textbf{3.48} & 5.7 & 6+21\n\\\\\\hline\nVienna Cathedral& 770 & 87876 &\n\n8.13 & 1.92 & 28.3 & 10+13 &\n8.07 & \\textbf{1.76} & 45.4 & 10+23 &\n\\textbf{6.91} & 2.63 & \\textbf{13.1} & 6 &\n7.21 & 2.83 & 42.6 & 6 +19\n\n\\\\\\hline\nYorkminster & 410 & 20298 &\n\n2.60 & 1.59 & \\textbf{2.4} & 10+7 &\n\\textbf{2.45} & 1.53 & 3.3 & 10+9 &\n2.49 & \\textbf{1.37} & 2.8 & 6 &\n2.47 & 1.45 & 3.9 & 6+7\n\\\\\\hline\n\n\n\\end{tabular}}\n\\caption{Performance on the Photo Tourism datasets: $n$ and $m$ are the number of nodes and edges, respectively; $\\tilde e$ and $\\hat e$ indicate mean and median errors in degrees, respectively.; runtime is in seconds; and numbers of iterations (explained in the main text).}\\label{tab:real}\n\\end{table*}\n\n\n\\subsubsection{Self-Consistent Corruption}\nIn order to simulate self-consistent corruption, we independently draw from Haar($SO(3)$) two classes of rotations: $\\{\\boldsymbol{R}_i^*\\}_{i\\in [n]}$ and $\\{\\tilde \\boldsymbol{R}_i\\}_{i\\in [n]}$. We denote their corresponding relative rotations by $\\boldsymbol{R}_{ij}^*=\\boldsymbol{R}_i^*\\boldsymbol{R}_j^{*\\intercal}$ and $\\tilde\\boldsymbol{R}_{ij}=\\tilde\\boldsymbol{R}_i\\tilde\\boldsymbol{R}_j^{\\intercal}$ for $ij\\in E$. \nThe idea is to assign to edges in $E_g$ and $E_b$ relative rotations from two different classes, so cycle-consistency occurs in both\n$G([n],E_g)$ and $G([n],E_b)$. We also add noise to these relative rotations and assign them with Bernoulli model to the two classes, so one class is more significant. \nMore specifically, for $ij \\in E$\n\\begin{equation}\n \\boldsymbol{R}_{ij}=\\begin{cases}\n \\text{Proj}(\\boldsymbol{R}^*_{ij}+\\sigma \\boldsymbol{W}_{ij}),&\\text{w.p. } 1-q;\\\\\n \\text{Proj}(\\tilde\\boldsymbol{R}_{ij}+\\sigma \\boldsymbol{W}_{ij}),& \\text{w.p. } q,\n \\end{cases}\n\\end{equation}\nwhere $q$, $\\sigma$, and $\\boldsymbol{W}_{ij}$ are the same as in the above uniform corruption model. We remark that an information-theoretic threshold for the exact recovery when $\\sigma =0$ is $q=0.5$. That is, for $q\\geq 0.5$ there is no hope of exactly recovering $\\{\\boldsymbol{R}_{i}^*\\}_{i\\in [n]}$.\n\nWe test the algorithms with four values of $\\sigma:$ $0$, $0.1$, $0.5$, and $1$. We average the mean error over 10 random samples from the self-consistent model and report it as a function of $q$ in Figure \\ref{fig:adv}.\nWe focus on values of $q$ approaching the information-theoretic bound $0.5$ ($q=0.4$, $0.45$ and $0.48$). We note that MPLS consistently outperforms the other algorithm and that when $\\sigma=0$ it can exactly recover the ground truth rotations when $q=0.48$.\n\n\n\n\n\\subsection{Real Data}\nWe compare the performance of the different algorithms on the Photo Tourism datasets \\citep{1dsfm14}. Each of the 14 datasets consists of hundreds of 2D images of a 3D scene taken by cameras with different orientations and locations. For each pair of images of the same scene, we use the pipeline proposed by \\citet{ozyesil2015robust} to estimate the relative 3D rotations. The ground truth camera orientations are also provided. Table \\ref{tab:real}\ncompares the performance of IRLS-GM, IRLS-$\\ell_{1\/2}$, CEMP+MST and MPLS, while reporting \nmean and median errors, runtime and number of iterations. The number of iterations is the sum of the number of iterations to initialize the rotations and the number of iterations of the rest of the algorithm, where CEMP+MST only has iterations in the initialization step.\n\nMPLS achieves the lowest mean and median error on $9$ out of $14$ datasets with runtime comparable to both IRLS, while IRLS-GM only outperforms MPLS on the Gendarmenmarkt dataset. This dataset is relatively sparse and lacks cycle information. It contains a large amount of self-consistent corruption and none of the methods solve it reasonably well. Among the tested 4 methods, the fastest approach is CEMP+MST.\nIt achieves shortest runtime on 13 out of 14 dataset. Moreover, CEMP+MST is 3 times faster than other tested methods on the largest dataset (Piccadilly). We remark that CEMP+MST is able to achieve comparable results to common IRLS on most datasets, and has superior performance on 2 datasets, which have some perfectly estimated edges.\nIn summary, for most of the datasets, MPLS provides the highest accuracy and CEMP+MST obtains the fastest runtime. \n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe proposed a framework for solving group synchronization under high corruption and noise. \nThis general framework requires a successful solution of the weighted least squares problem, which depends on the group. For $SO(3)$, we explained how a well-known solution integrates well with our framework. We demonstrated state-of-the-art performance of our framework for $SO(3)$ synchronization. We have motivated our method as an alternative to IRLS and explained how it may overcome the limitations of IRLS when applied to group synchronization.\n\nThere are many directions to expand \nour work. One can carefully adapt and implement our proposed framework to other groups that occur in practice. \nOne may develop certain theoretical guarantees for convergence and exact recovery of MPLS.\n\n\\section*{Acknowledgement}\nThis work was supported by NSF award DMS-18-21266. We thank Tyler Maunu for his valuable comments\non an earlier version of this manuscript.\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe linear fractional maps from the complex plane into itself are among the very first objects of study in one-variable complex analysis since they have many good geometric properties (e.g. mapping real lines and circles among themselves) and they are also the only biholomorphisms of the unit disk and the Riemann sphere. It is thus very natural to look at the linear fractional maps in several complex variables and such explorations have been made by, for instance, Cowen-MacCluer~\\cite{CM} and Bisi-Bracci~\\cite{BB}. In their works, they have focused on the linear fractional maps that map the unit ball into itself. They obtained a variety of results for those linear fractional maps, including the classification, normal forms, and also fixed point theorems for such maps. From the point of view of function theory, every linear fractional self map of the unit ball also give rise to a composition operator in various function spaces of the unit ball and such an operator has been studied by many people, e.g. Cowen~\\cite{Co}, Bayart~\\cite{Ba} and Chen-Jiang~\\cite{CJ}.\n\nIn this article, we will try to extend the study of linear fractional self maps to a much more general class of domains, which in particular include the classical bounded symmetric domains of type I and also the so-called \\textit{generalized balls}. These two classes both contain the usual complex unit balls as special cases. Although here our major focus is on the generalized balls, our results should shed some light on the behavior of the linear fractional maps defined on the classical bounded symmetric domains of type-I since they are determined by the same set of matrices (as we will see in Theorem~\\ref{expansion matrix}).\n\nFor any two positive integers $p$ and $q$, let $H_{p,q}$ be the standard non-degenerate\nHermitian form of signature $(p,q)$ on $\\mathbb C^{p+q}$ where $p$ eigenvalues are $1$\nand $q$ eigenvalues are $-1$, represented by the matrix $\\begin{pmatrix}I_p&0\\\\0&-I_q\\end{pmatrix}$ under the standard coordinates.\nFor a positive integer $r< p+q$, denote by $Gr(r, \\mathbb C^{p+q})$ the Grassmannian of $r$-dimensional complex linear subspaces\n(or simply $r$-\\textit{planes}) of $\\mathbb C^{p+q}$.\nWhen $1\\leq r\\leq p$, we define the domain $D^r_{p,q}$ in $Gr(r, \\mathbb C^{p+q})$ \nto be the set of positive definite $r$-planes in $\\mathbb C^{p+q}$ with respect to $H_{p,q}$.\nWe call $D^r_{p,q}$ a {\\it generalized type-I domain}.\nThe generalized type-I domain $D_{p,q}^r$ is an $SU(p,q)$-orbit on $Gr(r, \\mathbb C^{p+q})$ under the natural action induced by that of $SL(p+q;\\mathbb C)$ on $Gr(r,\\mathbb C^{p+q})$. It is an example of \\textit{flag domains} in a general context, of which one can find a comprehensive reference \\cite{FHW}. Recently $D_{p,q}^r$ have been studied by Ng~\\cite{ng1, ng2} regarding the proper holomorphic mappings between them\nand by Kim~\\cite{Kim} regarding the CR maps on some CR manifolds in $\\partial D_{p,q}^r$. We remark that the case $r=p$ corresponds to the type-I classical bounded symmetric domains~\\cite{ng2}. On the other extreme, when $r=1$, which will be the case of special interest to us, corresponds to the domains $D_{p,q}:=D^1_{p,q}$, which are called the generalized balls.\nIt follows immediately from our definition that $D_{p,q}$ can be also defined as the following domain on $\\mathbb P^{p+q-1}$:\n$$\nD_{p,q} = \\left\\{ [z_1,\\dots, z_{p+q}] \\in \\PP^{p+q-1} : |z_1|^2 +\\dots + |z_p|^2 > |z_{p+1}|^2 + \\dots + |z_{p+q}|^2\\right\\}.\n$$\nWhen $p=1$, it is biholomorphic to the unit ball in the Euclidean space $\\CC^q$.\n\nThe generalized ball is one of the simplest kinds of domains on the projective space and their boundaries are smooth Levi non-degenerate (but not pseudoconvex in general) CR manifolds, of which detailed studies have been carried out by Baouendi-Huang~\\cite{BH} and Baouendi-Ebenfelt-Huang~\\cite{BEH}. More recently, Ng~\\cite{ng2} discovered that the proper holomorphic mapping problem for the generalized balls is deeply linked to that of the classical bounded symmetric domains of type-I.\n\nHere we recall that a linear fractional map on $\\mathbb C^n$ is in fact just a restriction of a linear map on $\\mathbb P^n$ expressed in terms of the inhomogeneous coordinates of a Euclidean coordinate chart $\\mathbb C^n$ in $\\mathbb P^n$. Similarly, if we follow our notations, a linear fractional self map of the unit ball $D_{1,q}$ simply comes from a linear map on $\\mathbb P^{q}$ that maps $D_{1,q}$ into itself. We have defined our generalized type-I domains as domains on the Grassmannians on which, just like the case of the projective space, there are homogeneous coordinates and the associated linear maps. Since we will always work with homogeneous coordinates, we will thus call a self map of a generalized type-I domain a \\textit{linear self map} if it is the restriction of a linear map of the ambient Grassmannian. \n\\newline\n\n\\noindent\\textbf{Remark.} Hence, according to our terminology, \\textit{a linear self map and a linear fractional self map are the same.} ``Fractions\" appear only because inhomogeneous coordinates are used.\n\\newline\n\nWe call a linear self map of a generalized type-I domain $D^r_{p,q}$ \\textit{non-minimal} if its range is not of minimal dimension (see Definition~\\ref{minimality}). In the case of the unit balls, a non-minimal linear self map is nothing but a non-constant linear self map. Roughly speaking, the minimal linear self maps, just like the constant maps for the unit balls, are the cases for which most statements become trivialities or non-applicable. In Section~\\ref{type-I domain}, we will first establish a fundamental result for the study of linear self maps of generalized type-I domains. Namely, we will prove (Theorem~\\ref{expansion matrix}) that every non-minimal linear self map of a generalized type-I domain can be represented by a matrix $M$ satisfying the inequality\n\n\\begin{equation}\nM^H H_{p,q} M - H_{p,q}>0, \\label{expansioneq}\n\\end{equation}\nwhere $H_{p,q} = \\begin{pmatrix}I_p & 0 \\\\0 & -I_q\\end{pmatrix}$, in which $I_p$, $I_q$ are identity matrices of rank $p$ and $q$, and $\\cdot^H$ denotes Hermitian transposition, and ``$\\geq$\" means Hermitian semi-positivity. Furthermore, we will prove (Theorem~\\ref{isometrythm}) that a surjective linear self map can be represented by a matrix which makes the equality hold, i.e. by a matrix in the indefinite unitary group $U(p,q)$. For the case of the unit balls, these results have been established by Cowen-MacCluer~\\cite{CM} and we will modify their proof to work for any generalized type-I domain. As simple as it may look, such a matrix inequality is extremely useful in obtaining various kinds of results for the linear self maps of generalized type-I domains. In particular, we will use it to show that any non-minimal linear self map extends to a neighborhood of the closure of the domain (Theorem~\\ref{extension thm}).\n\nIn Section~\\ref{automorphisms}, we will study in detail the automorphism groups of the generalized balls, including the partial double transitivity on the boundary, fixed point theorems and normal forms. Here we remark that a generalized ball cannot be realized as a bounded convex domain in the Euclidean space unless it is a usual unit ball (see e.g.~\\cite{gn}) and hence one cannot apply Brouwer's fixed point theorem to get a fixed point in its closure. We will establish the existence of fixed points (in the closure) in Theorem~\\ref{existence of fixed points} and give a number of results regarding the behavior of the fixed points (Theorem~\\ref{fixed point number thm} and~Corollaries~\\ref{at most}, \\ref{hs generalization}). For obtaining a normal form for automorphisms, we will show that the subgroup $U(p)\\times U(q)$ and the ``non-isotropic dilations\" generate the full automorphism group $Aut(D_{p,q})$ (Theorem~\\ref{normal form}).\n\nAfter studying the automorphisms, we will then look at arbitrary linear self maps of the generalized balls in Section~\\ref{linear maps section}. We will again prove some results regarding their fixed points, including especially Theorem~\\ref{bb generalization}, which is about how the number of fixed points on the boundary of a generalized ball is related to the existence of interior fixed points. This generalizes a result for the unit ball (see Bisi-Bracci~\\cite{BB}) saying that any linear self map of the unit ball with more than two boundary fixed points must have an interior fixed point. We will also obtain in this section a relation between the linear self maps of the \\textit{real} generalized balls of $D_{p,q}$ and those of $D_{p,q}$ (Theorem~\\ref{real generalized ball}).\n\nFinally, in Section~\\ref{examples} we will collect some illustrating or extremal examples for the results obtained in the previous sections.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bigskip\n{\\bf Acknowledgements.} The first author was partially supported by the Key Program of NSFC, (No. 11531107). The second author\nwas partially supported by Thousand Talents Program of the Organization Department of the CPC Central Committee, and Science and Technology Commission of Shanghai Municipality (STCSM) (No. 13dz2260400). The third author was partially supported by Basic Science Research Program through the National Research \nFoundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1F1A1060175). \n\n\n\\section{Generalized type-I domains and their linear self maps}\\label{type-I domain}\n\n\\noindent\\textbf{Notations.} For what follows, for any $p,q\\in\\mathbb N^+$, we will equip $\\mathbb C^{p+q}$ with the standard non-degenerate indefinite Hermitian form $H_{p,q}$ and denote the resulting indefinite inner product space by $\\mathbb C^{p,q}$.\n\\newline\n\n\nDenote $M_n$ the set of $n$-by-$n$ matrices with complex entries. Let $M\\in M_{p+q}$ and consider the linear map, which will also be denoted by $M$, from $\\mathbb C^{p,q}$ into itself, given by $z\\mapsto Mz$, where $z\\in\\mathbb C^{p,q}$ is regarded as a column vector.\nLet the null space of $M$ be $ker(M):=\\{z\\in\\mathbb C^{p,q}:Mz=0\\}$. Then the image of every positive definite $r$-plane in $\\mathbb C^{p,q}$ under $M$ is still an $r$-plane if and only if $ker(M)$ is negative semi-definite with respect to $H_{p,q}$. In such case, $M$ gives rise to a holomorphic map from each $D^r_{p,q}$ to the Grassmannian $Gr(r,\\mathbb C^{p,q})$. It is clear that two matrices $M, M'\\in M_{p+q}$ induce the same map on $D^r_{p,q}$ if $M=\\lambda M'$ for some $\\lambda\\in\\mathbb C^*$.\nThe question of interest is whether or not such a map is actually a \\textit{self map} of $D^r_{p,q}$.\n\n\\begin{definition}\nA self map of $D^r_{p,q}$ is called a \\textit{linear self map}, if it is given by a matrix $M\\in M_{p+q}$ in the way described above. If there is no danger of confusion, we will also denote any such self map by the same symbol $M$. Conversely, any matrix $M\\in M_{p+q}$ inducing a given linear self map of $D^r_{p,q}$ is called a \\textit{matrix representation} of the linear self map.\n\\end{definition}\n\nLet $M\\in M_{p+q}$ be a matrix representation of a linear self map of some $D^r_{p,q}$. Then, we must have rank$(M)\\geq p$ since otherwise $\\dim_{\\mathbb C}(ker(M))>q$ and $ker(M)$ would not be negative semi-definite in $\\mathbb C^{p,q}$, contradicting the fact that $M$ induces a linear map on $D^r_{p,q}$. We \nmake the following definition in relation to this.\n\n\\begin{definition}\\label{minimality}\nA linear self map of $D^r_{p,q}$ is called \\textit{minimal} if it is given by a matrix $M\\in M_{p+q}$ with rank$(M)=p$. Otherwise, we say that the linear self map is \\textit{non-minimal}. \n\\end{definition}\n\n\\noindent\\textbf{Remark.} For the unit balls $D_{1,q}$, a non-minimal linear self map is simply a non-constant linear self map.\n\\newline\n\n\n\n\nIf $M\\in M_n$ satisfies Inequality~\\eqref{expansioneq}, then it follows immediately that the image of any positive definite $r$-plane is again a positive definite $r$-plane and thus $M$ induces a linear self map on each $D^r_{p,q}$. \nWe are now going to show that conversely any \\textit{non-minimal} linear self map of $D^r_{p,q}$ can be represented by a matrix satisfying Inequality~(\\ref{expansioneq}). For this purpose, we will use some terminologies and results by Cowen-MacCluer~\\cite{CM}. Let $(V,[\\cdot,\\cdot])$ be a finite dimensional complex vector space equipped with an indefinite Hermitian form $[\\cdot,\\cdot]$. Following~\\cite{CM}, we will say that a linear map $T$ of $V$ into itself is an \\textit{expansion} if $[Tv,Tv]\\geq [v,v]$ for all $v\\in V$, and an \\textit{isometry} if $[Tv,Tv]=[v,v]$ for all $v\\in V$. In particular, if we identify the linear maps of $\\mathbb C^{p,q}$ into itself with their matrix representations with respect to the standard basis, then $M\\in M_{p+q}$ is an expansion of $\\mathbb C^{p,q}$ if and only if it satisfies the inequality~(\\ref{expansioneq}) and is an isometry if $M$ makes the equality hold. We are going to show that the non-minimal linear self maps of $D^r_{p,q}$ are precisely those given by the expansions of $\\mathbb C^{p,q}$, and the surjective linear self maps of $D^r_{p,q}$ are given by the isometries of $\\mathbb C^{p,q}$.\n\n\n\n\nThere following lemma can be found in \\cite{CM} but we rewrite it (reversing the signs) in a way more suitable for our purpose.\n\n\n\\begin{lem}[\\cite{CM}] \\label{expansion}\nSuppose $[\\cdot,\\cdot]_1$ and $[\\cdot,\\cdot]_2$ are indefinite Hermitian forms on the complex vector space $V$ such that $[x,x]_1=0$ implies $[x,x]_2 \\geq 0$. Then,\n$$\\lambda:=-\\inf_{[y,y]_1=-1}[y,y]_2 < \\infty $$\nand $$[x,x]_2\\geq \\lambda[x,x]_1$$\nfor all $x\\in V$.\n\\end{lem}\n\n\n\n\nWe are now in a position to show that the non-minimal linear self maps of $D^r_{p,q}$ are all given by the matrices satisfying Inequality~(\\ref{expansioneq}). (The case for the unit ball was obtained by Cowen-MacCluer~\\cite{CM} and part of our proof is taken from there.)\n\n\\begin{thm}\\label{expansion matrix}\nEvery non-minimal linear self map of $D^r_{p,q}$ can be represented by a matrix satisfying Inequality~(\\ref{expansioneq}).\n\\end{thm}\n\n\\begin{proof}\nLet $M\\in M_{p+q}$ be a matrix such that it induces linear self map (also denoted by $M$) on some $D^r_{p,q}$. For $x,y\\in\\mathbb C^{p,q}$, let $[x,y]_1=y^H H_{p,q} x$ and let $[x,y]_2=(My)^HH_{p,q} Mx$. The hypothesis that $M$ maps\n $D^r_{p,q}$ into $D^r_{p,q}$ means that whenever $[x,x]_1>0$, we have $[x,x]_2>0$. By continuity, we get that if $[x,x]_1 \\geq 0$, then $[x,x]_2 \\geq 0$.\n Thus, the hypotheses of Lemma \\ref{expansion} are satisfied. So we only need to show that $\\la > 0$. Then $\\la^{-1\/2}M$ is an expansion in $\\mathbb C^{p,q}$.\n\nTo see that $\\lambda >0$, we let the range of $M$ be $R(M):=\\left\\{y\\in\\mathbb C^{p,q}: y=Mz \\textrm{\\,\\,for some\\,\\,} z\\right\\}$. Now if $M$ is non-minimal, then\n$\\dim_{\\mathbb C}(R(M))\\geq p+1$ (Definition~\\ref{minimality}). Thus, $R(M)$ must contain a negative vector $y$ and any preimage $z$ of $y$ must also be a negative vector since we have already seen in the previous paragraph that $[z,z]_1 \\geq 0$ would imply that $[y,y]_1=[z,z]_2 \\geq 0$.\nTherefore, we can find $z\\in \\mathbb{C}^{p,q}$ such that $[z,z]_1<0$ as well as $[z,z]_2<0$ and hence $\\la > 0$. \n\n\\end{proof}\n\n\\begin{thm}\\label{isometrythm}\nEvery surjective linear self map of $D^r_{p,q}$ can be represented by a matrix satisfying the equality in~(\\ref{expansioneq}). In particular, every surjective linear self map of $D^r_{p,q}$ is an automorphism.\n\\end{thm}\n\\begin{proof}\nIf a linear self map $M$ of a given $D^r_{p,q}$ is surjective, then $M$ is surjective as a linear map on $\\mathbb C^{p,q}$ since its image contains all the positive vectors (which constitute an open set in $\\mathbb C^{p,q}$). The inverse linear map $M^{-1}$ also maps positive vectors to positive vectors since each of the 1-planes generated by positive vectors can be regarded as the intersection of a set of positive $r$-planes and $M$ is surjective (as a self map of $D^r_{p,q}$). Hence, we see that $M^{-1}$ is also a surjective linear self map of $D^r_{p,q}$ and is thus an expansion. Therefore, there are non-zero scalars $\\alpha$ and $\\beta$ such that $\\alpha M$ and $\\beta M^{-1}$ are expansions. Thus, for every $z\\in\\mathbb C^{p,q}$, if we write $\\|z\\|^2_{p,q}:=z^H H_{p,q} z$, then\n$$\n\t\\|\\beta z\\|^2_{p,q}\\geq \\|Mz\\|^2_{p,q}\\geq \\|\\alpha^{-1} z\\|^2_{p,q}\n$$\nand hence\n$$\n\t|\\alpha\\beta|^2\\|z\\|^2_{p,q}\\geq\\|\\alpha Mz\\|^2_{p,q}\\geq\\|z\\|^2_{p,q}.\n$$\n\nSince the inequality is true for both positive vectors and negative vectors, we deduce that $|\\alpha\\beta|=1$ and $\\|\\alpha Mz\\|^2_{p,q}=\\|z\\|^2_{p,q}$ for every $z$. That is, $\\alpha M$ is an isometry.\n\\end{proof}\n\nA linear self map $M$ of $D^r_{p,q}$ originally comes from a linear map $\\tilde M$ defined on the ambient Grassmannian $Gr(r,\\mathbb C^{p,q})$. If $M$ is not surjective, then there is a set of indeterminacy $Z\\subset Gr(r,\\mathbb C^{p,q})$ on which $\\tilde M$ is not defined. The set $Z$ is outside $D^r_{p,q}$ and consists of the points corresponding to the $r$-planes that intersect the kernel of a matrix representation of $\\tilde M$. A priori, $Z$ can intersect the boundary $\\partial D^r_{p,q}$ and obstructs the extension of $M$ across $\\partial D^r_{p,q}$, but we are now going to show that this does not happen for non-minimal linear self maps. \n\n\\begin{thm}\\label{extension thm}\nEvery non-minimal linear self map of $D^r_{p,q}$ extends holomorphically to an open neighborhood of the closure $\\overline{D^r_{p,q}}:=D^r_{p,q}\\cup\\partial D^r_{p,q}$.\n\\end{thm}\n\\begin{proof}\nLet $M$ be a non-minimal linear self map of $D^r_{p,q}$ and denote also by $M$ a matrix representation of it which satisfies the inequality $M^HH_{p,q}M-H_{p,q}\\geq 0$. As mentioned at the beginning of this section, $ker(M)$ must be negative semi-definite with respect to $H_{p,q}$. We are now going to show that $ker(M)$ does not contain any non-zero null vector if $M$ is non-minimal. Suppose on the contrary there is a non-zero null vector $\\eta\\in ker(M)$. Let $v\\in\\mathbb C^{p,q}$ and write $\\|v\\|_{p,q}^2=v^HH_{p,q}v$. Then for any $r\\in\\mathbb R$, we have $M(v+r\\eta)=Mv$ and\n$$\n\\|Mv\\|_{p,q}^2=\\|M(v+r\\eta)\\|_{p,q}^2\\geq \\|v+r\\eta\\|_{p,q}^2=\\|v\\|_{p,q}^2+r(\\eta^HH_{p,q}v+v^HH_{p,q}\\eta).\n$$\n\nNow if $v$ is chosen such that $\\textrm{Re}(v^HH_{p,q}\\eta)\\neq 0$, then the above inequality cannot hold for every $r\\in\\mathbb R$ and hence we get a contradiction. Consequently, $ker(M)$ does not contain any non-zero null vector and therefore the image of any positive semi-definite $r$-plane under $M$ is still an $r$-plane and the set of indeterminacy $Z\\subset Gr(r,\\mathbb C^{p,q})$ of $M$ (as a linear map on $Gr(r,\\mathbb C^{p,q}))$ is disjoint from $\\overline{D^r_{p,q}}$. Since both $Z$ and $\\overline{D^r_{p,q}}$ are closed in $Gr(r,\\mathbb C^{p,q})$ (and hence compact), there is an open neighborhood of $\\overline{D^r_{p,q}}$ disjoint from $Z$ and now the theorem follows.\n\\end{proof}\n\n\\noindent\\textbf{Remark.} The non-minimality is indeed necessary to guarantee the extension across the entire boundary. See Example~\\ref{no extension} for a minimal linear self map which does not extend across some boundary point.\n\n\n\\section{Automorphisms on generalized balls}\\label{automorphisms}\n\nIn this section, we are going to study in detail the automorphisms on the generalized balls $D_{p,q}$, regarding their fixed point sets and also their normal form. We begin by determining the automorphism group of $D_{p,q}$.\n\n\\begin{thm}\nEvery automorphism of $D_{p,q}$ is a linear self map and thus extends to an automorphism of the ambient projective space $\\mathbb P^{p+q-1}$. The automorphism group $\\textrm{Aut}(D_{p,q})$ of $D_{p,q}$ is isomorphic to $PU(p,q)$, the projectivization of the indefinite unitary group $U(p,q)$. In particular, every automorphism of $D_{p,q}$ can be represented by a matrix in $U(p,q)$.\n\\end{thm}\n\\begin{proof}\nThe statements are well-known for the complex unit balls $D_{1,q}$ and also for the complements of the complex unit balls $D_{p,1}$. Suppose now $p,q\\geq 2$. Then, it has been shown by Baouendi-Huang~\\cite{BH} and (for a more geometric proof, see Ng~\\cite{ng1}) that every automorphism of $D_{p,q}$ is necessarily a linear map. Now by Theorem~\\ref{isometrythm} here (or Lemma 2.13 in~\\cite{ng1}), we see that every automorphism can be represented by a matrix in $U(p,q)$. Since two elements in $U(p,q)$ represent the same automorphism of $D_{p,q}$ if and only if they are scalar multiples of each other, it follows now that $\\textrm{Aut}(D_{p,q})\\cong PU(p,q)$.\n\\end{proof}\n\n\\begin{cor}\\label{extension cor}\nThe action of $\\textrm{Aut}(D_{p,q})$ extends real-analytically to $\\partial D_{p,q}$.\n\\end{cor}\n\n\n\nThe following version of Witt's theorem is very useful in studying the various transitivities of $\\textrm{Aut}(D_{p,q})$.\n\n\\begin{lem}[Witt \\cite{Witt}] \\label{cor_Witt theorem}\nLet $X$ be a complex vector space equipped with a non-degenerate Hermitian form and $Y \\subset X$ be any complex vector subspace. Then any isometric embedding $f : Y \\rightarrow X$ extends to an isometry $F$ of $X$.\n\\end{lem}\n\n\n\\begin{thm}\\label{doubly transitive} For $u\\in \\mathbb C^{p,q}$, let $[u]$ be its projectivization in $\\mathbb P^{p+q-1}$.\n\\begin{enumerate}\n\\item\n $\\text{Aut}(D_{p,q})$ is transitive on $D_{p,q}$ and also on $\\partial D_{p,q}$.\n\\item\nLet $[v_1]$, $[v_2]$, $[w_1]$, $[w_2]\\in \\partial D_{p,q}$, where $[v_1]\\neq [v_2]$ and $[w_1]\\neq [w_2]$. Then, there exists $M\\in \\textrm{Aut}(D_{p,q})$ such that $M([v_j])=[w_j]$ for $j=1,2$ if and only if there exists non-zero $\\alpha\\in\\mathbb C$ such that $v_1^HH_{p,q}v_2=\\alpha w_1^HH_{p,q}w_2$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\n{ 1. Let $[u_1]$, $[u_2]\\in D_{p,q}$. Then we choose some $k>0$ such that the linear map $f:\\mathbb Cu_1\\rightarrow \\mathbb C^{p,q}$, defined by $u_1\\mapsto ku_2$ is an isometric embedding. By Lemma~\\ref{cor_Witt theorem} $f$ extends to an isometry of $\\mathbb C^{p,q}$ and therefore there is an automorphism of $D_{p,q}$ mapping $[u_1]$ to $[u_2]$. Similarly, for any two points $[v_1]$, $[v_2] \\in \\partial D_{p,q}$. The map\n$i\\colon \\mathbb C v_1\\rightarrow \\mathbb C^{p,q}$ defined by $v_1\\mapsto v_2$ is an isometric embedding and hence there exists an automorphism of $D_{p,q}$ such mapping $[v_1]$ to $[v_2]$.\n}\n\n2. Suppose that there exists $\\alpha\\in\\mathbb C^*$ such that $v_1^HH_{p,q}v_2=\\alpha w_1^HH_{p,q}w_2$. Let $\\phi : Y \\rightarrow \\CC^{p,q}$, where $Y=\\textrm{Span}\\{v_1, v_2\\}$ be the linear embedding defined by $\\phi(v_1)=w_1$ and $\\phi(v_2)=w_2$. By replacing $w_1$ by some scalar multiple, we can make $v_1^HH_{p,q}v_2=w_1^HH_{p,q}w_2$. Then $\\phi$ is an isometric embedding and hence $\\phi$ extends to an isometry of $\\mathbb C^{p,q}$ by Lemma \\ref{cor_Witt theorem} and the desired result follows. The converse is trivial.\n\\end{proof}\n\n\\noindent\\textbf{Remark.}\nTheorem~\\ref{doubly transitive} is a generalization of the double transitivity of the automorphism groups of the complex unit balls on their boundaries. This is because for $[v_1], [v_2]\\in\\partial D_{1,q}$ with $[v_1]\\neq [v_2]$, we always have $v_1^HH_{p,q}v_2\\neq 0$ since otherwise we would get a two dimensional isotropic subspace in $\\mathbb C^{1,q}$.\n\n\n\n\n\\subsection{Fixed points on $D_{p,q}$ and $\\partial D_{p,q}$}\n\nLet $A\\in U(p,q)$ and denote also by $A$ the corresponding automorphism of $D_{p,q}$.\nIt follows directly from the definition of the matrix representation of a linear self map that the fixed points of $A$ (as an automorphism on $D_{p,q}$) correspond precisely to the one-dimensional eigenspaces (or projectivized eigenvectors) of $A$ associated to the non-zero eigenvalues. The following simple observation regarding eigenvalues and eigenvectors of matrices in $U(p,q)$ will be very useful in studying the fixed points of the automorphisms of $D_{p,q}$.\n\n\\begin{lem}\\label{ortho lemma}\nLet $A\\in U(p,q)$. If $\\lambda_1$, $\\lambda_2$ are eigenvalues of $A$ and $v_1$, $v_2$ are two eigenvectors associated to them respectively, then either $v_1$ and $v_2$ are orthogonal with respect to $H_{p,q}$ or $\\overline{\\lambda_2}\\lambda_1=1$.\n\\end{lem}\n\\begin{proof}\nThe result follows from\n$\nv_2^H H_{p,q} v_1= v_2^HA^H H_{p,q} Av_1=\\left(\\ov{\\lambda}_2\\lambda_1\\right)\\,v^H_2 H_{p,q} v_1.\n$\n\\end{proof}\n\nLet $\\ov{D_{p,q}}:=D_{p,q}\\cup\\partial D_{p,q}$ be the closure of $D_{p,q}$. Since the closed complex unit balls $\\overline{\\mathbb B^q}\\cong\\overline{D_{1,q}}$ is a convex compact set in $\\mathbb C^q\\cong \\mathbb R^{2n}$, it follows from Corollary~\\ref{extension cor} and Brouwer's fixed point theorem that every element in $Aut(D_{1,q})$ has a fixed point in $\\overline{D_{1,q}}$. \nWhen $p\\geq 2$, any $D_{p,q}$ cannot be embedded as a convex compact set in some Euclidean space since it contains positive dimensional projective subspaces (see~\\cite{ng1}). Nevertheless, with a bit of ``detour\", we will still be able to use Brouwer's fixed point theorem to get a fixed point in the closure for \\textit{any} linear self map of $D_{p,q}$. The proof will be given in Section~\\ref{linear maps section}. Hence, we have\n\n\\begin{thm}\\label{existence of fixed points}\nEvery element in $\\textrm{Aut}(D_{p,q})$ has a fixed point on $\\ov{D_{p,q}}:=D_{p,q}\\cup\\partial D_{p,q}$.\n\\end{thm}\n\\begin{proof}\nFollows directly from Theorem~\\ref{general existence of fixed points}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\nWe now recall some elementary linear algebra. Let $M\\in M_n$ and $\\lambda$ be an eigenvalue of $M$. For some $r\\in \\mathbb N$ a vector $v\\in\\mathbb C^n$ is called a generalized eigenvector of rank $r$ of $M$ associated to the eigenvalue $\\lambda$ if $(M-\\lambda I)^rv=0$ but $(M-\\lambda I)^{r-1}v\\neq 0$.\nIt turns out that both the absolute values of the eigenvalues and the existence of generalized eigenvectors of higher rank give information about the fixed-point set of the linear self map on a generalized ball represented by $M$.\n\nAs before, for a vector $v\\in\\mathbb C^{p,q}$, we will denote by $[v]\\in\\mathbb P^{p+q-1}$ its projectivization. Similarly, for any complex vector subspace $W\\subset\\mathbb C^{p,q}$, we denote its projectivization by $[W]$.\n\n\\begin{prop}\\label{fixed point}\nLet $A\\in \\text{Aut}(D_{p,q})$ and choose a matrix representation in $U(p,q)$ and denote it also by $A$. Let $\\lambda$ be an eigenvalue of $A$ and $v$ be an associated generalized eigenvector of rank $r$.\n\\begin{enumerate}\\label{projective subspace on the boundary}\n\\item If $|\\lambda|\\neq 1$, then $[(A-\\lambda I)^{r-1}v]$ is a fixed point of $A$ on $\\partial D_{p,q}$. Furthermore, if $r\\geq 2$, then there is an $(r-1)$-dimensional projective linear subspace $[W]$ in $\\partial D_{p,q}$\ninvariant under $A$ and $[(A-\\lambda I)^{r-1}v]\\in [W]$ is a unique fixed point of $A$ in $[W]$.\n\\item If $|\\lambda|=1$ and $r\\geq 2$,\nthen $[(A-\\lambda I)^{r-1}v]$ is a fixed point of $A$ on $\\partial D_{p,q}$.\nFurthermore, there is an $\\left(\\left[ \\frac{r}{2} \\right]-1\\right) $-dimensional projective subspace\n$[W]$ in $\\partial D_{p,q}$ invariant under $A$ and $[(A-\\lambda I)^{r-1}v]$ is a unique fixed point of $A$ in $[W]$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n1. Let $A\\in U(p,q)$, $\\lambda$ be an eigenvalue of $A$ and $v$ be an associated generalized eigenvector of rank $r$. Let $v_r:=v$ and define inductively $v_{j-1}:=(A-\\lambda I)v_j$ for $j\\in\\{r,r-1,\\ldots, 2\\}$. In particular $v_1$ is an eigenvector of $A$ associated to $\\lambda$.\n\\begin{eqnarray*}\nAv_1&=&\\lambda v_1, \\\\\nAv_2 &=& v_1 + \\lambda v_2,\\\\\n&&\\vdots\\\\\nA v_r &=&v_{r-1} + \\lambda v_r.\n\\end{eqnarray*}\nSuppose that $|\\lambda|\\neq 1$.\nWe claim that\n\\begin{equation}\nv_i^HH_{p,q} v_j=0 \\text{ for all } 1\\leq i,j\\leq r.\n\\end{equation}\nWe will prove it using induction.\nBecause of $A^H H_{p,q} A = H_{p,q}$, we have\n$v_1^H H_{p,q}v_1 =0$\nand this implies that $[v_1]\\in \\partial D_{p,q}$.\nSuppose that $r\\geq 2$ and $v_1^H H_{p,q}v_{j'}=0$ for every $j'0$ for all $i$.\n\nAs $\\|Av\\|_{p,q}^2=\\|v\\|_{p,q}^2$, we have $v^H (A^H H_{p,q}A-H_{p,q})v=0$. \nLet $v'=P^Hv=({x'}_1,x'_2,\\ldots, x'_{p+q})$. Then $x'_1=x'_2=\\cdots=x'_r=0$ since $v'^H \\text{diag}(\\lambda_1, \\lambda_2, \\ldots,\\lambda_{r},0,\\ldots,0)v'=v^H (A^H H_{p,q}A-H_{p,q})v=0$. It follows that $P^H(A^H H_{p,q}A-H_{p,q})PP^Hv=0$ and hence $(A^H H_{p,q}A-H_{p,q})v=0$. Then $u^H (A^H H_{p,q}A-H_{p,q})v=0$ for any $u\\in \\mathbb{C}^{p,q}$.\n Therefore $(Au)^HH_{p,q}Av=u^HH_{p,q}v$. So $A(v^\\perp)\\subset v^\\perp$.\n\\end{proof}\n\n\n\nThe following generalization of the fixed point theorem of Hayden-Suffridge to linear fractional maps of the complex unit ball was accomplished by Bisi-Bracci~\\cite{BB}: \\textit{If a linear fractional map of the complex unit ball has more than\ntwo fixed points on the boundary, then it has fixed points in the interior.} Just like what happens for automorphisms (as studied in Section~\\ref{automorphisms}), we will have to assume that no projective line in the boundary is fixed everywhere by the linear self map before we can generalize this result to $D_{p,q}$.\n\n\n\nWe have proven in Theorem~\\ref{extension thm} that any non-minimal linear self map of a generalized ball extends holomorphically across the boundary. For minimal linear self maps, we still have such an extension for a \\textit{general} boundary point since the set of indeterminacy cannot contain the entire boundary. But even so, a minimal linear self map cannot have any boundary fixed point. In order to see this, let $F$ be a minimal linear self map of a generalized ball. The range $R$ of its matrix representation (also denoted by $F$) is of dimension $p$ (see Definition~\\ref{minimality}). On the other hand, since $F$ maps positive vectors to positive vectors, for any positive $p$-plane $P\\subset\\mathbb C^{p,q}$, we must have $\\dim_\\mathbb C F(P)=p$ and hence $F(P)=R$, which implies that $R$ is a positive $p$-plane. Hence, the image of any null vector under $F$ is either a positive vector or the zero vector. Therefore, a minimal linear self map cannot have any fixed point on the boundary.\n\n\nWe will first establish our fixed point theorem for the complex unit ball and its exterior, i.e. for $D_{1,q}$ and $D_{p,1}$. In particular, this will give a new proof for the result of Bisi-Bracci~\\cite{BB}.\n\n\n\\begin{thm}\\label{ball}\n\nLet $q\\geq 1$ and $p\\geq 2$. For $D_{1,q}$ (resp. $D_{p,1}$), any linear self map with at least three (resp. two ) boundary fixed points has a fixed point in the interior.\n\n\\end{thm}\n\n\n\\begin{proof}\n\n\n\nWe will first prove the theorem for $D_{1,q}$. Let $F$ be a linear self map of $D_{1,q}$ with at least three fixed points in $\\partial D_{1,q}$. Let $A$ be a matrix representation of $F$ and by the hypotheses we can find three null vectors $v_1,v_2,v_3\\in\\mathbb C^{1,q}$ such that any two of them are not proportional and they are eigenvectors of $A$ associated to some non-zero eigenvalues.\n\nFirst of all, when $q=1$, then $\\dim_\\mathbb C(\\mathbb C^{1,1})=2$. But $v_1,v_2,v_3$ are pairwise non-proportional eigenvectors and so we deduce that $A$ can only be a non-zero multiple of the identity matrix and the desired result follows in this case.\n\nNow we can assume that $q\\geq 2$. Let $E=span_\\mathbb C(v_1,v_2,v_3)$. Then $E$ is an invariant subspace of $A$. Moreover, the restriction $H_{1,q}|_E$ is non-degenerate since there is no isotropic subspace in $\\mathbb C^{1,q}$ of dimension greater than one.\n\n If $\\dim_\\mathbb C(E)=2$, then the $H_{1,q}|_E$ must be of signature $(1,1)$ and we are back to the case $q=1$. Suppose now $\\dim_\\mathbb C(E)=3$. Then $H_{1,q}|_E$ is of signature $(1,2)$. Let $E_{12}=span_\\mathbb C(v_1,v_2)$. Then, the signature of $H_{1,q}|_E$ is again $(1,1)$. Hence, the orthogonal complement of $E_{12}$ in $E$, denoted by $N_{12}$, is complementary to $E_{12}$ and $\\dim_\\mathbb C(N_{12})=1$. Therefore, $E=E_{12}\\oplus N_{12}$. Note that since $\\|Av_j\\|^2_{p,q}=\\|v_j\\|_{p,q}^2=0$ for $j=1,2$, by Lemma~\\ref{semipositive}, we see that $N_{12}=v_1^\\perp\\cap v_2^\\perp$ is invariant by $A$ and thus is a one dimensional eigenspace of $A$. Choose now an eigenvector $v_4\\in N_{12}$. But on the other hand, since $H_{1,q}|_E$ is of signature $(1,2)$, we see that $H_{1,q}|_{N_{12}}$ is negative definite. This implies that $v_3$ and $v_4$ are not proportional. We now have four eigenvectors $v_1,v_2,v_3,v_4\\in E$ which are pairwise non-proportional. But $\\dim_\\mathbb C(E)=3$, so this is possible only if the restriction of $A$ on $E$ has at most two different eigenvalues. Suppose $v_j$ and $v_k$ are associated to the same eigenvalue $\\lambda$ (which must be non-zero), then for $E_{jk}:=span_\\mathbb C\\{v_j,v_k\\}$, we have that $H_{1,q}|_{E_{jk}}$ is of signature $(1,1)$ and hence we get an eigenvector $v\\in E_{jk}$ which is also a positive vector and the desired result now follows.\n\nFor the case of $D_{p,1}$ with $p\\geq 2$, we similarly let $G$ be a linear self map of $D_{p,1}$ with at least two fixed points in $\\partial D_{p,1}$ and let $B$ be a matrix representation of $G$. By the hypotheses, we can find two null vectors $u_1,u_2\\in\\mathbb C^{p,1}$ such that they are not proportional and are eigenvectors of $B$ associated to some non-zero eigenvalues. Let $F_{12}:=span_\\mathbb C(u_1,u_2)$. Again, since $\\|Bu_j\\|^2_{p,q}=\\|u_j\\|_{p,q}^2=0$ for $j=1,2$, we see from Lemma~\\ref{semipositive} that $Q_{12}:=F_{12}^\\perp=u_1^\\perp\\cap u_2^\\perp$ is invariant by $B$. The signature of $H_{p,1}|_{F_{12}}$ is $(1,1)$ and thus $H_{p,1}|_{Q_{12}}$ is of signature $(p-1,0)$. But $\\dim_\\mathbb C(Q_{12})=p-1$ and hence $Q_{12}$ is a positive definite invariant subspace of $A$. It now follows that we can find an eigenvector of $A$ which is a positive vector. Therefore, we get a fixed point in $D_{p,1}$.\n\n\n\\end{proof}\n\n\n\n\nWe can now generalize our result to an arbitrary generalized ball.\n\n\\begin{thm}\\label{bb generalization}\nLet $F$ be a linear self map of $D_{p,q}$. Suppose there is no projective line in $\\partial D_{p,q}$ on which every point is fixed by $F$.\n\\begin{enumerate}\n\\item If $p\\leq q$ and $F$ has at least $2p+1$ fixed points on $\\partial D_{p,q}$, then $F$ must have a fixed point in $D_{p,q}$. Furthermore, there is a projective line fixed everywhere by $F$.\n\\item If $p>q$ and $F$ has at least $2q$ fixed points on $\\partial D_{p,q}$, then $F$ must have a fixed point in $D_{p,q}$.\n\\end{enumerate}\n\n\\end{thm}\n\n\\noindent\\textbf{Remark.} Note that every linear self map of the ball (or its exterior) satisfies the condition given in Theorem~\\ref{bb generalization}\nsince there is no projective line in the boundary.\nHowever, in the case of $D_{p,q}$ with $p,q>1$, the condition is necessary. For instance, there is an automorphism\nof $D_{2,2}$ which has infinitely many fixed points on the boundary but no fixed point in $D_{2,2}$ (Example~\\ref{hs example}). \nFurthermore, the number $2p+1$ in Theorem~\\ref{bb generalization} (1) is sharp since there is an automorphism having four fixed points on $\\partial D_{2,2}$ and no fixed point in $D_{2,2}$ and there is no projective line in $\\partial D_{2,2}$ on which every point is fixed (Example~\\ref{sharp example 2}).\n\n\\begin{proof}[Proof of Theorem~\\ref{bb generalization}]\nLet $A$ be a matrix representation of $F$. \nAs explained before Theorem~\\ref{ball}, we can assume that $F$ is non-minimal. Thus, we can choose $A$ such that $A^H H_{p,q}A-H_{p,q}\\geq 0$ by Theorem~\\ref{expansion matrix}. \n\n\nLet $s\\geq 2$ be a positive integer and for every $k\\in\\{1,\\ldots,s\\}$, let $v_k\\in\\mathbb C^{p,q}$ be a null vector which is also an eigenvector of $A$ associated to a non-zero eigenvalue $\\lambda_k$. Suppose that $\\{v_1,\\ldots,v_s\\}$ are linearly independent. Thus, $\\{[v_1],\\ldots,[v_s]\\}\\subset\\partial D_{p,q}$ are $s$ distinct boundary fixed points of $F$. For $i\\neq j$, define $E_{ij}:=span_\\mathbb C\\{v_i, v_j\\}\\subset\\mathbb C^{p,q}$.\n\n Assume that $\\lambda_k=\\lambda_\\ell$ for some $k\\neq\\ell$. Then, as $v_k$, $v_\\ell$ are two linearly independent null vectors, $H_{p,q}|_{E_{k\\ell}}$ must be of signature $(1,1)$ or $E_{k\\ell}$ is isotropic. In the first situation, we can find a positive vector in $E_{k\\ell}$ while in the latter situation, the projectivization of $E_{k\\ell}$ gives a projective line in $\\partial D_{p,q}$. But $E_{k\\ell}$ is an eigenspace of $A$ associated to $\\lambda_k=\\lambda_\\ell$, so we either get a fixed point of $A$ in $D_{p,q}$ or we get a projective line in $\\partial D_{p,q}$ on which every point is fixed by $A$.\n\nSo now we only need to consider the case where $\\lambda_1,\\ldots,\\lambda_s$ are distinct.\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,$(\\star)$\n\n\nWe now divide the proof into two cases: \n\\begin{enumerate}\n\\item Every $E_{ij}$ is isotropic; \\label{i}\n\\item At least one $E_{ij}$ is not isotropic. \\label{ii}\n\\end{enumerate}\n\nIn case \\eqref{i}, $v_i$ is orthogonal (with respect to $H_{p,q}$) to $ v_j$ for every $i,j$ and it follows that $span_\\mathbb C \\{ v_1,\\ldots, v_{s}\\}$ is isotropic.\nIf $s\\geq 2p+1$ or $s\\geq 2q$, we get a contradiction since the maximal isotropic subspace in $\\mathbb C^{p,q}$ is of dimension $\\min(p,q)$.\n\nIn case \\eqref{ii}, without loss of generality, we may assume that $E_{12}$ is non-isotropic. \nThe restriction of $H_{p,q}$ on $E_{12}$ must be of signature $(1,1)$ (hence non-degenerate) as the null vectors $v_1$ and $v_2$ are linearly independent. \n Now take any $v_k$, where $k\\neq 1, 2$. Let $E_{12k}:=span_\\mathbb C\\{v_1, v_2, v_k\\}$, which is a 3-dimensional invariant subspace of $A$. Let $N$ be the orthogonal complement of $E_{12}$ in $E_{12k}$. Then $N\\cap E_{12}=\\{0\\}$ since $H_{p,q}|_{E_{12}}$ is non-degenerate. In particular, $\\dim_\\mathbb C(N)=1$. Moreover, as $\\|Av_1\\|_{p,q}^2=\\|v_1\\|_{p,q}^2=\\|Av_2\\|_{p,q}^2=\\|v_2\\|_{p,q}^2=0$, by Lemma \\ref{semipositive}, $N=v_1^\\perp\\cap v_2^\\perp$ is invariant under $A$. Hence, $N$ is a one-dimensional eigenspace of $A$. If $v_k\\not\\in N$, then we get four distinct one-dimensional eigenspaces in $E_{12k}$. Since $\\dim_\\mathbb C(E_{12k})=3$, at least three of these one-dimensional eigenspaces are associated to the same eigenvalue. \nIt implies that at least two elements of $\\{\\lambda_1,\\lambda_2,\\lambda_k\\}$ are equal, which contradicts ($\\star$).\n\nFinally, we just need to settle the situation where $v_k$ belongs to the orthogonal complement of $E_{12}$ for every $k\\neq 1,2$.\nLet $N_{12}$ be the orthogonal complement of $E_{12}$ in $\\mathbb C^{p,q}$. Then, as before, since $H_{p,q}|_{E_{12}}$ is non-degenerate, we have $\\mathbb C^{p,q}=E_{12}\\oplus N_{12}$ and hence the restriction of $H_{p,q}$ on $N_{12}$ is of signature $(p-1,q-1)$. As argued previously, $N_{12}$ is an invariant subspace of $A$ by Lemma~\\ref{semipositive}.\nFurthermore, $[N_{12}]\\cap D_{p,q}\\cong D_{p-1,q-1}$, where $[N_{12}]$ denotes the projectivization of $N_{12}$ and for every $k\\neq 1,2$, we have $[v_k]\\in [N_{12}]\\cap\\partial D_{p,q}\\cong\\partial D_{p-1,q-1}$. That is, we have $s-2$ fixed points on $\\partial D_{p-1,q-1}$ and the desired result now follows by induction together with Theorem \\ref{ball}, which serves as the initial step for the induction. The proof is complete.\n\\end{proof}\n\n\n\n\n\\subsection{Real generalized balls}\n\n\nLet $D_{p,q}^\\mathbb R$ be the subspace of $D_{p,q}$\ndefined by \n$$\nD_{p,q}^\\mathbb R = \\left\\{ [x_1,\\ldots, x_{p+q}]\\in D_{p,q} : x_i\/x_j\\in \\mathbb R \\, \\text{ for all } i,j \\textrm{ whenever } x_j\\neq 0\\,\\right\\}.\n$$\n\nThus, for a point in $D_{p,q}^\\mathbb R$, we can choose homogeneous coordinates which are all real. We call $D_{p,q}^\\mathbb R$ a \\textit{real generalized ball}.\nWhen $p=1$, Cowen-MacCluer \\cite[Theorem 10]{CM} proved that any linear fractional map with real coefficients\nmaps $D_{1,q}^\\mathbb R$ into itself if and only if it maps $D_{1,q}$ into itself. We now show that the same holds true for any $p\\geq 1$.\n\n\n\\begin{thm}\\label{real generalized ball}\nLet $M \\in M_{p+q}$ be a matrix with real entries.\nThen $M$ defines a linear self map of $D_{p,q}$ if and only if it defines a linear self map of $D_{p,q}^\\mathbb R$ in the same manner.\n\\end{thm}\n\\begin{proof}\nSuppose that $M$ defines a linear self map of $D_{p,q}$.\nSince the entries of $M$ are all real, $D_{p,q}^\\mathbb R$ is mapped into $D_{p,q}^\\mathbb R$.\n\nSuppose that $M$ defines a linear self map of $D_{p,q}^\\mathbb R$. For $[z]\\in D_{p,q}$, write $z = x+iy$ with\n$x=(x', x'')\\in\\mathbb R^{p+q}$, $y=(y', y'')\\in\\mathbb R^{p+q}$\nfor $x'$, $y'\\in \\mathbb R^p$, $x''$, $y''\\in \\mathbb R^q$.\nNote that $|x'|^2 + |y'|^2 > |x''|^2 + |y''|^2$.\nIf $|x'|^2 >|x''|^2$ and $|y'|^2 >|y''|^2$, then $Mz = Mx+iMy$ belongs to $D_{p,q}$.\nIf otherwise, without loss of generality, we may assume that\n$|x'|^2 \\leq|x''|^2$ and $|y'|^2 > |y''|^2$.\nNow, for any $\\theta\\in\\mathbb R$, we have $[e^{i\\theta}z]=[z]$. And since $e^{i\\theta}z = x \\cos\\theta - y \\sin\\theta + i(y\\cos\\theta + x\\sin\\theta) $ and the expression\n\\begin{equation}\\label{real part}\n|x'\\cos\\theta - y'\\sin\\theta|^2 - |x'' \\cos\\theta - y'' \\sin\\theta|^2\n\\end{equation}\nequals $|x'|^2 - |x''|^2\\leq 0$ when $\\theta=0$ and\n$|y'|^2 - |y''|^2> 0$ when $\\theta=\\pi\/2$, by continuity there exists\n$\\theta$ such that equation \\eqref{real part} becomes zero.\nThis implies that for $[z]\\in D_{p,q}$, we can always choose homogeneous coordinates $z=x+iy$ such that $|x'|^2 = |x''|^2$ and $|y'|^2 > |y''|^2$. Here the last inequality is due to the fact that $e^{i\\theta }z\\in D_{p,q}$.\nHence we have $Mz\\in D_{p,q}$ and in particular $M$ induces a linear self map of $D_{p,q}$.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Examples}\\label{examples}\n\nIn this section, we collect a number of examples that the reader have been referred to from various places in the article.\n\n\\begin{example}\\label{no extension}\nLet $F$ be the rational map on $\\mathbb P^3$ defined by $F[z_1,z_2,z_3,z_4]=[z_1+z_3, z_4, 0, 0]$. Then the indeterminacy of $F$ is the projective subspace spanned by $[1,0,-1,0]$ and $[0,0,0,1]$. As elements in $\\mathbb C^{2,2}$, these two vectors spanned a negative semi-definite 2-dimensional subspace and hence we see that the set of indeterminacy of $F$ intersects $\\partial D_{2,2}$ at $[1,0,-1,0]$ but lies outside $D_{2,2}$. Thus, $F$ is a holomorphic on $D_{2,2}$ but it cannot extend across the boundary point $[1,0,-1,0]$. It is a minimal linear self map of $D_{2,2}$ because the range of $F$ (as a linear map on $\\mathbb C^{2,2}$) is positive definite and of dimension 2. \n\n\\end{example}\n\n\\begin{example}\\label{hs example}\nLet $$A = \\left(\n \\begin{array}{cc}\n \\sqrt{2}I_2& I_2 \\\\\n I_2 & \\sqrt{2}I_2\\\\\n\n \\end{array}\n \\right)\\in U(2,2).$$\n$A$ induces an automorphism of $D_{2,2}$. The characteristic polynomial of $A$ is $(x-\\sqrt{2}+1)^2(x-\\sqrt{2}-1)^2$ and $A$ has two eigenvalues $\\sqrt{2} \\pm 1$. \nThe eigenspace of the eigenvalue $\\sqrt{2}-1$ is spanned by $v_1=(1,0,-1,0)^t$ and $v_2 = (0,1,0,-1)^t$ and that of $\\sqrt{2}+1$ is spanned by $v_3=(1,0,1,0)^t$ and $v_4=(0,1,0,1)^t$. One also sees immediately that both eigenspaces are isotropic in $\\mathbb C^{2,2}$ and thus their projectivizations lie inside $\\partial D_{2,2}$. This implies that $A$ has infinitely many fixed points on the boundary but no fixed point in $D_{2,2}$. \nThis example can be generalized to $D_{p,p}$ in a straightforward way.\n\\end{example}\n\n\\begin{example}\\label{sharp example 3}\nLet\t$$A = \\left(\n \\begin{array}{cccc}\n 1& 0 & 0& 0 \\\\\n 0 & -1 & 0 & 0\\\\\n 0 & 0 & i & 0\\\\\n 0 &0&0& -i\\\\\n \\end{array}\n \\right)\\in U(2,2).$$\nTrivially $A$ has four different eigenvalues and two of them correspond to fixed points in $D_{2,2}$\nand the other two of them corresponds to fixed points in $\\mathbb P^3 \\setminus \\overline{D_{2,2}}$. There is no projective line on which every point is fixed.\n\\end{example}\n\n\\begin{example}\\label{sharp example 2}\nLet\t$$A = \\left(\n \\begin{array}{cccc}\n 1& 1 & 1& 0 \\\\\n 1 & -1 & 0 & 1\\\\\n 1 & 0 & 1 & 1\\\\\n 0 &1&1& -1\\\\\n \\end{array}\n \\right)\\in U(2,2).$$\n\nThe matrix $A$ has four different eigenvalues. The eigenvalues and the corresponding eigenvectors are the following:\n\\begin{equation}\n\\begin{array}{ll}\n\\lambda_1=1+\\sqrt{2}, & \\alpha_1 =(\\frac{1}{4}+\\frac{\\sqrt{2}}{8},\\frac{\\sqrt{2}}{8},\\frac{1}{4}+\\frac{\\sqrt{2}}{8},\\frac{\\sqrt{2}}{8})^t \\\\\n\\lambda_2=-1+\\sqrt{2},& \\alpha_2=(\\frac{1}{4}+\\frac{\\sqrt{2}}{8},\\frac{\\sqrt{2}}{8},-\\frac{1}{4}-\\frac{\\sqrt{2}}{8},-\\frac{\\sqrt{2}}{8})^t \\\\\n\\lambda_3=1-\\sqrt{2},& \\alpha_3=(\\frac{1}{4}-\\frac{\\sqrt{2}}{8},-\\frac{\\sqrt{2}}{8}, \\frac{1}{4}-\\frac{\\sqrt{2}}{8},-\\frac{\\sqrt{2}}{8})^t \\\\\n\\lambda_4=-1-\\sqrt{2},& \\alpha_4=(\\frac{1}{4}-\\frac{\\sqrt{2}}{8},-\\frac{\\sqrt{2}}{8},-\\frac{1}{4}+\\frac{\\sqrt{2}}{8},\\frac{\\sqrt{2}}{8})^t \\\\\n\\end{array}\n\\end{equation}\nThe automorphism of $D_{2,2}$ induced from $A$ has four fixed points on $\\partial D_{2,2}$ but it has no fixed point in $D_{2,2}$. Moreover there is no projective line in $\\partial D_{2,2}$\non which $A$ fixes every point. \n\\end{example}\n\n\n\n\n\n\n\n\n\n\\begin{example} \\label{nondiagonalizable}\nFor any non-zero real number $\\alpha$ let $$A = \\left(\n \\begin{array}{cccc}\n 1& \\alpha & 0& \\alpha \\\\\n -\\alpha & 1 & \\alpha & 0\\\\\n 0 & \\alpha & 1 & \\alpha\\\\\n \\alpha &0&-\\alpha& 1\\\\\n \\end{array}\n \\right)\\in U(2,2).$$\nThen the characteristic polynomial of $A$ is $(1-x)^4$ and the minimal polynomial of $A$ is $(1-x)^2$. Thus, in any Jordan canonical form, the Jordan blocks of $A$ are at most of rank 2. By direct computation, one sees that $A$ only has two linearly independent eigenvectors, which are chosen to be $ (1,0,1,0)^t$ and $(0,1,0,-1)^t$, associated to the eigenvalue 1. In particular, there are two Jordan blocks of rank 2.\n\n\n\n\\end{example}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe study of opinion dynamics focuses on decision making process in\nmulti-agent systems. This line of research dates back to 1950s\n\\cite{french1956formal}. Since then, the study of opinion dynamics has\nattracted researchers from diverse areas, and hence various approaches have been\nproposed for modeling of the evolution of opinions \\cite{castellano2009statistical}. \nIt is natural to assume that an agent's decision making process is influenced by the information he\/she receives from the society. The influence of society has been considered in the form of the agent's interactions with his\/her ``neighbors.'' An agent's neighbors can be chosen based on the agent's opinion \\cite{stamoulas2018convergence,blondel2010continuous,hegselmann2002opinion,hendrickx2016symmetric,ceragioli2012continuous} or based on a given communication graph independent of the opinions \\cite{degroot1974reaching,jia2015opinion,friedkin1990social,ceragioli2012continuous,bartolozzi2005stochastic,holyst2000phase,yildiz2013binary,sznajd2000opinion}. \nThe set of possible opinions of a given \nagent at a given time may be regarded as a finite discrete set without any \nadditional structure, the simplest example being a binary set \\cite{bartolozzi2005stochastic,holyst2000phase,yildiz2013binary,sznajd2000opinion,holley1975ergodic,san2005binary,galam2013modeling,zanette2006opinion,mobilia2007role,lyst2002social,palombi2014stochastic}, or alternatively \nas a continuum of values in $\\mathbb{R}$ or $\\mathbb{R}^d$ \\cite{stamoulas2018convergence,blondel2010continuous,hegselmann2002opinion,degroot1974reaching,weisbuch2002meet}. The structure of the $\\mathbb{R}^d$ imposes a concept of nearby opinions \nand leads to the notion of confidence bounds where an agent may only interact \nwith other agents within his\/her confidence bound. See for instance \\cite{hegselmann2002opinion,blondel2010continuous,hendrickx2016symmetric,stamoulas2018convergence,ceragioli2012continuous}. In this paper, we shall \nfocus on the binary opinion case and as such the notion of bounded confidence\ndoes not apply. See \\cite{bartolozzi2005stochastic,holyst2000phase,yildiz2013binary,sznajd2000opinion,holley1975ergodic,san2005binary,galam2013modeling,zanette2006opinion,mobilia2007role,lyst2002social} for binary opinion models.\n\nOne simple way an agent can choose to update his\/her opinion would be by interacting with his\/her neighbor(s) and adopting the opinion of the neighbor or the leading opinion of the group of neighbors\n \\cite{holley1975ergodic,san2005binary,galam2013modeling,zanette2006opinion}. \nIt is important to note that this opinion updating rule assumes that agents conform to the neighbor(s) with whom they are interacting, and hence it does not take agents' personalities into account.\nIn this respect, \\cite{mobilia2007role,yildiz2013binary,galam2013modeling,sznajd2011phase,schneider2004influence,de2005spontaneous} and references therein introduce agents with \\emph{personalities}. \\cite{yildiz2013binary,galam2013modeling} include \\emph{stubborn} agents who do not change their opinion, however, influence other agents. \\emph{Independent} agents who change their opinions independent of the interactions are considered in \\cite{sznajd2011phase}. As a special case of independent agents, \\emph{zealots} are introduced in \\cite{mobilia2007role}. Zealots are independent agents who may favor one opinion. \\cite{galam2013modeling,schneider2004influence,de2005spontaneous} study \\emph{contrarian} agents who adopt the opposite of the leading opinion with a certain probability. The effect of contrarians and independent agents to the group's limiting behavior are discussed in \\cite{nyczka2013anticonformity}. The presence of \\emph{leaders} is considered in \\cite{ellero2013stochastic,lyst2002social,holyst2000phase}. \n\nIt is natural to focus on two possible extreme outcomes:\nbalance of opinions and consensus where all agents agree. In reality, the situation is not ``black\nor white''; for instance, one may find that in the long-term, one opinion is likely\nto be held by say $70\\%$ of the agents resulting in dominance of one opinion. \nThe idea of balance of opinions and dominance of opinions are explored in \\cite{galam2013modeling,mobilia2007role,sznajd2000opinion} in discrete time setting in relation to the personalities of agents. \n\nIn this study, we propose a continuous time binary opinion model (say $0$ or $1$) for a group in which agents\nare considered to have personalities defined by a monotonic function and a\nspontaneity coefficient. An agent's personality determines the effect of\nsocial influence on the agent (e.g., conformists, rebellious). Moreover, in our model, contrary to the pair-wise\ninteraction of agents that are defined to be neighbors, agents are considered\nto be informed on the distribution of opinions in the entire group at each time\n$t$. We investigate various personality traits and their effect on the group's limiting behavior for a large number of agents. \nIn particular, the personality traits that lead to dominance of one opinion is our main interest.\nWe note that, our model can be thought of as the result of a situation \nwhere agents do not change their mind after one (pairwise or group) interaction, but rather after \nseveral interactions. In this case, assuming all agents can interact with all \nothers, in a large population an agent interacts with\nsufficiently many others before changing their mind and the sample from the \nsufficiently many can be taken as a good approximation of sampling the entire\npopulation. \n\nWe also assume (as is done in other models) that there is no ``natural bias'' towards one of the opinions. \nAs an example, if the opinion is about whether the earth is flat or not, one\nwould expect that in an informed society, agents are more likely to believe\nthe truth. We are focused on the long-term probability distribution for \nthe number of agents with opinion $1$ when the number $N$ of total agents is\nvery large. In particular, we study the effects of agents' personalities on balance \nand dominance of the opinions. We found that the shape of the so called conformity function plays an important role.\n\n\nThe paper is organized as follows. In Section \\ref{sec:model} we introduce our\nbinary model and focus on a \\emph{homogeneous} group. In section \\ref{sec:C1}\nwe study the effects of personality of the (homogeneous) group on the group's\nlimiting behavior. We extend our model to \\emph{heterogeneous} groups and\nexamine limiting decision behavior when the group is formed by two extreme\npersonality classes in Section \\ref{sec:heterogeneous}. The concluding remarks follow in Section\n\\ref{sec:conclusions}. \n\n\n\n\n\n\n\\section{\\label{sec:model}The model and the homogeneous case}\n\nWe consider a group of $N$ agents where each agent holds an opinion from the set $\\{0,1\\}$. \nAn agent flips his\/her opinion based on the group's current configuration and his\/her \\emph{personality}. Here, \\emph{personality} of an agent $i$, $i = 1,\\dots,N$, is given by the pair $(\\phi_i, \\beta_i)$ where \n $\\phi_i: [0,1]\\to[0,\\infty)$ is a monotonic function that accounts for {\\em conformity}, $\\beta_i$ is a \nnonnegative quantity that accounts for {\\em spontaneity}. We call a group\n\\emph{homogeneous} if all agents in the group share the same personality,\n$\\phi = \\phi_i, \\beta = \\beta_i \\; \\forall i= 1,\\dots,N$. We first look at the\ncase when the group is homogeneous. \n\nWe define\n$X^N(t)$ to be the number of agents holding opinion 1\n at time $t \\in [0,\\infty)$ and assume that a given agent \nflips his\/her opinion during a time interval $(t,t+h]$ with a \nprobability\n\\begin{equation}\n\\label{eq:opinion-change-rate}\n(\\phi(n\/N) + \\beta) h + o(h) \\;\\; h \\to 0+,\n\\end{equation}\nwhere $n$ is the number of\nagents with opposite opinion and $N$ is the total number of agents at time\n$t$.\nThus we note that $\\phi(x)$ determines the rate of conformity where $x$ is the\nfraction of the population holding the opposite view. We note that the\npairwise interaction model is a special case of our model where $\\phi$ is\nlinear, that is\n$\\phi(x)=\\alpha x$ for some $\\alpha>0$. \n \nThis results in a Markov process model for $X^N(t)$ with the state space $\\{0,1,2,\\ldots,N\\}$. \n Moreover, we regard $X^N(t)$ as a birth-death process since when an agent flips his\/her opinion, the\nprocess increases\/decreases by one. \nUsing the rate of opinion change for an agent defined by \\eqref{eq:opinion-change-rate}, \n the birth rate $\\lambda_i^N$ and the death rate $\\mu_i^N$ at the state $X^N(t) = i$ can be written as follows:\n\\begin{eqnarray}\n\\label{eq:jump-up}\n\\lambda_i^N\n&=& \\left(\\phi \\left(\\frac{i}{N} \\right) + \\beta \\right)(N-i)\n= (N-i) \\phi \\left(\\frac{i}{N} \\right)+\\beta (N-i), \\nonumber \\\\\n\\mu_i^N \n\\label{eq:jump-down}\n&=& \\left( \\phi \\left(\\frac{N-i}{N} \\right) + \\beta \\right) i\n= i \\phi \\left( \\frac{N-i}{N} \\right) + \\beta i.\n\\end{eqnarray}\nSince the state space $\\{0,1,2,\\ldots,N\\}$ is finite, $\\lambda_N^N=0$ and $\\mu_0^N=0$. \nUsing these transition rates one can construct a transition rate matrix\n$Q^N=[q_{ij}^N]$, where $q_{i( i+1)}^N = \\lambda_i^N$, $q_{i( i-1)}^N = \\mu_i^N$, $q^N_{ii}= - \\sum_{j\\ne i} q^N_{ij}$\nand $q_{ij}^N = 0 \\; \\forall j \\notin \\{i-1,i, i+1\\}$. \n\nWe may rewrite the birth rate $ \\lambda_i^N $ and the death rate $\\mu_i^N$ as follows:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:bd-rates}\n \\lambda_i^N &= N \\bar{\\lambda} \\left(\\frac{i}{N} \\right), \\quad \\\\\n \\mu_i^N &= N\\bar{\\mu}\\left(\\frac{i}{N} \\right) \\quad i =0, 1,\\dots, N,\n\\end{aligned}\n\\end{equation}\nwhere $\\bar{\\lambda},\\bar{\\mu}:[0,1] \\rightarrow [0,\\infty)$ are\n\n given by\n\\begin{equation}\n\\begin{aligned}\n\\label{lamda_mu_bar}\n\t\\bar{\\lambda}(x) =& (1-x) \\phi(x)+ \\beta (1-x), \\\\\n\t\\bar{\\mu}(x) =& x \\phi(1-x)+ \\beta x.\n\t\\end{aligned}\n\\end{equation} \nWe define the probability $p^N(t)=(p^N_0(t), p^N_1(t), \\ldots, p^N_N(t) )$,\nwhere $p^N_i(t)=\\mathbb{P}[X^N(t)=i]$ for each $i=0,1,\\ldots,N$. The\nprobability mass function satisfies the Kolmogorov's forward equation \n\\begin{equation}\n\\label{eq:p-derivative}\n\\dot{p}^N_j(t) = \\sum_k p^N_k(t)q^N_{kj}.\n\\end{equation}\nIt should be noted that when \\emph{spontaneity} coefficient $\\beta =0$, \nthe states $i =0$ and $i= N$ are absorbing states since \n$\\lambda_0^N = \\mu_N^N\n= 0$. When $\\beta>0$, the birth-death process $X^N(t)$ is an irreducible\nMarkov process with the finite state space $\\{0,1,2,\\ldots,N\\}$ and thus,\n$X^N(t)$ is ergodic and attains a unique stationary probability distribution\nas $t \\to \\infty.$ The probability vector $p^N(t) \\to \\pi^N =\n(\\pi^N_0,\\pi^N_1,\\ldots,\\pi^N_N)$ as $t\\rightarrow \\infty$ and $\\pi^N$ does\nnot depend on the initial state $X^N(0)$. Using the detailed balance condition at stationarity, one can obtain\n\\begin{equation*}\n \\pi^N_n= \\frac{\\lambda^N_{n-1}\\lambda^N_{n-2}\\ldots\\lambda^N_0}\n {\\mu^N_n \\mu^N_{n-1}\\ldots \\mu^N_1} \\pi_0^N, \\quad n=1,2,\\ldots,N.\n\\end{equation*}\nHence,\n\\begin{eqnarray}\n\\label{eq:pi_n}\n \\pi_n^N &=& \\frac{R^N_n}{\\sum_{k=0}^N R^N_k}, \\quad n=0,1,\\ldots,N. \\\\\n\\label{eq:pi_0}\n \\pi_0^N &=& \\frac{1}{\\sum_{k=0}^N R^N_k}, \n\\end{eqnarray}\nwhere \n\\begin{equation}\n\\label{eq:R_n^N}\nR_n^N = r^N_n r^N_{n-1}\\ldots r^N_1, \\quad n = 1,2,\\ldots,N\n\\end{equation}\n for $r_n^N = \\frac{\\lambda^N_{n-1}}{\\mu^N_n}$ and $R^N_0 = 1$.\n \n\n\\section{\\label{sec:C1}The effects of the conformity function : The homogeneous case }\n\nIn order to study $X^N(t)$ for large $N$ and $t$, we shall consider the normalized process $X_N(t) = \\frac{X^N(t)}{N}$. As $N \\to \\infty$, in the fluid limit, one expects $X_N$ to converge to $x$ \nwhere $x$ satisfies the ODE \n\\begin{equation}\n\\label{eq:binary-ode-gen}\n\\dot{x}(t)= F(x(t)) = \\bar{\\lambda}\\big(x(t)\\big) - \\bar{\\mu}\\big(x(t)\\big),\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:vectorfield-homo}\nF (x) = \\phi\\left(x\\right) \\left( 1-x \\right) - x \\phi\\left(1-x \\right) +\n \\beta (1 - 2 x).\n\\end{equation}\nIntuitively, when $N$ and $t$ are both large, one expects the peaks of the probability \ndistribution of $X_N(t)$ to occur near the stable equilibria of this ODE. \nThis observation will motivate the rest of the analysis in this paper. \n\nWhile we do not make new claims about rigorous limits as $t \\to \\infty$ and $N \\to \\infty$ jointly, some rigorous limits exist in literature that we \nmention here. A major result is that if $F$ is $C^1$ (continuously differentiable), then \ngiven any finite time interval $[0,T]$, as $N \\to \\infty$ $X_N \\to x$ uniformly \non $[0,T]$ with probability one, and moreover a diffusion approximation\nfor $X_N$ is also available \\cite{ethier2009markov}. Since this result\nonly considers the limit as $N \\to \\infty$ over finite intervals of time, \none needs to be cautious in interpreting the large $N$ and large $t$\napproximation. In particular, if one fixes any large final time $t$, and considers \nincreasing $N$, then one expects distributions at time $t$ to have peaks\naround the stable equilibria of the ODE. \nWhen $F$ has a unique globally attractive \nequilibrium $\\overline{x}$, under suitable conditions, as $N \\to \\infty$ one can\nrigorously justify a Gaussian approximation with mean $\\overline{x}$ for the\nstationary probability distribution (see Theorem 2.7 in \\cite{kurtz1976limit}).\n\nHenceforth, we shall study the system \\eqref{eq:binary-ode-gen} for its\nstable equilibria. We note that $F(0)>0$, $F(1)<0$ and $\\overline{x}=\\frac{1}{2}$ is always an\nequilibrium for the dynamics \\eqref{eq:binary-ode-gen}. If\n$\\overline{x}=\\frac{1}{2}$ is the unique equilibrium, it will be globally\nattractive on $[0,1]$. Then, for very large $N$, the stationary probability\ndistribution will be a narrow Gaussian with mean $\\overline{x}$ and hence, the\nmodel leads to balance of opinions. \n\nWe shall see that the shape of the conformity function $\\phi$ plays an\nimportant role in deciding if dominance of an opinion is likely. \nIntuitively, one may expect that greater conformity leads to dominance of one\nopinion while greater spontaneity leads to balance of opinions via a law of\nlarge number effect. However, our examples suggest that the dependence on \n$\\phi$ is more subtle in that the shape of the function plays a crucial role. \nWe see that when $\\phi$ is strictly convex, for sufficiently small $\\beta$ \ndominance is observed. When $\\phi$ is not strictly convex or if it is concave, we do not \nsee dominance in our examples. \n\n{\\bf Remark:} For the sake of precision, we shall use the term {\\em balance} to\nmean the situation where there is only one stable equilibrium of\n\\eqref{eq:binary-ode-gen} which is $\\overline{x}=1\/2$. We shall use the \nterm {\\em dominance} rather loosely to stand for lack of balance. \n\nIn order to investigate the effects of $\\phi$, it makes sense to make some\nnatural assumptions. The most natural \nconditions on $\\phi:[0,1] \\to [0,\\infty)$ are that $\\phi(0)=0$ and $\\phi$ is\n increasing for conformity, and $\\phi(1)=0$ and decreasing for {\\em rebelliousness}\n (opposite of conformity). \n\n \n\\begin{thm}\nConsider conformity function $\\phi(x): [0,1] \\to [0, \\infty)$ such that\n $\\phi'(x)$ strictly increasing on $(0,1)$ and suppose that $\\phi(0)=0$.\nThen for sufficiently small $\\beta$, the equilibrium $\\overline{x}=1\/2$ is unstable and hence the model leads to dominance of one opinion for large $N$ and large $t$. \n\\end{thm} \n\\begin{proof}\nDefine\n$\nG(x) = \\phi(x) (1-x) - x \\phi(1-x).\n$\nThen, the vector field \\eqref{eq:vectorfield-homo} is \n\\[\nF(x) = G(x) + \\beta (1-2x).\n\\]\nWe note that $G(0)=G(1\/2)=G(1)=0$. \nWe note that $F'(1\/2) = G'(1\/2) - 2 \\beta$, and that \n \\[\n \tG'(1\/2) = \\phi'(1\/2) - 2 \\phi(1\/2).\n \\]\nUsing the mean value theorem for $\\phi(x)$ on $ [0,1\/2]$, we conclude that for some $c \\in (0,1\/2)$, \n\\[\n\t\\phi'(c) = \\frac {\\phi(1\/2)-\\phi(0) }{1\/2} = 2 \\phi(1\/2),\n\\]\nsince $\\phi(0) = 0$. Thus,\n\\[\n\tG'(1\/2) = \\phi'(1\/2) -\\phi'(c) >0, \\quad c\\in(0,1\/2)\n\\]\nsince $\\phi '(x)$ is strictly increasing on $(0,1)$. Then $F'(1\/2)=G'(1\/2) - 2 \\beta >0$ for sufficiently small $\\beta$ and thus $1\/2$ is unstable. On the other hand since $F(0)=\\beta>0$, $F(1)=-\\beta<0$ there must be at least two (symmetrically placed) equilibria, one in\n$(0,1\/2)$, \nand the other in $(1\/2,1)$. Moreover, generically, these equilibria \nwill be stable, and hence the model leads to dominance of one opinion for\nlarge $N$ and large $t$. \n\n\\end{proof}\n \n Next, we provide examples with various conformity functions $\\phi$ and investigate the stable\nequilibria to predict dominance or balance. We shall also check the prediction\nfrom the stable equilibria of the ODE model \nagainst computational results of the stationary probability distributions. \nWe note that, there are two methods to compute the stationary\ndistributions. One is to use the formulas \\eqref{eq:pi_n} and \\eqref{eq:pi_0}\nand the other is to use an ODE solver to compute the solution to\n\\eqref{eq:p-derivative}. For very large $N$ values, our numerical experiments\nsuggest that using \\eqref{eq:pi_n} and \\eqref{eq:pi_0} provide more accurate\nresults compared to the ODE solver. Hence, throughout this study, we refer to\n\\eqref{eq:pi_n} and \\eqref{eq:pi_0} to verify our predictions from the \nstable equilibria of \\eqref{eq:binary-ode-gen}.\n\n\n \n\\textbf{Example 1}\nConsider the simplest example of $\\phi(x) =x$. This is convex, but not\nstrictly so. \nIn this case, the rate that an agent changes his\/her opinion is\n$\\frac{n}{N} + \\beta$. \nUsing \\eqref{eq:binary-ode-gen}, we can conclude that as $N$ gets large $X_N(t)$ converges to $x(t)$, where \n\\begin{equation}\n\\label{eq:ode-x}\n\t\\dot{x}(t) = \\beta ( 1 -2 x(t) )\n\\end{equation}\nSince this ODE has a unique equilibrium at $\\overline{x} = \\frac{1}{2}$ that is globally attractive, regardless of spontaneity coefficient $ \\beta>0$, it is expected that the group will reach balance of opinions as can be observed in Fig.~\\ref{fig:exact_gen1_x}. We note that in Fig.~\\ref{fig:exact_gen1_x}(b) the bell shape curve becomes narrower as N gets larger. \n\nIt is thus interesting to note that as long as $\\beta>0$, no matter how small, one\nexpects balance of opinions. \n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width = 8.4cm]{exact_gen1.eps}\n\\label{fig:exact_gen1}\n\\end{center}\n\\quad\n\\begin{center}\n\\includegraphics[width = 8.4cm]{exact_gen1_second.eps} \n\\label{fig:exact_gen1_second}\n\\caption{Exact values of the stationary probabilities calculated using \\eqref{eq:pi_n} and \\eqref{eq:pi_0} for $\\phi(x) = x$ and $N = 1000$. (a) $\\beta = 5$. (b) $\\beta = 0.01$. }\n\\label{fig:exact_gen1_x}\n\\end{center}\n\\end{figure}\n\n\\textbf{Example 2}\nConsider $\\phi(x) = x^2$ which is strictly convex. \nThe limiting ODE is\n\\begin{equation}\n\\label{eq:binary-ode_x2}\n\t\\dot{x}(t) =\\left(1-2x(t) \\right) ( x(t)^2- x(t) + \\beta).\n\\end{equation}\nIt is easy to see that \\eqref{eq:binary-ode_x2} has three possible\nequilibria; $\\overline{x}_1 = \\frac{1}{2}$ and $\\overline{x}_{2,3} =\n\\frac{1}{2} \\pm \\frac{\\sqrt{1 -4 \\beta}}{2 }$. Hence, based on the choice of\nspontaneity coefficient $ \\beta$, different scenarios are expected. When\n$\\beta \\geq \\frac{1}{4}$, the stable equilibrium is $\\overline{x}_1=\\frac{1}{2}$ and the model leads to balance of opinions. On the other hand, when $ \\beta < \\frac{1}{4}$, the stable\nequilibria are $ \\overline{x}_{2,3}$ and the model leads to dominance. In Fig.~\\ref{fig:exact_gen2_x2}(a) one can observe that the model leads to balance of opinions for $\\beta = 5$. On the other hand, when $\\beta =0.2$, as can be seen in Fig.~\\ref{fig:exact_gen2_x2}(b), the model leads to dominance of one opinion. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width = 8.4cm]{exact_gen2.eps}\n\\end{center}\n\\quad\n\\begin{center}\n\\includegraphics[width = 8.4cm]{exact_gen2_second.eps} \n\\caption{ Exact values of the stationary probabilities calculated using \\eqref{eq:pi_n} and \\eqref{eq:pi_0} for $\\phi(x) = x^2$ and $N = 1000$. (a) $\\beta = 5 $. (b) $\\beta = 0.2$. }\n\\label{fig:exact_gen2_x2}\n\\end{center}\n\\end{figure}\n\n\\textbf{Example 3}\nWe consider $\\phi(x) = 1-x^2$ to explore the impact of rebelliousness in our model. In this case, the rate an agent changes his\/her opinion is $ \\frac{N^2-n^2}{N^2} + \\beta$. Hence, as the number of agents with the opposite opinion increases, the rate of opinion change decreases. \nThe limiting ODE for this model is\n\\begin{equation}\n\\label{eq:binary-ode_(1-x2)}\n\t\\dot{x}(t) =\\big( x(t)^2 - x(t) - 1 - \\beta \\big) \\big( 2x(t)-1 \\big).\n\\end{equation}\nOne can see that \\eqref{eq:binary-ode_(1-x2)} has a unique equilibrium at\n$\\overline{x} = \\frac{1}{2}$ regardless of the spontaneity coefficient $\\beta\n> 0$. Thus, we can conclude that the model leads to balance of opinions as \ncan also be observed in Fig.~\\ref{fig:exact_gen_(1-x2)}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width = 8.4cm]{exact500_1-x2.eps}\n\\label{fig:exact500_(1-x2)}\n\\end{center}\n\\quad\n\\begin{center}\n\\includegraphics[width = 8.4cm]{exact500_1-x2_second.eps} \n\\label{fig:exact500_(1-x2)_second}\n\\caption{ Exact values of the stationary probabilities calculated using \\eqref{eq:pi_n} for $\\phi(x) = 1-x^2$ and and $N = 500$. (a) $\\beta = 7$. (b) $\\beta = 0.001$.}\n\\label{fig:exact_gen_(1-x2)}\n\\end{center}\n\\end{figure}\n\n\\textbf{Example 4}\n Let $\\phi(x) = \\sqrt{x}$ which is concave. Then \nthe corresponding ODE is \n\\begin{equation}\n\\label{eq:binary-ode_(sqrtx)}\n \\dot{x}(t) = (1-2x(t)) \\left( \\frac{ \\sqrt{x(t)}\\sqrt{1-x(t)} }{ \\sqrt{1-x(t)}+\\sqrt{x(t)} } + \\beta \\right) .\n\\end{equation}\nWe first observe that $F$ is not $C^1$ on $[0,1]$ and the fluid limit theorem \ndoes not apply. Nevertheless, we proceed heuristically to look for the stable equilibria. \nOne can observe that $\\overline{x} = 1\/2$ is the only equilibrium for \\eqref{eq:binary-ode_(sqrtx)}.\nThus, one may expect that the model will lead to balance of opinions regardless of the choice of the spontaneity coefficient $\\beta$ as can be observed in Fig.~\\ref{fig:exact_gen_sqrt}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width = 8.4cm]{exact_gen1_sqrt.eps}\n\\end{center}\n\\quad\n\\begin{center}\n\\includegraphics[width = 8.4cm]{exact_gen2_sqrt.eps} \n\\caption{ Exact values of the stationary probabilities calculated using \\eqref{eq:pi_n} and \\eqref{eq:pi_0} for $\\phi(x) = \\sqrt{x}$ and $N = 1000$. (a) $\\beta = 10 $. (b) $\\beta = 0.1$. }\n\\label{fig:exact_gen_sqrt}\n\\end{center}\n\\end{figure}\n\n\\textbf{Example 5} Consider the monotone increasing, convex conformity\nfunction $\\phi(x) = \\frac{x}{1-x}$, so that the conformity function is the\nratio of the fraction of agents with opposite opinion to those with the same opinion. Then, the rate of change of one's opinion is \n$n\/(N-n) +\\beta$.\nWe note that $\\phi(x)$ has a singularity at $x=1$ and that \\eqref{eq:bd-rates} does not hold for $i = 0,N$, \nand hence the fluid limit theorem does not hold. Nevertheless, we proceed \nheuristically to \nconsider $F$ in \\eqref{eq:vectorfield-homo} which is given by\n\\[\nF(x) = (\\beta -1)(1-2x),\n\\]\nwhich has only one equilibrium $\\overline{x}=1\/2$ and it is (asymptotically) stable if and only\nif $\\beta >1$. Thus we expect balance for $\\beta>1$. When $\\beta <1$, we \nexpect dominance of one opinion. \n\nThis heuristic is verified both by our numerical computation of the stationary\nprobabilities as well as the asymptotic formulas for the stationary\nprobabilities \\eqref{eq:pi_n} and \\eqref{eq:pi_0} that we derive in Appendix \\ref{sec:asymptotic approximations}.\n\nIn Fig.~\\ref{fig:figure-ab}, we present an example for $\\beta <1$ case. In this experiment we use $\\beta = 0.2$ and $N=100$. Fig.~\\ref{fig:figure-ab}. shows the asymptotic approximation $\\tilde{\\pi}^N =(\\tilde{\\pi}^N_1, \\tilde{\\pi}^N_2,\\ldots,\\tilde{\\pi}^N_N)$ calculated in \\eqref{eq:asym_pi_n}\nAs shown in Fig.~\\ref{fig:figure-ab} our model leads to dominance of one opinion. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width = 8.4cm]{asymp.eps} \n\\label{fig:asymp_pi}\n\\caption{ Asymptotic approximation $\\tilde{\\pi}^N$ when $\\phi(x) = \\frac{x}{1-x}$, $\\beta = 0.2$ and $N=100$. }\n\\label{fig:figure-ab}\n\\end{center}\n\\end{figure}\nOn the other hand, an example of $\\beta>1$ case is shown in Fig.~\\ref{fig:figure-ab_second} and it displays\nthe approximations for stationary probabilities, $\\tilde{\\pi}^N$,\ncalculated using \\eqref {eq:asym_pi_n_second}.\nAs seen in Fig.~\\ref{fig:figure-ab_second} the model leads to balance of opinions.\n \\begin{figure}\n\\begin{center}\n\\includegraphics[width = 8.4cm]{asymp100_second.eps} \n\\caption{Asymptotic approximation for $\\tilde{\\pi}^N$ when $\\phi(x) = \\frac{x}{1-x}$, $\\beta = 10$ and $N=100$.}\n\\label{fig:figure-ab_second}\n\\end{center}\n\\end{figure}\n\\section{\\label{sec:heterogeneous}Heterogeneous binary opinion dynamics}\nLet us consider the case where the group is \\emph{heterogeneous}. Namely, suppose we have $m$ personality classes of agents such that all agents \nwithin a class $i$ (where $i=1,\\dots, m$) have the same personality\n$(\\phi_i,\\beta_i)$, but personalities differ among the classes. \nThis results in a Markov process model where the state is \na vector $x=(x_1,\\dots, x_m)$ where $0 \\leq x_i \\leq N_i$ is the\nnumber of class $i$ agents who hold opinion $1$ with $N_i$ being the \ntotal number of class $i$ agents. We assume that the personalities of agents \nis fixed in time, thus $N_i$ is a constant for each $i$ and $N=N_1+\\dots+N_m$\nis the total number of all agents. We note that during a time interval\n$(t,t+h]$ an agent from class $i$ will flip with probability\n\\[\n(\\phi_i(n\/N) + \\beta_i)h + o(h) \\;\\; h \\to 0+,\n\\]\nwhere $n$ is the total number of all agents who have the opposite opinion \nto that of the given agent. We shall be concerned with the case of large $N$ \nwith the fractions $k_i=N_i\/N$ within classes being held constant. \n\nThis results in a family of Markov process $X^N(t)$ which undergo\na jump $e_i$ or $-e_i$ for $i=1,\\dots,m$ (here $e_i \\in \\mathbb{R}^m$ is the vector\nwith $i$th component equal to one and all others equal to zero) with corresponding class $i$ \nbirth and death rates given by\n\\begin{equation}\\label{eq-hetero-lambdamu}\n\\begin{aligned}\n\\lambda^N_i(x) &= \\phi_i\\left(\\frac{|x|}{N}\\right) (N_i-x_i) + \\beta_i (N_i - x_i),\\\\ \n\\mu^N_i(x) &= \\phi_i\\left(1-\\frac{|x|}{N}\\right) x_i + \\beta_i x_i, \n\\end{aligned}\n\\end{equation}\nwhere given the state $x=(x_1,\\dots,x_m)$ we denote by $|x|$ the total \nnumber of agents holding opinion $1$:\n\\[\n|x| = \\sum_{i=1}^m x_i.\n\\]\nWe note that $0 \\leq x_i \\leq N_i$. We shall consider the normalized process $X_N(t)=X^N(t)\/N$, \nwhere $X_{N,i}(t)$ is the fraction of class $i$ agents with opinion $1$ where the fraction is\nnormalized by $N$ and not $N_i$. We may write\n\\[\n\\begin{aligned}\n\\lambda^N_i(x) &= N \\bar{\\lambda}_i\\left( \\frac{x}{N}\\right), \\\\ \n\\mu^N_i(x) &= N \\bar{\\mu}_i \\left( \\frac{x}{N} \\right) \n\\end{aligned}\n\\]\nwhere \n\\begin{equation}\\label{eq-hetero-barlambdamu}\n\\begin{aligned}\n\\bar{\\lambda}_i(x) &= \\phi_i(|x|) (k_i-x_i) + \\beta_i (k_i-x_i),\\\\ \n\\bar{\\mu}_i(x) &= \\phi_i\\left(1-|x|\\right) x_i + \\beta_i x_i, \n\\end{aligned}\n\\end{equation}\nwhere as before $|x|=x_1 + \\dots x_m$.\n\nIn the fluid limit, as $N \\to \\infty$, one expects $X_N$ to converge to $x$ \nwhere $x$ satisfies the ODE \n\\[\n\\dot{x}(t)= F(x(t)),\n\\]\nwhere the $m$ dimensional vector field $F$ is given by\n\\[\nF_i(x) = \\bar{\\lambda}_i(x)-\\bar{\\mu}_i(x), \\quad i=1,\\dots,m,\n\\]\nwhich simplifies to\n\\begin{equation}\\label{eq_hetero_F}\nF_i(x) = \\beta_i (k_i - 2 x_i) + \\phi(|x|) (k_i-x_i) - \\phi(1-|x|) x_i\n\\end{equation}\nfor $ i=1,\\dots,m.$ When $N$ and $t$ are both large, we expect to see the peaks of the probability \ndistribution of $X_N(t)$ to occur near the stable equilibria of this ODE. \nWe note that $\\bar{x}=(k_1\/2,\\dots,k_m\/2)$ is always an equilibrium. \n\n{\\bf Example with two extreme personality classes}\n\nWe consider the case of two extreme personality classes $(\\phi_i,\\beta_i)$ \nfor $i=1,2$ where \n\\begin{equation}\\label{eq_two_types}\n\\begin{aligned}\n\\phi_1(\\xi) &= \\phi(\\xi), & \\beta_1&=0,\\\\\n\\phi_2(\\xi) &= 0, & \\beta_2&=\\beta>0,\\\\ \n\\end{aligned}\n\\end{equation}\nwhere $\\phi: [0,1] \\to \\mathbb{R}$ is a monotonic function.\nWe note that class $1$ corresponds to total conformity and class $2$ \ncorresponds to total spontaneity. Let us write $k_1= 2k$ (thus $k$ is half the\nfraction of class $1$) and thus $k_2=1- 2k$. This results in \n\\begin{equation}\\label{eq:vectorfields}\n\\begin{aligned}\nF_1(x) &= \\phi(x_1+x_2)(2k-x_1) - \\phi(1-x_1-x_2) x_1,\\\\\nF_2(x) &= \\beta (1-2k - 2 x_2).\n\\end{aligned}\n\\end{equation}\nAt an equilibrium, clearly $x_2=\\frac{1}{2}-k$ and $ \\bar{x}=(k,1\/2-k)$ is always an\nequilibrium. Additional equilibria are found by solving the equation\n\\begin{equation}\\label{eq:eq-x1bar}\nF_1(x) = \\phi(x_1 + \\frac{1}{2}-k) (2k-x_1) - \\phi(\\frac{1}{2}-x_1+k) x_1\n\\end{equation}\nfor $x_1$. We note that class 2 (spontaneous class) is always expected to reach a balance since at an equilibrium $x_2 = \\frac{1-2k}{2}=\\frac{k_2}{2}$. \n\n\\begin{thm}\nConsider the group with two extreme personality classes \\eqref{eq_two_types} such that the derivative of conformity function, $\\phi'(x)$, is strictly increasing on $(0,1)$ and the fraction of the class of conformists $2k > \\frac{2\\phi(1\/2)}{ \\phi'(1\/2)}$. Then the equilibrium $\\bar{x}= (k,1\/2-k)$ is unstable and hence the model \\eqref{eq_two_types} leads to dominance of one opinion for large $N$ and large $t$. \n\\end{thm} \n\\begin{proof}\nNote that it is sufficient to study the limiting behavior of the class of conformists to conclude the whole group's behavior for large $N$ and $t$. \nNext, we analyze the stability of the equilibrium $(k, 1\/2-k)$. The Jacobian at the equilibrium,\n\\[\n\tJ(1\/2-k, k) = \n \\begin{bmatrix}\n \\frac{\\partial F_1}{\\partial x_1} & \\frac{\\partial F_1}{\\partial x_2} \\\\\n \\frac{\\partial F_2}{\\partial x_1} & \\frac{\\partial F_2}{\\partial x_2}.\n \\end{bmatrix}.\n\\]\nSince $\\frac{\\partial F_2}{\\partial x_1} =0$, eigenvalues at the equilibrium are \n\\[\n\\epsilon_1 = \\frac{\\partial F_1}{\\partial x_1}, \\quad \\epsilon_2 = \\frac{\\partial F_2}{\\partial x_2} = -2\\beta<0,\n\\]\nwhere\n\\[\n\\epsilon_1 =2k \\phi'(1\/2) -2 \\phi(1\/2).\n\\]\nSolving for $k$ to study the sign of $\\epsilon_1$, we conclude that when $2k < \\frac{2\\phi(1\/2)}{ \\phi'(1\/2)}$, $\\epsilon_1 <0$ and hence the equilibrium $(k,1\/2-k)$ is stable leading to balance for the conformists and thus for the entire group.\n\nHowever, when $2k > \\frac{2\\phi(1\/2)}{ \\phi'(1\/2)}$, the equilibrium $(k, 1\/2-k)$ is unstable. On the other hand, since $ 0 \\leq x_1+x_2 \\leq 1$, at an equilibrium $0 \\leq x_1 \\leq \\frac{1}{2}+k$ where $0 < k < 1\/2$. We note that there are no other equilibrium points except those on the line $x_2=1-2k$ as shown in Fig.~\\ref{fig:phase}.\n\nMoreover, from \\eqref{eq:eq-x1bar} one can conclude that $F_1(0,\n\\frac{1}{2}-k) = 2k\\phi(\\frac{1}{2} - k) >0$, $F_1(\\frac{1}{2}+k,\\frac{1}{2}-k\n) = (k-\\frac{1}{2}) \\phi(1) <0$. Thus, there have to be two other equilibria\non the line $x_2 = \\frac{1}{2}-k$ that are symmetrically placed. For one of\nsuch equilibrium point $ x_1 \\in(0,k)$ and for the other $x_1 \\in (k,\n\\frac{1}{2}+k)$. \nMoreover, generically, these equilibria will be stable and\nconformists as well as the entire group will reach dominance of one opinion for large $N$ and $t$. Table \\ref{Tab:class limits} summarizes the limiting behaviors of the classes with respect to the fraction of class populations.\n\\end{proof}\n\\begin{figure}\\centering\n\\begin{tikzpicture}[\n scale=3,\n axis\/.style={thick, ->, >=stealth'},\n important line\/.style={thick},\n dashed line\/.style={dashed, thin},\n pile\/.style={thick, ->, >=stealth', shorten <=2pt, shorten\n >=2pt},\n every node\/.style={color=black,}\n ]\n \n \\draw[axis] (-0.1,0) -- (1.2,0) node(xline)[right]\n {\\tiny$x_1$};\n \\draw[axis] (0,-0.1) -- (0,1.2) node(yline)[above] {\\tiny$x_2$};\n \n \\draw[important line] (0,1) coordinate (A) node[left, text width=0.5em] {\\tiny$1$} -- (1,0)\n coordinate (B) node[below, text width=0.5em] {\\tiny$1$};\n \\draw[important line] (0,0.6) coordinate (C) node[left, text width=2em] {\\tiny$1-2k$}-- (0.4,0.6)\n coordinate (D); \n \\draw[important line] (0.4,0.6) coordinate (D) -- (0.4,0)\n coordinate (E) node[below, text width=0.5em] {\\tiny$2k$}; \n \\fill[gray!20, opacity =1] (C)--(D)--(E)--(0,0)--cycle;\n \\draw[important line] (0,0.3) coordinate (F)node[left, text width=2em]{\\tiny$\\frac{1}{2}-k$} -- (0.8,0.3)\n coordinate (G) node[right, text width=5em]{\\tiny$x_2=\\frac{1}{2}-k$};\n \\filldraw[black] (0.2,0.3) circle (0.3pt);\n \\draw [shorten <=0pt, dashed] (0.2,0.3) coordinate (H)-- (0.2,0) node[below, text width = 0.5em]{\\tiny$k$};\n \\draw[->] (0.3,0.4)--(0.3,0.35); \n \\draw[->] (0.2,0.4)--(0.2,0.35); \n \\draw[->] (0.1,0.4)--(0.1,0.35);\n \\draw[->] (0.3,0.2)--(0.3,0.25); \n \n \\draw[->] (0.1,0.2)--(0.1,0.25);\n \\end{tikzpicture}\n \\caption{ Phase portrait of the system $\\dot{X}= F$ with \\eqref{eq:vectorfields}. The equilibria is on the line $x_2=\\frac{1}{2}-k$ and $(k, \\frac{1}{2}-k)$ is always an equilibrium. Values of $x_2$ evolves towards the line $x_2 = \\frac{1}{2} - k$ for any initial condition and regardless of $x_1$ values.}\n \\label{fig:phase}\n\\end{figure}\n\n\n\\begin{table}[H]\n\\begin{center}\n\\caption{Limiting behavior of group classes with respect to the fraction of conformists, 2k }\n\\begin{tabular}{ l | p{2.4cm} | p{2.4cm}}\n & $2k < \\frac{2\\phi(1\/2)}{ \\phi'(1\/2)}$ & $2k > \\frac{2\\phi(1\/2)}{ \\phi'(1\/2)}$ \\\\\n\\hline \nClass 1 (conformists)& balance & dominance \\\\\n\\hline\nClass 2 (spontaneous) & balance & balance \\\\\n\\hline\n\\textbf{Whole group} & \\textbf{balance} & \\textbf{dominance}\\\\\n\\hline\n \\end{tabular}\n \\label{Tab:class limits}\n \\end{center}\n\\end{table}\n\\textbf{Example 1} \nConsider $\\phi(x) = x$. The corresponding ODE has a unique equilibrium $(k, 1\/2-k)$ and it is stable ($\\epsilon_1 = 2k - 1$, $\\epsilon_2 = -2\\beta$). Thus regardless of the\nchoice of $\\beta$ and the fraction of the class of conformists, $2k$, this model leads to balance in both classes. Therefore, the whole group reaches balance. \n\n\\textbf{Example 2} Let $\\phi(x) = x^2$. In this case, when conformists are less than $50\\% $ of the population, $2k < \\frac{1}{2}$, the corresponding ODE has only one equilibria $(k, 1\/2-k)$ and it is stable $( \\epsilon_1 = 2k-1\/2)$. Thus, conformists reach balance as well as the spontaneous class, and hence the entire group reaches balance for large $N$ and large $t$. On the other hand, when conformists are more than $ 50\\% $ of the population, $2k >\\frac{1}{2}$, the equilibrium $(k, 1\/2-k)$ is unstable and the class of conformists reaches to dominance of one opinion. In fact, the other two equilibria are $(k \\pm \\frac{\\sqrt{4k-1}}{2}, 1\/2-k )$ and \n the stability analysis suggests that these equilibrium points are stable. Hence, the model leads to dominance for the whole group. One Example is given in Fig.~\\ref{fig:TwoTypes} where probabilities are computed using Monte Carlo simulations of 10.000 trajectories. Here, the total number of agents is $N = 120$ with $N_1=100$ ($2k = 5\/6$) being the population of the conformists where $\\phi(x) = x^2 $ and spontaneity coefficient $\\beta = 0.02$.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width =8.4cm]{TwoTypes.eps} \n\\caption{Empirical probability mass function for the number of agents with opinion 1 at time $T=100$ for the case of two extreme classes with $\\phi(x)=x^2$ and $\\beta = 0.02$.}\n\\label{fig:TwoTypes}\n\\end{center}\n\\end{figure}\n\\section{Conclusions}\n\\label{sec:conclusions}\nWe have proposed a simple binary model where agents hold an opinion from the\nset $\\{0,1\\}$ at any time $t \\geq 0$. An agent flips his\/her opinion based on\nthe number of agents with opinion $1$ in the entire population. The influence\nof the group on an agent is determined by his\/her personality. A personality\nis formed by a conformity function $\\phi$ and a spontaneity coefficient $\\beta\n$. When all agents in the group share the same personality, we call the group\nhomogeneous. \n\nInitially, focusing on a homogeneous group, we analyzed the\nlong time probabilities for a large population size for different personality characteristics of the group. The question\nof what personality characteristics lead to dominance of one opinion was\nstudied. We found that the shape of the conformity function, namely strict\nconvexity or lack thereof, seems to be an important determining factor in \nwhether dominance of one opinion occurs for sufficiently small $\\beta$. \n\nWe extended our model to a heterogeneous group, where the group consists of\ndifferent personality classes. In particular, when the group is formed by two\nextreme classes, complete conformity and complete spontaneity, the dominance\nof group opinion is analyzed. In this\nexample, we found that the fraction of the pure conformists was a key \ndetermining factor of dominance along \nwith the strict convexity of $\\phi$. \n\n\n\\bibliographystyle{plain} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe propagation of electromagnetic waves along periodic arrays of sub-wavelength apertures has been the subject of many studies in the last two decades. These structures have various applications, including sensors \\cite{yahiaoui2016terahertz,gordon2008new,dhawan2008plasmonic}, filters \\cite{panwar2017progress}, high refractive index metamaterials \\cite{shen2005mechanism}, negative refractive index metamaterials \\cite{torres2015accurate}, superlenses \\cite{huang2018superfocusing}, etc. Along with these novel applications, observation of the physical phenomena such as extraordinary transmission (EoT) \\cite{ebbesen1998extraordinary} and broadband Brewster transmission \\cite{edalatipour2015physics} has increased the importance of these structures.\n\nIn recent years, understanding and justifying the physical behaviors of the array of holes has been researched using various analytical methods \\cite{de2007colloquium,martin2001theory}. For instance, the extraordinary transmission at optical frequencies may be understood by coupling incident light with the surface plasmon polariton modes supported at the boundary between metals and dielectrics \\cite{liu2008microscopic}. On the other hand, this enhanced transmission at microwave frequencies through perforated PEC structures, has recently been linked to the role of proper complex modes \\cite{ansari2020local}. Providing an analytical method in the form of a circuit model can become a tool for designing and predicting these behaviors. This work focuses on microwave and terahertz frequencies, where the permittivity of the metals becomes high enough to justify perfect electric conductor (PEC) approximation. The transmission line model is considered in this paper due to its appropriate accuracy and high flexibility, which can be easily generalized to multilayer structures.\n\nModeling a waveguide structure in the form of a transmission line has been researched for a long time and has numerous applications in obtaining scattering parameters \\cite{marcuvitz1951waveguide,pozar2011microwave}. The transmission line model is used in a wide frequency range from microwave to infrared \\cite{costa2014overview,zhao2011homogenization,khavasi2014analytical}. With the help of the transmission line, the array of apertures in a PEC film has already been investigated \\cite{molero2021cross,borgese2020simple,kaipa2010circuit,medina2008extraordinary}.\nIn the first developed circuit model, the array of holes was analyzed by considering the thickness of the PEC film. In this case, the circuit model was created by considering only the dominant mode inside the apertures \\cite{medina2009extraordinary,khavasi2015corrections}.\n\nIn previous works, a circuit model has been developed for rectangular apertures by considering only $TE_{01}$ as the dominant mode \\cite{khavasi2015corrections,marques2009analytical}. In addition to two-dimensional arrays, one-dimensional arrays of slits have also been investigated by considering the $TEM$ mode as the dominant mode using the circuit model \\cite{medina2009extraordinary,yarmoghaddam2014circuit}. If the thickness of the aperture is large enough that the excited high-order modes inside the hole are damped from the first opening to the second opening of the hole, the model has high accuracy.\nIn these models, if the structure's thickness decreases, the error increases due to high-order modes; additionally, the error increases more rapidly for circular-shape apertures compared to rectangular apertures. Due to the widespread applications of metasurfaces and frequency selective surfaces, we need to create models for thin arrays.\n\nFrequency selective surfaces are thin periodic surfaces that are important both in terms of industrial applications as well as in terms of fundamental science. The array of apertures perforated in a PEC thin film is a kind of FSS, and many powerful circuit models have been proposed for these structures that are suitable for better design and understanding of the behavior of these structures \\cite{mesa2018efficient,alex2021exploring}. In these structures, the circuit model is created with the help of the spatial profile of the transverse electric field at the opening of the aperture. Previous works have provided profiles for rectangular \\cite{alex2021exploring}, cross \\cite{molero2021cross}, and annular holes \\cite{rodriguez2017annular,alex2021exploring}. Initially, the circuit model was developed for a structure consisting of a single layer of the intended FSS \\cite{costa2014overview}; then, the circuit model was improved for the case where the FSS is embedded in a layered medium \\cite{rodriguez2012analytical,rodriguez2015analytical}. Structures with multiple dielectric layers and multiple FSS layers were then examined using circuit models \\cite{molero2021cross,alex2021exploring}, but all of these circuit models were developed regardless of the FSS's thickness. It should be noted that effect of thickness can cause a considerable error; however, the thickness is about one-fiftieth of the structure's period.\n\nIn this paper, an extended circuit model is developed, which is much faster than full-wave simulations and is much more accurate than its predecessors. The circuit model is formulated in such a way that it takes into account high-order modes inside the holes; thus, it has high accuracy in low thickness structures. A closed-form expression is presented for the spatial profile of the transverse electrical field on the circular aperture which means that, unlike previous models, we do not need to extract the field profile from the full-wave simulators. It is shown that the previous electric field profile for the square hole has a large error when the width of the square hole is large (the width of the square is greater than half the period) \\cite{molero2021cross}, and this problem is well solved by presenting the new profile. Unlike square and circular holes, an array of PEC pillars always supports propagating mode that makes interesting properties \\cite{edalatipour2015physics}, and these structures are also analyzed analytically with the help of the proposed circuit model in this work.\n\nThe paper is organized as follows: Section 2 is devoted to describing how the proposed circuit model is developed, and three different cases: single-mode, multi-mode, and modified single-mode circuit models are described. In section 3, the low-thickness structures are investigated. In section 4, the proposed circuit model is extended to multilayered structures; finally, a conclusion is described in section 5.\n\n\\section{The proposed circuit model}\nPeriodic structures can be analyzed similar to other waveguide structures by considering a unit cell as a waveguide; thus, the formulation presented can be extended to any waveguide structure, but we focus only on the square array of sub-wavelength holes. In addition to the previous circuit models, environmental models for the desired structure have also been investigated \\cite{edalatipour2015physics,edalatipour2012creation}. Similarly, the isotropic\/anisotropic environmental model is created only by considering the dominant mode inside the aperture, which means that these methods also have a large error for a thin array of holes. The following formulation is presented by considering all the modes inside the hole.\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=13cm,height=5cm]{structures.pdf}\n\t\\caption{Figure (a) shows the array of perforated square apertures in a PEC film, and (b) shows the array of circular holes. In two structures, the period is P in both directions, the width of the square is W, and the circle's diameter is D. Refractive index of the upper and lower environments are $n_{1}$ and $n_{3}$, respectively, and the holes are filled with \n\t\trefractive index $n_{2}$.}\\label{periodic_array}\n\\end{figure}\n\nFigure \\ref{periodic_array} shows the schematic of the structures which are illuminated by a normal incident plane wave. The upper environment of the structure is region I ($n_{1}$), inside the holes is region II ($n_{2}$), and the lower environment is region III ($n_{3}$). For simplicity's sake, we first assume the structures depicted in Fig. \\ref{periodic_array} is semi-infinite, i.e., $L \\rightarrow \\infty$. Impinging of a plane-wave on a semi-infinite structure excites aperture modes in region II, and the Floquet modes in region I. The Floquet expansion of the transverse (x, y components) electric and magnetic field at the discontinuity ($z=0$) can be\nwritten as follow \\cite{molero2021cross,rodriguez2015analytical}:\n\\begin{equation}\\label{up_mode_expansion1}\n\t\\begin{cases}\n\t\tE^{1}(x,y)=(1 + R)e_{0}^{1}(x,y) + \\sum_{h}^{'}V_{h}^{1} e_{h}^{1}(x,y)\n\t\t\\vspace{.5cm}\\\\\n\t\tH^{1}(x,y)=(1 - R)Y_{0}^{1}(\\hat z \\times e_{0}^{1}(x,y)) - \\sum_{h}^{'}V_{h}^{1} Y_{h}^{1}(\\hat z \\times e_{h}^{1}(x,y))\n\t\\end{cases}\n\\end{equation}\nwhere the expression $e_{0}^{1}(x,y)$ is the tangential component of the incident wave whose reflection is R. The superscript refers to the region, and the prime in the series of Eq. (\\ref{up_mode_expansion1}) indicates that the incident wave is excluded from the summation. $e_{h}^{1}(x,y)$ is the normalized transverse electric field of $h$th Floquet harmonic ($h$ is associated with a pair of integer numbers $mn$). Similarly, expansion of aperture modes can be written as\n\\begin{equation}\\label{aperture_mode_expansion1}\n\t\\begin{cases}\n\t\tE^{2}(x,y)=V_{0}^{2}e_{0}^{2}(x,y) + \\sum_{h}^{'}V_{h}^{2} e_{h}^{2}(x,y)\n\t\t\\vspace{.5cm}\\\\\n\t\tH^{2}(x,y)=V_{0}^{2}Y_{0}^{2}(\\hat z \\times e_{0}^{2}(x,y)) + \\sum_{h}^{'}V_{h}^{2} Y_{h}^{2}(\\hat z \\times e_{h}^{2}(x,y)) \\hspace{.7cm}\n\t\\end{cases}\n\\end{equation}\nwhere $e_{0}^{2}(x,y)$ is the tangential component of the dominant mode inside the aperture. It should be noted that we want to use the dominant mode to create the transmission line; thus, we have written this mode separately from the other modes. Similarly, $e_{h}^{2}(x,y)$ is the transverse electric field of harmonic $h$ inside the aperture. Modes inside the aperture can be used in equations depending on the type of aperture but the Floquet modes in region I can be written as\n\\begin{equation}\\label{fl1}\n\t\\begin{aligned}\n\t\te_{h}^{1}(x,y)=\\frac{e^{-jK_{th}^{1}.\\hat{\\rho}}}{PP} \\hat{e_{h}}^{1} \\hspace{1cm} \\hat{\\rho}=x\\hat{x} + y\\hat{y}\n\t\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{fl2}\n\t\\begin{aligned}\n\t\tk_{th}^{1}=k_{xm}^{1}\\hat{x} + k_{yn}^{1}\\hat{y} = (k_{m}^{1} + k_{x0}^{1} )\\hat{x} + (k_{n}^{1} + k_{y0}^{1})\\hat{y}\n\t\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{fl3}\n\t\\begin{aligned}\n\t\tk_{x0}^{1}=k^{1}\\sin(\\theta)\\cos(\\phi) \\hspace{1cm} k_{y0}^{1}=k^{1}\\sin(\\theta)\\sin(\\phi)\n\t\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{fl4}\n\t\\begin{aligned}\n\t\tk_{m}^{1}=\\frac{2\\pi m}{P} \\hspace{1cm} k_{n}^{1}=\\frac{2\\pi n}{P}\n\t\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{fl5}\n\t\\hat{e_{h}}^{1}=\n\t\\begin{cases}\n\t\t\\hat{k_{th}}^{1} \\hspace{.5cm} & TM\n\t\t\\vspace{.5cm}\\\\\n\t\t(\\hat{k_{th}}^{1} \\times \\hat{z} ) \\hspace{.5cm} & TE\n\t\\end{cases}\n\t\\hspace{1cm} \\hat{k_{th}}^{1}=\\frac{k_{th}^{1}}{\\mid k_{th}^{1} \\mid} \\hspace{1cm}\n\\end{equation}\nThe modal admittances $Y_{h}^{1}$ are given by\n\\begin{equation}\\label{fl6}\n\tY_{h}^{1}=\\frac{1}{\\eta^{1}}\n\t\\begin{cases}\n\t\t\\frac{k^{1}}{k_{zh}^{1}} \\hspace{.5cm} & TM\n\t\t\\vspace{.5cm}\\\\\n\t\t\\frac{k_{zh}^{1}}{k^{1}} \\hspace{.5cm} & TE\n\t\\end{cases}\n\t\\hspace{.5cm} k_{zh}^{1}=\\sqrt{(k^{1})^{2} - \\mid k_{th}^{1} \\mid ^{2}}\n\\end{equation}\nwith \n\\begin{equation}\\label{f21}\n\t\\begin{aligned}\n\t\tk^{1}=n_{1}k_{0} \\hspace{1cm} \\eta^{1}=\\frac{\\eta^{0}}{n_{1}}\n\t\\end{aligned}\n\\end{equation}\nwhere $k_{0}$ is the vacuum wavenumber, $\\eta^{0}$ is the intrinsic impedance of free space. These equations are for any incident angle; however, we focus on normal incident waves in this article for simplicity in presentation. In the case of a normal incident wave ($\\theta = 0, \\phi = 0$), we can use modes of a waveguide with PEC and PMC walls instead of using the written Floquet modes which decrease computational volume significantly \\cite{marques2009analytical,kaipa2010circuit}. Aperture modes can be used in equations depending on the shape of the aperture. In this paper, square and circular aperture modes are used (the dominant modes are $TE_{01}$ and $TE_{11}$ for the square and circular aperture, respectively).\n\nTo create the circuit model, we use the transverse electric field at the opening of the aperture ($E_{a}$). The transverse electric field on the aperture can be written as $E_{a}(x,y)=f(\\omega)e_{a}(x,y)$. By changing the frequency, almost the spatial profile of the transverse electric field on the aperture doesn't change ($e_{a}(x,y)$), and only its factor changes ($f(\\omega)$). This assumption is the basis for creating many analytical models that were referred to in the introduction. To develop the circuit model, we must apply boundary conditions at the discontinuity. The boundary conditions of the tangential electric fields at the discontinuity are\n\\begin{equation}\\label{eq_Exa1}\n\t\\begin{aligned}\n\t\t(1 + R)e_{0}^{1}(x,y) + \\sum_{h}\\phantom{}^{'}V_{h}^{1} e_{h}^{1}(x,y)=E_{a}(x,y)\n\t\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{eq_Exa2}\n\t\\begin{aligned}\n\t\tV_{0}^{2}e_{0}^{2}(x,y) + \\sum_{h}\\phantom{}^{'}V_{h}^{2} e_{h}^{2}(x,y)=E_{a}(x,y)\n\t\\end{aligned}\n\\end{equation}\n\nWe can write the orthogonality of modes as\n\\begin{equation}\\label{ortho}\n\t\\iint\\limits_{c} (e_{h}^{1} \\times (\\hat{z}\\times e_{k}^{1})^{*}).\\hat{z} ds =0 \\hspace{.5cm} if \\hspace{.5cm} k\\neq h\n\\end{equation}\n\nIf we use the orthogonality of Floquet modes in Eq. (\\ref{eq_Exa1}), the coefficients of Floquet harmonics are calculated as:\n\\begin{equation}\\label{orth_e1}\n\t\\begin{cases}\n\t\t(1 + R)=\\frac{\\iint\\limits_{a} (E_{a}(x,y) \\times (\\hat{z}\\times e_{0}^{1})^{*}).\\hat{z} ds}{\\iint\\limits_{c} (e_{0}^{1} \\times (\\hat{z}\\times e_{0}^{1})^{*}).\\hat{z} ds}=\\frac{A_{0}}{C_{0}}\n\t\t\\vspace{.5cm}\\\\\n\t\tV_{h}^{1}=\\frac{\\iint\\limits_{a} (E_{a}(x,y) \\times (\\hat{z}\\times e_{h}^{1})^{*}).\\hat{z} ds}{\\iint\\limits_{c} (e_{h}^{1} \\times (\\hat{z}\\times e_{h}^{1})^{*}).\\hat{z} ds}=\\frac{A_{h}}{C_{h}}\n\t\t\\vspace{.5cm}\\\\\n\t\tV_{h}^{1}=(1 + R)\\frac{A_{h}C_{0}}{C_{h}A_{0}}\n\t\\end{cases}\n\\end{equation}\n\nSimilarly, using the orthogonality of aperture modes and Eq. (\\ref{eq_Exa2}) implies that:\n\\begin{equation}\n\t\\begin{cases}\n\t\tV_{0}^{2}=\\frac{\\iint\\limits_{a} (E_{a}(x,y) \\times (\\hat{z}\\times e_{0}^{2})^{*}).\\hat{z} ds}{\\iint\\limits_{a} (e_{0}^{2} \\times (\\hat{z}\\times e_{0}^{2})^{*}).\\hat{z} ds}=\\frac{B_{0}}{D_{0}}\n\t\t\\vspace{.5cm}\\\\\n\t\tV_{h}^{2}=\\frac{\\iint\\limits_{a} (E_{a}(x,y) \\times (\\hat{z}\\times e_{h}^{2})^{*}).\\hat{z} ds}{\\iint\\limits_{a} (e_{h}^{2} \\times (\\hat{z}\\times e_{h}^{2})^{*}).\\hat{z} ds}=\\frac{B_{h}}{D_{h}}\n\t\t\\vspace{.5cm}\\\\\n\t\tV_{h}^{2}=V_{0}^{2}\\frac{B_{h}D_{0}}{D_{h}B_{0}}\n\t\\end{cases}\n\\end{equation}\n\n\nThe continuity of the tangential magnetic field on the aperture's surface helps us to create the circuit model. The continuity of the magnetic field on the aperture can be written as:\n\\begin{equation}\\label{H_con}\n\t\\begin{aligned}\n\t\t(1 - R)Y_{0}^{1}(\\hat z \\times e_{0}^{1}(x,y)) - \\sum_{h}\\phantom{}^{'}V_{h}^{1} Y_{h}^{1}(\\hat z \\times e_{h}^{1}(x,y))= \n\t\tV_{0}^{2}Y_{0}^{2}(\\hat z \\times e_{0}^{2}(x,y)) + \\sum_{h}\\phantom{}^{'}V_{h}^{2} Y_{h}^{2}(\\hat z \\times e_{h}^{2}(x,y))\n\t\\end{aligned}\n\\end{equation}\n\nMultiplying both sides of Eq. (\\ref{H_con}) by $E_{a}(x,y)$ and a few mathematical operations yields\n\\begin{equation}\\label{H_con_2}\n\t\\begin{aligned}\n\t\tY_{0}^{1}(1 - R)A_{0}^{*} - \\sum_{h}\\phantom{}^{'}Y_{h}^{1} V_{h}^{1}A_{h}^{*} =V_{0}^{2}Y_{0}^{2} B_{0}^* + \\sum_{h}\\phantom{}^{'}Y_{h}^{2}V_{h}^{2} B_{h}^{*}\n\t\\end{aligned}\n\\end{equation}\n\nIf we now place $V_{h}^{1}$ and $V_{h}^{2}$ in Eq. (\\ref{H_con_2}), the following equation is obtained.\n\\begin{equation}\\label{circuit_eq}\n\t\\begin{aligned}\n\t\tY_{0}^{1}\\frac{1 - R}{1 + R} =Y_{0}^{2} \\frac{C_{0}B_{0}B_{0}^*}{D_{0}A_{0}A_{0}^*} +\\sum_{h}\\phantom{}^{'}Y_{h}^{2} \\frac{B_{h}B_{h}^{*}C_{0}}{A_{0}A_{0}^{*}D_{h}} +\\sum_{h}\\phantom{}^{'} Y_{h}^{1} \\frac{A_{h}A_{h}^{*}C_{0}}{A_{0}A_{0}^{*}C_{h}}\n\t\\end{aligned}\n\\end{equation}\n\nEquation \\ref{circuit_eq} can be converted to the circuit model shown in\nFig. \\ref{fig:circuit_1}, where the circuit model admittances are in the form of equations in Eq. (\\ref{circuit_parameter1}). In the obtained admittances, the frequency dependence of $E_{a}(x,y)$ is eliminated; Thus, the admittances depend only on the spatial profile of the transverse electric field at the opening of the aperture and the electric field value on the aperture does not affect the admittances. In this circuit model, $Y_{eq1}$ is the result of all the Floquet modes except the incident wave in region I (the incident mode is used as the transmission line $Y_{0}^{1}$). $Y_{st1}$ is the result of the dominant mode inside the aperture, and the effect of other modes inside the aperture is modeled as admittance $Y_{st2}$.\n\\begin{equation}\\label{circuit_parameter1}\n\t\\begin{cases}\n\t\tY_{eq1}=\\sum_{h}^{'} Y_{h}^{1} \\frac{A_{h}A_{h}^{*}C_{0}}{A_{0}A_{0}^{*}C_{h}}\n\t\t\\vspace{.5cm} \\\\\n\t\tY_{st1}=Y_{0}^{2} \\frac{C_{0}B_{0}B_{0}^*}{D_{0}A_{0}A_{0}^*}\n\t\t\\vspace{.5cm} \\\\\n\t\tY_{st2}=\\sum_{h}^{'}Y_{h}^{2} \\frac{B_{h}B_{h}^{*}C_{0}}{A_{0}A_{0}^{*}D_{h}} \\hspace{1.5cm}\n\t\\end{cases}\n\\end{equation}\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=4.5cm,height=3.5cm]{circuit_11.pdf}\n\t\\caption{Circuit model for a semi-infinite array of sub-wavelength apertures.\n\t\t\\label{fig:circuit_1}\n\t}\n\\end{figure}\n\n\nIn the developed circuit model, we assume that the structures are semi-infinite; therefore, we need to change the admittances to achieve a circuit model compatible with the finite-thickness array of holes. It should be noted that the evanescent modes are equivalent to the energy storage elements, so the TE modes are equivalent to the inductors, and the TM modes are equivalent to the capacitors. The corresponding admittance of each harmonic is proportional to the energy stored in that mode. Unlike the semi-infinite structure, the aperture modes do not extend to infinity in the finite-thickness array; therefore, we must modify the admittances related to aperture modes. We use the concept of energy to modify the admittances. Admittances related to aperture modes to represent the energy stored in the finite thickness must be multiplied by $(1 - e^{j2K_{zh}^{2}L})$. Admittance related to region I ($Y_{eq1}$) does not change; similarly, admittance related to region III is written. It should be noted that the dominant mode inside the hole does not need to be modified because it has become a transmission line. According to the above, the admittances of a finite-thickness array of holes in Fig. \\ref{periodic_array} are written as:\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=5cm,height=4.5cm]{circuit_22.pdf}\n\t\\caption{The final circuit model for finite-thickness arrays of sub-wavelength holes shown in Fig. \\ref{periodic_array}.}\\label{fig:circuit_3}\n\\end{figure}\n\n\\begin{equation}\\label{circuit_model_3}\n\t\\begin{cases}\n\t\tY_{eq1}=\\sum_{h}^{'} Y_{h}^{1} \\frac{A_{h}A_{h}^{*}C_{0}}{A_{0}A_{0}^{*}C_{h}}\n\t\t\\vspace{.5cm} \\\\\n\t\tY_{st1}=Y_{0}^{2} \\frac{C_{0}B_{0}B_{0}^*}{D_{0}A_{0}A_{0}^*}\n\t\t\\vspace{.5cm} \\\\\n\t\tY_{st2}=\\sum_{h}^{'}Y_{h}^{2} \\frac{B_{h}B_{h}^{*}C_{0}}{A_{0}A_{0}^{*}D_{h}}(1 - e^{j2K_{zh}^{2}L})\n\t\t\\vspace{.5cm} \\\\\n\t\tY_{eq3}=\\sum_{h}^{'} Y_{h}^{3} \\frac{A_{h}A_{h}^{*}C_{0}}{A_{0}A_{0}^{*}C_{h}}\n\t\\end{cases}\n\\end{equation}\n\nThe bottom opening of the aperture has the same profile as the top opening of the aperture, so in the bottom opening of the aperture, the admittance $Y_{st2}$ will be unchanged. Only the directions of the aperture modes at the top and bottom are opposite, which does not make a difference in the value of this admittance. Due to the similarity of the profiles at the top and bottom openings, $Y_{eq3}$ is computed like $Y_{eq1}$. The final circuit model is examined in 3 forms which are described below.\n\n\\subsection{Single-Mode Circuit Model (SMCM)}\nThis case has been discussed in previous works \\cite{khavasi2015corrections,marques2009analytical,medina2009extraordinary}, but additional explanations are provided to clarify the differences between the cases in this subsection. The SMCM considers all the Floquet modes in regions I and III, but it considers only the dominant mode inside the aperture ($Y_{st2}=0$). The formulation in previous articles has been written in such a way that the electric field profile at the aperture opening is considered the same as the electric field profile of the dominant mode. For instance, in square and rectangular holes, the electric field profile on the aperture is considered as follows:\n\\begin{equation}\\label{TE01_model}\n\tE_{a}(x,y)=e_{0}^{2}(x,y)=\\cos(\\frac{\\pi y}{W}) \\hat{x}\n\\end{equation}\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=8cm,height=6cm]{pillar_structure.pdf}\n\t\\caption{A periodic array of square PEC pillars with period P. The width of the pillars is W and the thickness of the pillars is L.\\label{PEC_pillar}}\n\\end{figure}\n\nIn general, to find the realistic dominant mode profile inside the hole, we can increase the thickness of the array in full-wave simulations, and the electric field profile in the middle of the thickness can be used. For apertures with shapes such as circles, squares, and rectangles, the dominant mode has an analytical closed-form expression; however, a simulation profile can be used for unusual shapes. This case is suitable for arrays with a high thickness (almost thickness greater than $\\frac{P}{4}$).\n\nOne of the structures that can be examined with the help of this model is the array of PEC pillars (Fig. \\ref{PEC_pillar}). For this structure, the dominant mode which propagates between the pillars was obtained analytically \\cite{edalatipour2015physics}. Using this profile, isotropic\/anisotropic environmental models were presented and also if this profile is used in the SMCM, it gives accurate results (Fig. \\ref{pill_results}).\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=13cm,height=5cm]{pillar_results.pdf}\n\t\\caption{The results obtained from the single-mode circuit model for two arrays of PEC pillars (Figure (a) geometrical parameters: $W=.9P$,$L=2P$ and (b) geometrical parameters: $W=.6P$,$L=1P$). The blue curves are the results of the circuit model, and the red dashed curves are the full-wave simulation results.\\label{pill_results}}\n\\end{figure}\n\n\\subsection{Multi-Mode Circuit Model (MMCM)}\nIn the MMCM, we consider all the Floquet modes in regions I and III, as well as all the modes inside the apertures. In the MMCM, the electric field profile on aperture is not considered similar to the dominant mode and this profile is obtained from the full-wave simulator. First, let us examine an array of square holes in which the width of the squares is $\\frac{P}{2}$, and the thickness of the holes is $\\frac{P}{40}$. The electric field on the aperture at the resonance frequency obtained from the Comsol simulation is used to compute circuit model admittances (it should be noted that\nwe have used field profiles at the first resonance frequency). In Fig. \\ref{yeq_yst2}, the values $Y_{st2}$ and $Y_{eq1}$ are depicted. TE and TM modes in the circuit model are considered inductors and capacitors, respectively. Due to the predominance of TE modes at low frequencies, they show an inductive effect, which gradually decreases with increasing frequency, and eventually, the capacitive effect will be dominant (the amount of transmitted power through the array is shown in Fig. \\ref{cm_1}).\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=8cm,height=5.5cm]{Yeq1.pdf}\n\t\\caption{\n\t\tThe red curve shows the admittance resulting from the modes inside the aperture except the dominant mode. The blue curve shows the admittance resulting from Floquet modes except the incident mode in region I.}\\label{yeq_yst2}\n\\end{figure}\n\nThe MMCM can be more efficient in multilayer structures. First, we simulate the array at a frequency close to the resonance; then, with the help of the obtained profile, the multilayer structures with any number of layers can be examined. One of the disadvantages of the MMCM is that it requires an accurate electric field profile on the aperture, so we must obtain the field profile at a frequency close to the resonance frequency from the full-wave simulator and use it to compute admittances (of course, we can use the electric field profile at an arbitrary frequency, but the closer to the frequencies of interest e.g. resonance, the more accurate the results). Another complexity of the MMCM is that it is difficult to obtain high-order modes for unusual shapes; thus, we present another case that efficiently solves this problem.\n\n\\subsection{Modified Single-Mode Circuit Model (MSMCM)}\nThe MSMCM similar to MMCM and SMCM considers all the Floquet modes in regions I and III, but considers only the dominant mode inside the aperture. The difference between the MSMCM and the SMCM is in the electric field profile. In the SMCM, the electric field profile at aperture opening is considered equivalent to the electric field profile of the dominant mode; However, in the MSMCM, we use the electric field profile on aperture obtained from the full-wave simulator (in the next section, for circular and square holes, closed-form expressions are presented). The modified single-mode circuit model gives results obtained in Fig. \\ref{cm_1} for the array of square holes which was examined in the previous subsection.\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=8cm,height=5.5cm]{All_method.pdf}\n\t\\caption{Comparison of the results of circuit models and full-wave simulation.}\n\t\\label{cm_1}\n\\end{figure}\n\nAs clearly seen in Fig. \\ref{cm_1}, the circuit models predict extraordinary transmission (EoT). As can be seen, the accuracy of the MSMCM is less than the MMCM, but it is more accurate than SMCM. In the MSMCM, modifying the aperture profile has changed $Y_{eq1}$ and $Y_{st1}$ compared to the SMCM, which has significantly increased the accuracy. Since the MSMCM uses only one dominant mode, which makes this model more flexible than the MMCM, so we use this model in the following sections.\n\n\\section{Thin arrays of sub-wavelength apertures}\nDue to the importance and applications of arrays of sub-wavelength apertures perforated in a thin film, we focus on these structures in this section. In the previous section, we saw that the two cases MMCM and MSMCM have high accuracy. In these models, we need to use the aperture field profile obtained from the full-wave simulation. If the structure's thickness is zero, closed-form expressions are presented as electric field profiles of both circle and square shapes; however, these profiles also produce good results for non-zero thickness arrays. It should be noted that in order to use the MMCM, we must have an accurate electric field profile on the aperture; therefore, we use the approximated profiles in the MSMCM.\n\n\\subsection{Circular aperture}\nApertures with different shapes have different properties. Changing the shape and size of the aperture can help us to control electromagnetic waves. Experimental results have been investigated for holes with different shapes, one of which is circular holes.\nA closed-form expression has not been provided for the electric field profile of circular holes. Equation \\ref{app_cr} is an excellent approximation of the electric field profile for circular apertures. The actual (top row) vs approximated (bottom row) profiles are depicted in Fig. \\ref{pro_cr}. Compared to rectangular apertures, in circular apertures, changes in the angle of incident waves have little effect on the reflection or transmission power; thus, these structures can be considered from this perspective. In the case of circular apertures, the dominant mode within the aperture is $TE_{11}$. \n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=8cm,height=7cm]{profile_circle.pdf}\n\t\\caption{Transverse electric field profiles obtained from the full-wave simulation and Eq. (\\ref{app_cr})}\\label{pro_cr}\n\\end{figure}\n\n\\begin{equation}\\label{app_cr}\n\t\\begin{cases}\n\t\tE_{xa}=.1936\\frac{1}{\\sqrt{1 - 3.85(\\frac{r}{D})^2}} \\frac{\\cos(\\frac{\\pi r\\sin(\\phi)}{D})}{\\sqrt{1 - 1.12(\\frac{r\\sin(\\phi)}{D})^2}}\n\t\t\\vspace{.5cm} \\\\\n\t\tE_{ya}=.1074\\frac{(\\frac{2r}{D})^2}{\\sqrt{1 - 3.68(\\frac{r}{D})^2}} \\sin(2\\phi)\n\t\\end{cases}\n\\end{equation}\n\nUsing the approximated profile for the circular aperture, the result in Fig. \\ref{cr_ll} is obtained. As can be seen, the thickness $L=\\frac{P}{40}$ in the full-wave results can make a significant difference at high frequencies compared to the zero thickness full-wave simulation. However the MSMCM has predicted the non-zero thickness result quite accurately, not previously possible with other methods. Since we use the MSMCM, the admittance resulting from the high-order modes inside the aperture is zero. If the thickness is zero, the middle transmission line resulting from the dominant mode inside the aperture is also removed.\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=8cm,height=6cm]{circle_results.pdf}\n\t\\caption{Results of full-wave simulations for two different thicknesses and the MSMCM result (geometrical parameters used in the circuit model: $D=.5P$, $L=\\frac{P}{40}$).}\n\t\\label{cr_ll}\n\\end{figure}\n\n\\subsection{Square aperture}\nPrevious profiles for square shapes only consider field changes in one direction, which can cause errors, especially when the aperture is large (the width is more than half a period). The profiles used in previous studies produced accurate results for narrow rectangular apertures because the field changes along the breadth could be ignored. For square holes, we can use Eq. (\\ref{app_sq}) as an electric field profile. The comparison of the electric field profile obtained from the full-wave simulation and closed-form expression is provided in Fig. \\ref{pro_sq} and using the approximated profile yields the results in Fig. \\ref{sq_l0}.\n\\begin{equation}\\label{app_sq}\n\t\\begin{cases}\n\t\tE_{xa}=.1936\\frac{\\cos(\\frac{\\pi y}{W})}{\\sqrt{1 - 4(\\frac{y}{W})^2}} \\frac{1}{\\sqrt{1 - 3.85(\\frac{x}{W})^2}}\n\t\t\\vspace{.5cm} \\\\\n\t\tE_{ya}=.0928\\frac{\\frac{y}{W}}{\\sqrt{1 - 3.85(\\frac{y}{W})^2}} \\sin(\\frac{2\\pi x}{W})\n\t\\end{cases}\n\\end{equation}\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=8cm,height=7cm]{profile_square.pdf}\n\t\\caption{Transverse electric field profiles obtained from the full-wave simulation and Eq. (\\ref{app_sq})} \\label{pro_sq}\n\\end{figure}\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=13cm,height=5cm]{square_results1.pdf}\n\t\\caption{Figure (a) compares the results from the previous profile and the new profile ($L=0,W=.75P$). Figure (b) is the results obtained from the modified single-mode circuit model using the new profile.}\n\t\\label{sq_l0}\n\\end{figure}\n\n\n\\section{multilayer structure}\nUsing an array of sub-wavelength apertures perforated in a film, in practical applications requires placing the array between one or more dielectric layers. For example, to use these structures for applications such as sensors, typically a substrate is used, and eventually, this two-layer structure is placed on a layer of glass for testing \\cite{yahiaoui2016terahertz,gordon2008new,dhawan2008plasmonic}. Adding dielectric layers increases the full-wave simulation time. Moreover, since the thickness of the dielectric layers may be many times larger than the other dimensions, the optimization will be very time-consuming. One of the advantages of the circuit model is its flexibility and simplicity in analyzing multilayer structures. This feature further highlights the importance of developing accurate circuit models. In this part, other layers are added to the upper and lower environment of the periodic structure. In this case, we do not change the admittance related to the inside of the hole, but for the top and bottom layer, we have to modify the admittances as follows:\n\\begin{equation}\n\t\\begin{cases}\n\t\tY_{eq12}=\\sum_{h}^{'}Y_{h}^{2} \\frac{(Y_{h}^{1} - jY_{h}^{2}\\tan(K_{zh}^{2}T1))}{(Y_{h}^{2} - jY_{h}^{1}\\tan(K_{zh}^{2}T1))} \\frac{A_{h}A_{h}^{*}C_{0}}{A_{0}A_{0}^{*}C_{h}}\n\t\t\\vspace{.5cm} \\\\\n\t\tY_{eq45}=\\sum_{h}^{'} Y_{h}^{4}\\frac{(Y_{h}^{5} - jY_{h}^{4}\\tan(K_{zh}^{4}T2))}{(Y_{h}^{4} - jY_{h}^{5}\\tan(K_{zh}^{4}T2))} \\frac{A_{h}A_{h}^{*}C_{0}}{A_{0}A_{0}^{*}C_{h}}\n\t\\end{cases}\n\\end{equation}\n\nFor a multilayer structure with an array of circular apertures like Fig. \\ref{multi_layer_1}, the result of the circuit model is shown in Fig. \\ref{mult1}. As can be seen, the reflection curve has undergone many changes compared to the monolayer structure, and many resonances have occurred.\nIn the examined multilayer structure, a layer was added to the upper environment of the structure, which can represent the sample under test within the sensor. As we know, in sensors, the goal is to detect the change in the refractive index of layer 2, which is the reason for adding this layer.\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=7cm,height=6cm]{multi_circuit.pdf}\n\t\\caption{Modified single-mode circuit model for the array of circular apertures (Fig. \\ref{periodic_array}(b)) with two layers added to the top and bottom. The thickness of the circular holes is $\\frac{P}{40}$, and the thickness of the added dielectric layers is shown in the figure ( $n_{1}= n_{5}=1,n_{2}=1.8,n_{3}=2, n_{4}=1.4 , L=\\frac{P}{40} , D=\\frac{P}{2}$). \\label{multi_layer_1} }\n\\end{figure}\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=8.5cm,height=6cm]{multi_result.pdf}\n\t\\caption{The reflection curve corresponding to the circuit model shown in Fig. \\ref{multi_layer_1}. \\label{mult1}}\n\\end{figure}\n\n\n\n\n\n\\section{Conclusion}\nIn this paper, an extended circuit model was proposed for the arrays of sub-wavelength holes in a PEC film, by considering all the modes inside the aperture. The circuit model was examined in 3 general cases. The SMCM, due to the lack of need for electric fields profile at hole opening (dominant mode profile is considered as the profile at hole opening), is more useful than the others if the structure's thickness is large. If the thickness is large, for holes with common shapes such as circles, squares, and rectangles, the use of the SMCM is recommended. MMCM and MSMCM provide superior and accurate results, which is especially unprecedented for thin arrays. The MSMCM has higher performance than other models in thin arrays because it has high accuracy and, unlike the MMCM, does not need to obtain high-order modes inside the aperture, which means that the MSMCM has more flexibility. Using the approximated profiles and MSMCM, we obtained accurate results for thin arrays of circular and square holes.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzdgf b/data_all_eng_slimpj/shuffled/split2/finalzdgf new file mode 100644 index 0000000000000000000000000000000000000000..e2dea1c9cedffcfb5b882bc84298e60e17f4bcbb --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzdgf @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nIn condensed matter research there is a strong desire for new experimental tools that can measure the local electrical properties of materials and buried layers at high frequencies and with high spatial resolution \\cite{basov2011electrodynamics}.\nFor this goal, in recent years a variety of powerful scanning probe techniques have emerged which cover different parts of the electromagnetic spectrum: on the one hand, scanning near-field optical microscopy (SNOM) has enabled imaging with\ninfrared and far-infrared \\cite{qazilbash2007mott,liu2016nanoscale,huber2008terahertz} frequencies down to a few THz by utilizing the optical toolbox, i.e. free-space radiation, lasers and fiber technology; on the other hand, at GHz frequencies (typically 1-20 GHz) co-axial probes and shielded cantilevers have made possible quantitative local imaging by making use of commercially available microwave electronics for high performance signal processing (scanning microwave impedance microscopy, SMIM) \\cite{lai2010mesoscopic,de2016spatial,gramse2017nondestructive,anlage2007principles,steinhauer1998quantitative,buchter2018scanning,tselev2007broadband,huber2012calibrated}. \nThe frequency band in between, however, ranging approximately from 100 GHz to a few THz (also referred to as \\textit{sub-mm waves}), is a technological challenge. In the field of astronomy detection major progress has been made in sub-mm technology, for instance in the development of phase preserving instruments based on superconducting tunnel junctions such as for the Herschel Space Telescope, the Atacama Pathfinder Experiment (APEX) and the Atacama Large Millimeter array (ALMA)\\cite{eht2019black,siegel2002terahertz}.\nThese advances are being picked up to promote technological progress also in other research fields. In condensed matter physics this is expected to have a strong impact on measurement instrument development which will help understanding of a variety of important problems, in particular for disordered and unconventional superconductors, as well as for the so-called quantum materials where strong electron-electron interactions in the THz energy range give rise to a number of puzzling, unconventional and often spatially inhomogeneous electrical properties \\cite{liu2016nanoscale,basov2011electrodynamics,dordevic2006electrodynamics}. \nRealizing an experimental tool for probing these properties, however, remains challenging. Only recently first scattering SNOM measurements below 1 THz have been reported \\cite{liewald2018all}. \nIn the sub-mm band on-chip electronic circuits suffer from high losses (at room temperature) which strongly complicates the fabrication of more complex circuitry, required for signal control and processing. An alternative technology, much less prone to losses, consists of quasi-optical and metallic waveguide components. However, this technology increases the size of the measurement instrument compared with on-chip circuitry and thus also imposes certain boundaries when more complex signal handling is needed.\nIn order to overcome these technological hurdles, there is an ongoing development to combine quasi optical and on-chip electronics in hybrid devices \\cite{westig2011balanced}, but also to push the performance of microwave electronics into the sub-mm-band \\cite{zmuidzinas2004superconducting}. Picking up on this development, we have recently reported on a microstrip (MS) fabrication technology based on PECVD SiN$_x$, that is compatible with thin film membranes. For this technology losses are sufficiently well controlled at frequencies around 0.3 THz, such that the realization of room-temperature THz on-chip components is feasible \\cite{finkel2017performance}. \nHere, we use this technology to extend scanning impedance microscopy from microwaves into the THz frequency range. We present a shielded THz cantilever suitable for scanning probe microscopy that enables quantitative measurement of the impedance of the cantilever tip at around 0.3 THz. A key ingredient is a branchline coupler which is patterned on the cantilever and which acts as an interferometer for the THz signal, thereby providing high sensitivity of the circuit to small impedance changes at the cantilever tip. \n\nFirst, we will revisit the concept of scanning near field microscopy with shielded cantilevers as it is currently being used in microwave microscopy and we will identify the key features a THz cantilever should comprise. We then present the concept of our THz on-chip interferometer. Finally, we demonstrate how this concept enables impedance measurements with a THz cantilever that is compatible with conventional atomic force microscopy (AFM).\n\n\n\n\\section{Principle of shielded cantilever microscopy}\n\n \n The principle of scanning near field microscopy with a shielded microstrip (MS) cantilever is illustrated in Fig.~\\ref{fig:principle}a) \\cite{anlage2007principles,lai2011nanoscale}. The cantilever consists of a dielectric membrane of which the bottom side is covered with a thin metal layer, serving as a transmission line ground plane. \n The signal line of width \\textit{w} is patterned on the top side of the same dielectric. A cross section of the resulting MS transmission line geometry is sketched in Fig.~\\ref{fig:principle}b). Because the high frequency fields are mostly confined within the dielectric, such a transmission line geometry allows for delivering the signal to the cantilever tip in a controlled way while the ground plane screens the environment and prevents radiation losses. At the end of the cantilever the signal line terminates in a metallic tip. When a high frequency tone is launched to the MS, the tip acts as a capacitive termination, reflecting the signal back into the cantilever. This is quantified by the reflection coefficient $\\Gamma$, which is given by the mismatch between the generally complex valued tip impedance $Z$ and the characteristic MS line impedance $Z_0 = 50~\\Omega$: $\\Gamma={(Z-Z_0)}\/{(Z+Z_0)}$. When the cantilever is lifted far away from the sample surface, $Z$ is given by the capacitance $C_t$ between the tip and the cantilever ground plane (see Fig. \\ref{fig:principle} a). When the tip is on a sample, $Z$ is modified by contributions from the tip-sample capacitance $C_{s,tip}$, the capacitance between the sample and the ground plane $C_{s,gnd}$ and from resistive losses inside the sample, $R_s$. Measuring changes in $\\Gamma$ by detecting phase and amplitude of the reflected signal while scanning the tip over the sample provides a quantitative image of the spatial distribution of the conducting and dielectric properties of the sample \\cite{lai2010mesoscopic}. Since the electric field becomes strongly enhanced at the sharp tip, these local contributions dominate the total response, thus enabling spatial resolution down to 100 nm, i.e. three orders of magnitude below the signal wavelength \\cite{,lai2010mesoscopic,gramse2017nondestructive}. \n \n The tip-ground plane capacitance C$_t$ is generally given by the size and geometry of the cantilever close to the tip, which results in values of the order of $C_t \\sim 10^{-15}F$ . Since $Z\\propto 1\/i\\omega C$, at GHz frequencies ($\\omega = 2\\pi f$, $f \\sim 10^9$ Hz) the terminating impedance is large $Z \\sim 10^6 \\Omega$ $ \\gg Z_0$ and therefore the reflection coefficient becomes $\\Gamma \\simeq 1$. This means that most of the signal is reflected back into the cantilever when the tip is floating over the sample. We will refer to this part of the signal as scattered signal because it does not carry information about the sample itself. When the tip is in contact with the sample, the desired contributions from the tip-sample interaction thus only lead to small variations on top of an otherwise large $\\Gamma$, which is obviously difficult to detect. It is therefore highly desirable to minimize the scattered signal in the detector line and to become sensitive to those contributions only, which originate from the tip-sample interaction. At microwave frequencies this problem has been addressed by making the cantilever and the tip part of a resonator \\cite{cui2016quartz,gramse2017nondestructive,huber2012calibrated,anlage2007principles,tselev2007broadband} or by adding an impedance matching circuit which matches the open tip impedance to $Z_0$ \\cite{lai2010mesoscopic}. Both solutions create a narrow band resonance condition which enhances the sensitivity of the circuit to changes in the tip impedance. Furthermore, a common mode cancellation loop is typically included into the microwave readout circuit \\cite{huber2012calibrated,lai2010mesoscopic} which further reduces the scattered signal level at the detector.\n \n \\begin{figure}\n \t\\includegraphics[width=1\\linewidth]{Fig1.pdf}\n \t\\caption{\\textbf{a)} Sketch of a shielded cantilever and the corresponding lumped element circuit for measuring the complex reflection coefficient $\\Gamma$ of the cantilever tip. The cantilever consists of a metallic signal line and groundplane, separated by a dielectric, which determines the mechanical properties of the cantilever. The probe-cantilever interaction can be described with a lumped element circuit. \\textbf{b)} Cross section of the microstrip shielded cantilever. $w$ and $h$ denote the width of the signal line and the height of dielectric layer, respectively. \\textbf{c)} Equivalent lumped element circtuit of the situation depicted in a). \\textbf{d)} Distributed element circuit diagram for measuring the cantilever impedance using a balanced branchline coupler. The reflection coefficient $\\Gamma$ at the cantilever tip is determined by measuring the scattering parameter $S_{41}$ of the branchline coupler. This is achieved by intereference of the reflected signal from the tip with an unknown phase shift $\\Delta \\Phi_2$ with that of a balanced cancellation arm with known phase shift $\\Delta \\Phi_{3}$. $Z_0$ denotes the characteristic line impedance and $\\lambda$ is the signal wavelength. }\n \t\\label{fig:principle}\n \\end{figure}\n \n\n While at microwave frequencies such circuitry can be incorporated rather easily, this is not straight forward at THz frequencies because many required technologies are not readily available. In order to realize a shielded impedance microscope cantilever at THz frequencies it is therefore plausible to aim for an on-chip THz circuit solution that can be patterned close to the tip with lithographical means. We identify the following key properties such a circuit should provide: 1) separation of the in-going and reflected signal to facilitate signal processing. 2) Cancellation of the scattered signal in the detector line. 3) Sensitive response when the tip is brought into contact with a dielectric or a metallic sample. 4) Short signal lines to minimize losses. \n In the following section we will present and demonstrate a circuit that fulfills all of these requirements.\n \n\n\\section{Balanced branchline coupler as on-chip interferometer} \nFigure \\ref{fig:principle}d) depicts the diagram of a circuit designed to accomplish the above criteria. \nA key component is the balanced branchline coupler. It consists of 4 ports (labelled 1 - 4) which are connected through transmission line segments of a quarter wavelength $\\lambda\/4$ of the aimed for measurement frequency. By properly designing the impedance of each branch of the coupler (i.e. by choosing the appropriate signal line width $w$ for a constant thickness of the dielectric layer, cf. Fig.\\ref{fig:principle}b), one can control the transmission coefficients between the ports. \n The key idea of the concept we introduce here derives from analogies between a branchline coupler and an optical beam splitter: When the branch impedances $Z$ are chosen such that for two opposite branches $Z=Z_0$ ($w = 3.75~\\mu$m), while for the other two $Z = Z_0\/\\sqrt{2}$ ($w = 7.5~\\mu$m), an incoming signal at, for example, port 1, is split in equal parts between ports 2 and 3, and it acquires an additional phase shift of $-\\pi\/2$ between the these ports, while no signal arrives at port 4 \\cite{pozar2009microwave}. Since the coupler is designed symmetrically, the signal is split in the same fashion when injected at any other port.\n\n\n\n We can now use these properties, signal splitting and phase delay, to build an on-chip interferometer that is highly sensitive to impedance changes at the cantilever tip: We attach transmission lines of finite length $L_2$ and $L_3 = L_2 + \\Delta L$ at ports 2 and 3, respectively, as shown in Fig. \\ref{fig:principle}d (which we will refer to as \\textit{arms}, in analogy to an optical Michelson-interferometer). As the signal gets reflected at the end of each arm, it picks up a phase shift and gets re-injected into the coupler. For simplicity, assuming an ideal coupler with perfect isolation \\cite{pozar2009microwave} and neglecting losses, the signal at port 4 (detector line) is then given by the sum of the reflected signals re-injected at port 2 and 3,\n \n \\begin{equation}\n \tS_{41} = \\frac{A}{2} e^{i (\\Phi_c + \\Phi_{L2} + \\Delta \\Phi_2)} + \\frac{A}{2} e^{i (\\Phi_c + \\Phi_{L3} +\\Delta \\Phi_3)},\n \t\\label{eq:coupler} \n \\end{equation}\n where $A$ corresponds to the total signal amplitude, $\\Phi_c = 3\\pi\/2$ is the total phase accumulated in the coupler, $\\Phi _{L2,3}$ refer to the phase picked up due to the signal traveling down the respective arms and $\\Delta \\Phi_{2,3}$ is the phase picked up due to reflection at the terminations of arms 2 and 3, respectively. For our purposes it is convenient to express Eq.\\ref{eq:coupler} as \n \n \\begin{equation}\n S_{41} = \\frac{A}{2} e^{i (\\Phi_c + \\Phi_{L2})} (e^{i \\Delta \\Phi_2} + e^{i (\\Delta \\Phi_3 + \\Phi_{\\Delta L})}).\n \\label{eq:coupler_DeltaLeq0} \n \\end{equation}\n \n which indicates that signal cancellation in the detector line is achieved for \n \\begin{equation}\n \\Delta \\Phi_2 = \\Delta \\Phi_{3} + \\Phi_{\\Delta L} - \\pi .\n \\label{eq:phase}\n \\end{equation}\n \n\n\n As shown in Fig.~\\ref{fig:principle}d), in our case arm 2 terminates in the cantilever tip, which, in a first approximation ($C_t \\rightarrow 0$), acts as an \\textit{open} termination ($\\Delta \\Phi_2 \\rightarrow -\\pi$) when the tip is lifted off the sample. It is therefore convenient to terminate arm 3 with a \\textit{short} ($\\Delta \\Phi_3 = 0$) and to choose $\\Phi_{\\Delta L} = 0$ to achieve good signal cancellation at port 4. \n When scanning, changes in the dielectric or metallic environment of the tip lead to a phase mismatch at the detector line due to an enhanced capacitance at the tip, according to Fig.~\\ref{fig:principle}c. This results in a measurable signal which can be directly related to the phase change due to modified reflection conditions at the tip, using Eqs. \\ref{eq:coupler_DeltaLeq0} and \\ref{eq:phase}. When the tip is landed on a fully metallic sample, $(1\/C_{s,tip} + 1\/C_{s,gnd})^{-1} \\gg C_t$. As can be seen from the circuit in Fig.~\\ref{fig:principle}c) this corresponds to arm 2 being effectively shorted. In this case the reflected signals will interfere constructively and the full signal is detected at port 4. \n We note that in a real cantilever $C_t$ is finite ($\\Delta \\Phi_2 \\gtrsim -\\pi$) in which case $\\Delta L$ can be used as an additional phase matching parameter to achieve cancellation of the scattered signal. \n\n\\begin{figure}\n\t\\includegraphics[width=0.7\\linewidth]{Fig2.pdf}\n\t\\caption{\\textbf{a)} Optical image of the on-chip interferometer (device A), containing a balanced branchline coupler with branch lengths $\\lambda\/4$. The ports of the coupler are denoted 1-4. At a distance $L_2$ and $L_3$ from port 2 and 3 the transmission lines terminate in an open and short circuit, respectively. The signal (indicated with blue arrows) is injected at port 1 and detected at port 4. The scale bar corresponds to $100~\\mu$m. \\textbf{b)} S-parameter magnitude of a transmission measurement (symbols) and the corresponding analytic calculation (solid line) of device A (blue) and B (black). $L_2 = L_3 = 49 ~\\mu$m.}\n\t\\label{fig:branchline}\n\\end{figure}\n \n In order to test this concept, we have realized a series of balanced branchline couplers on a Si substrate using the technology described by Finkel \\textit{et al.}\\cite{finkel2017performance} All structures consist of $3~\\mu$m thin SiN$_x$ ($\\epsilon_r = 5.9$) serving as a MS dielectric and 2\/300 nm Ti\/Au as ground plane and stripline. As THz source and detector we use a vector network analyser together with frequency multipliers that cover the WR-03 band (220 to 325 GHz) and a GSG landing probe setup (for details see Finkel et al, Ref.\\cite{finkel2017performance}). Note however, that our concept is also compatible with other THz sources and detectors, for instance photomixers \\cite{mayorga2012first,westig2019waveguide}. We first demonstrate conceptually the basic idea. For this we have fabricated two samples (device A and B) for which $\\Delta L = 0$ and which realize two different arm terminations \\textit{open\/short} and \\textit{open\/open} at ports 2 and 3, respectively. An optical image of device A is shown in Fig.~\\ref{fig:branchline}a. The signal is launched and picked up from the circuit via the ports labeled 1 and 4 in Fig.~\\ref{fig:branchline}a, which consist of co-planar waveguide type fixtures\\cite{finkel2017performance} (not visible) that enable coupling of THz signals into the circuit with the landing probes. \n The $\\lambda\/4$-branches of the coupler have a length of 130 $\\mu$m, corresponding to a branchline coupler center frequency of $f_c = 270$ GHz. For the arm lengths we choose $L_2= L_3 = 49~\\mu$m. \n \n\nFigure \\ref{fig:branchline} b shows the measured scattering parameter $S_{41}$ obtained for device A and B (blue and black symbols, respectively). As expected, for device A we observe a low transmission ($\\sim -30$~dB) between port 1 and 4 with a minimum at $f =280$ GHz, which is close to the branchline coupler's center frequency $f_c = 270$ GHz. For device B both arms terminate in an open, i.e. $\\Delta \\Phi_2 = \\Delta \\Phi_3$. As a result constructive interference leads to a high transmission ($\\sim -5$~dB) over the full frequency range. \n\nNext we demonstrate how, owing to the sharp interferometer cancellation conditions, the circuit is highly sensitive to contributions from $\\Phi_{\\Delta L}$. Figure \\ref{fig:varyL} shows the measured $S_{41}$ parameter obtained from a series of devices for which we have varied $L_\\text{3} = 49~\\mu$m$ + \\Delta L$ by $\\Delta L = (1, 0, -1...-5)~ \\mu$m, while leaving $L_\\text{2} = 49~\\mu$m fixed. This leads to a small phase imbalance for the signal paths along arms 2 and 3. The experimental data reveal that indeed the position of the dip in frequency as well as its depth sensitively depend on $\\Delta L$ (dotted line). In Fig.~\\ref{fig:Hyb_IQ}a we have extracted magnitude and phase (symbols) for each $\\Delta L$ at fixed frequency $f = 280$ GHz (dashed line in Fig.\\ref{fig:varyL}). The data show that signal cancellation improves for small $\\Delta L$ with an optimal configuration at $\\Delta L = -1~\\mu m$. For even larger length difference it levels off. As discussed above this behavior reflects the termination of arm 2 with a finite capacitance, leading to phase shift slightly different from $-\\pi$, which gets compensated for by a slightly shorter $L_3$.\nThis has been confirmed quantitatively within a textbook analytical model of the circuit \\cite{pozar2009microwave} (for details see Appendix and Supplementary Material) that nicely reproduces all of our experimental data consistently (solid lines in Fig.~\\ref{fig:varyL} and Fig.~\\ref{fig:Hyb_IQ}a). In addition to a small dissipative contribution in the via, $R_\\text{short} = 1.6~\\Omega$, we have taken into account a finite terminating capacitance $C_{t} = 0.163$~fF, consistent with a standard text book approximation for an open MS line (see Appendix).\n\n\\begin{figure}\n\t\\includegraphics[width=0.6\\linewidth]{Fig3.pdf}\n\t\\caption{ Transmission measurements (symbols) and calculation (solid line) for a set of \\textit{open\/short} circuits with $L_3 = L_2 + \\Delta L$ and $\\Delta L = +1,0,...,-5~\\mu$m and $L_2 = 49~ \\mu$m. The curves are offset by -30 dB for clarity. }\n\t\\label{fig:varyL}\n\\end{figure}\n\n\\begin{figure}\n\t\\includegraphics[width=0.8\\linewidth]{Fig4.pdf}\n\t\\caption{\\textbf{a)} Magnitude (left) and phase (right) as function of $\\Delta L$ (bottom axis). Symbols: measurements. solid line: calculation. Dashed line and top axis: Calculated phase and magnitude for $L_2=L_3 = 49~\\mu$m and with varying $C_{load}$ connected in parallel to the termination of interferometer arm 2. \\textbf{b)} In-phase (I, red) and quadrature (Q, black) representation of the measured (circles and squares) and calculated (solid and dashed line) response versus $\\Delta L$. Dashed lines and top axis: Calculated I (red) and Q (black) as a function of $C_\\text{load}.$}\n\t\\label{fig:Hyb_IQ}\n\\end{figure}\n\nWe can further use the analytical model to analyse theoretically the circuit's response to a load capacitance $C_\\text{load}$ connected in parallel to $C_t$, representing a sample in a scanning probe experiment (cf. lumped element diagram in Fig. \\ref{fig:principle}c). The resulting amplitude and phase are plotted in Fig.~\\ref{fig:Hyb_IQ}a) as dashed lines. The corresponding $C_\\text{load}$ is given in the top axis. As expected this yields a fairly similar behaviour as a variation of $\\Delta L$. \nFigure ~\\ref{fig:Hyb_IQ}b) plots the data as in-phase (I) and quadrature (Q) amplitudes, for a variation of $\\Delta L$ (bottom axis, solid lines) and $C_\\text{load}$ (top axis, dashed lines), respectively. In this representation $I$ can be directly related to dissipative contributions to the signal while $Q$ represents the imaginary part of the reflection coefficient which is related to capacitive (and, in principle, also inductive) contributions. This is consistent with the observed linear behaviour of $Q$ and a constant $I$. From these plots we estimate our circuit to be sensitive to a capacitance change smaller than 0.25 fF. \n\n \n \\section{Cantilever implementation}\n \n We will now describe how this detection scheme can be implemented and used in a scanning probe cantilever to detect impedance changes at the probe tip.\n Figure \\ref{fig:cantilever}b shows an optical microscope image of the shielded cantilever containing the THz circuit, patterned on its top side. The \\textit{signal in} and \\textit{signal out} lines (corresponding to ports 1 and 4 in Fig.~\\ref{fig:branchline}a) are connected via landing probes with the source and detector (not visible). Since the dimensions of the cantilever (300 $\\mu$m long, 75 $ \\mu$m wide) are too small to host a circuit as shown in Fig.~\\ref{fig:branchline}a, we have re-designed the branchline coupler such that the cross-branches are now folded inwards to fit the lateral dimensions of the cantilever. This slightly modifies the coupler properties. However, it does not change its basic functionality. \n As discussed previously for the branchline coupler devices, one of the interferometer arms terminates in a \\textit{short}. The other one, previously terminating in an \\textit{open}, is now connected to the tip. We will keep the notation of the arms as introduced above, referring to the arm terminating in the tip as arm 2 with length $L_2$, and to the arm terminating in a short to ground as arm 3 with length $L_3$. In order to balance the coupler such that scattered signal cancellation is achieved, we have to take into account the finite capacitance of the open tip ($C_t \\sim 2$ fF, obtained from finite element (FE) simulations) and adjust $L_3$ by $\\Delta L$ accordingly. However, due to the folded geometry of the coupler and a resulting unwanted cross-coupling between the branches, significant leakage currents within the coupler result in a non-trivial relation between $\\Delta L$ and signal cancellation at the detector line. Therefore, we use FE simulations to empirically determine a well-balanced configuration for the given $C_t$, for which we obtain $L_2 = 44~ \\mu m$ and $L_3 = 54~ \\mu m$, i.e. $\\Delta L = 8~\\mu$m . \n \n \\begin{figure}\n \t\\includegraphics[width=1\\linewidth]{Fig5.pdf}\n \t\\caption{\\textbf{a)} Main fabrication steps of the THz cantilever. \\textbf{b)} Optical image of the final, released cantilever. The scale bar corresponds to $75~\\mu$m. \\textit{Signal in} and \\textit{signal out} denote the transmission lines connected to THz source and detector, respectively. The folded hybrid coupler patterned on the back of the cantilever is indicated. One arm of the coupler terminates in a short circuit, the other one terminates in the tip. \\textbf{c)} Transmission amplitude from \\textit{signal in} to \\textit{signal out}. The transmission dip at 250 GHz indicates cancellation of the signal reflected from the open tip. The dashed curve shows the result obtained from finite element (FE) modeling.}\n \t\\label{fig:cantilever}\n \\end{figure}\n\n \\subsection{Fabrication}\nIn fig. \\ref{fig:cantilever}a the fabrication flow for the cantilever is sketched.\nIn a first step (1) a pyramid shaped pit ($5~\\mu$m deep) is etched into the Si wafer using KOH etching. This defines the position and shape of the tip. Next (2) we deposit (10+300)nm Ti\/Au which serves as a ground plane. During this step also the pit is filled with a Ti\/Au layer, which will become the metallic tip. The area around the pit is protected with an optical mask. The wafer is then (3) covered with 3 $\\mu$m of PECVD SiNx that is subsequently etched with a Bosch process to define the geometry of the cantilever, $300~\\mu$m long and $75~\\mu$m wide. In a separate step $5\\times 5~\\mu$m$^2$ sized \\textit{vias} are etched into the SiN$_x$ layer. These \\textit{vias} will serve as an electrical connection between the MS top layer and the ground plane (to form a short) or with the cantilever tip, respectively. \nAfter a cleaning step, we pattern the strip lines with electron beam lithography and lift off techniques (5) and, in a separate step, connection of the\\textit{ via} is established through angled deposition of Au. We use (2+300)nm Ti\/Au bilayers. This step concludes the patterning of the transmission lines on the cantilever. Next, we release the cantilevers. In order to avoid exposure of the striplines to chemicals, we protect the surface of the wafer by gluing a Sp wafer on top of it with \"black wax\" (Apiezon W100).\nWe take particular care that no air bubbles remain in the wax to ensure a complete and efficient protection. The release step is prepared by patterning a SiN$_x$ mask on the backside of the wafer which contains windows at those positions where the cantilevers have been patterned on the front side of the wafer. The two wafers are then subjected to a KOH etch which etches through the windows on the backside of the wafer until the Ti\/Au ground layer is reached. At this point the cantilever gets released from the Si wafer. Note, however, that on its front side it is still glued to the protection wafer. When the KOH etch is complete, the wafer is carefully immersed in Toluene to dissolve the black wax and to fully release the cantilever chips. \nSubsequently the cantilever is mounted on the landing probe setup. \n \n \\subsection{Experimental results}\n \n\nThe measured THz response of the cantilever is shown in Fig.\\ref{fig:cantilever}c. We clearly observe a dip in transmission ($\\sim -30$ dB) indicating a suppression of the scattered signal that gets reflected from the open tip into the detector line. We note that compared to the previously discussed branchline couplers without the tip (Fig.~\\ref{fig:branchline}), the position of the dip is slightly shifted towards lower frequencies ($f = 250$ GHz). This is most likely a result of the folded geometry of the branchline coupler, consistent with a cross-capacitive coupling between neighbouring parts of the circuit. Moreover, the frequency shift as well as the relatively low transmission at higher and lower frequencies, suggest that the terminating capacitance of the tip slightly differs form the assumed value ($C_t = 2$ fF) such that the chosen $\\Delta L = 8~\\mu$m turns out to be not yet the best match. The FE model for the THz response (dashed line) yields good agreement with the experiment if we assume $C_t = 2.9~$fF and $R_{short} = 5~\\Omega$. \n\nOur cantilever can be used to detect changes in the tip impedance when landed on a dielectric or metallic sample. This is demonstrated in Fig.~\\ref{fig:TIM}. We have mounted the cantilever on the landing probe setup and we get the tip in contact with different materials, approached from below via a mechanical height control. Fig.~\\ref{fig:TIM}a compares the measured response for the tip floating near the sample surface (red) and landed on 3 different materials, Au (green), Si (black) and SrTiO$_3$ (blue). \nWe clearly observe distinct responses for each material. When brought into contact with a dielectric (Si, $\\epsilon_r = 11.9$ and SrTiO$_3$, $\\epsilon_r = 300$) the dip shifts towards lower frequencies by $\\Delta f_\\text{(Si)} = 2$ GHz for Si and $\\Delta f_\\text{(STO)} = 10$ GHz for SrTiO$_3$. Notably, the overall line shape remains fairly similar. In contrast, upon contact with highly conductive Au ($\\rho = 2\\mu \\Omega $cm), the dip vanishes and transmission is high over the full frequency range, as expected for a shorted tip. \n\n\n \\begin{figure}\n \t\\includegraphics[width=1\\linewidth]{Fig6.pdf}\n \t\\caption{ \\textbf{a)} THz response of the cantilever when open (red) and landed on Si (black), SrTiO$_3$ (blue) and Au (green) samples. Solid lines correspond to measurements, dashed lines indicate finite element (FE) simulations with the following parameters: $R_\\text{via} = 5 \\Omega$, Tip capacitance: $C_t= 2.9~$fF; $L_2 = 44~\\mu$m, $L_3 = 52~\\mu$m, conductivity signal line and gnd plane: $\\sigma_{au} = 19.2\\times 10^{-6}$~S\/m, $\\epsilon_r(\\text{SiN}_x) = 5.9$. $C_\\text{load}$ (corresponding to $(1\/C_{s,tip} + 1\/C_{s,gnd})^{-1}$ in Fig. 1, $R_s = 0$) for Off: 0 fF; Si: 0.25fF; SrTiO$_3$: 0.75fF; Au: 15fF. \\textbf{b)} I (dashed line, red) and Q (solid line, black) of the cantilever response at 260 GHz for different $C_{load}$ obtained from FE simulations. Symbols: experimental data extracted from a). \\textbf{c)} $20~\\mu$m$\\times 20~\\mu$m topography image of a scratched Si wafer, obtained with a cantilever similar to the one used in a) utilizing the beam deflection mode in a commercial AFM. Inset: SEM image of the cantilever tip. } \n \t\\label{fig:TIM}\n \\end{figure}\n \t \n\t \n\n\n\\subsection{Discussion}\n\nIn order to quantitatively understand the cantilever response we use FE modelling of the full circuit and we include a load capacitance $C_{load}$ in parallel to $C_{t}$ to take into account contributions from the sample materials (cf. fig. \\ref{fig:principle}a, neglecting Ohmic dissipation in the sample, $R_s = 0$). We find that the curves can be reproduced very well if we use $C_{Si} = 0.25 ~$fF, $C_{STO} = 0.75 ~$fF, and $C_{Au} = 15 ~$fF as the only adjustable parameter for each material. \nIn Fig.~\\ref{fig:TIM}b) we compare $I$ and $Q$ of the FE response for various $C_{load}$ with the experimental values obtained at $f = 260$ GHz for each sample. Since the response of the folded branchline coupler connected to the tip is slightly off resonance ($\\sim 250$ GHz) we do not expect a simple linear behaviour as for the branchline coupler without the tip discussed previously. Figure \\ref{fig:TIM}b shows that the response becomes more sensitive, i.e. the slope of the curves for $I$ and $Q$ becomes steep, for larger $C_{load}$ ($\\sim 10$ fF). Since this is in the range of capacitance which we obtained for the Au sample, this indicates that our THz cantilever becomes more sensitive for metallic samples. In contrast to shielded microwave cantilevers, where sensitivity is highest around the metal-insulator transition \\cite{lai2010mesoscopic}, for our cantilever the working point is shifted towards samples with higher conductivity. It may thus be used to detect electronic variations at high frequencies within a metal or even in superconductors in future scanning experiments.\n\nOur THz cantilever is also compatible with AFM topography imaging. This is shown in Fig.\\ref{fig:TIM} c) where a topography image of a scratched Si wafer surface is displayed, obtained using a THz cantilever mounted on a commercial Asylum Cypher AFM with a laser deflection read-out. Using a de-convoluting tip geometry modeling algorithm (\\textit{Gwyddion} blind tip estimation algorithm \\cite{nevcas2012gwyddion}) we estimate the tip apex to be $\\approx$100 nm. An SEM image of a cantilever tip is shown in the inset in Fig. \\ref{fig:TIM} c). \n\nFinally, we like to point out some aspects that we aim to improve for future THz cantilever generations. Firstly, even though our fabrication technique provides useful devices, the current yield is rather low ($\\approx 10 \\%$). This is mostly related to the use of the black wax, which is needed to protect the THz circuitry from chemicals during the release step, but which also induces mechanical stress, resulting in loss of a large number of cantilevers. Secondly, in its current design the substrate, which serves as a handling wafer, faces in the same direction as the tip. This limits the surface region on the sample, that can be reached by the cantilever to approximately the cantilever length ($\\sim 300 ~\\mu$m). In order to lift this constraint, developing a flip-chip technology may provide the most suitable means to bond a handling wafer to the top side of the cantilever chip. At the same time, however, it will be important to maintain access to the circuitry with landing probes. Thirdly, our experiments have shown that due to the folded geometry of the balanced branchline coupler the circuit response deviates slightly from that of the un-folded geometry tested on a substrate (Fig.~\\ref{fig:cantilever}b in comparison to Fig.~\\ref{fig:branchline}a). Therefore, FE simulations are required for a quantitative analysis of the measurement signal while a simple analytical equation would be more desirable. This may motivate a re-design of the cantilever such that it can host the balanced coupler without the need to modify its layout. In this case the measurements could be modeled within a textbook analytical description, which will strongly facilitate a quantitative interpretation of the measurement signal.\n\n\n\n\\section{Conclusion}\nWe have presented a shielded THz probe suitable for impedance microscopy in the sub-mm band (0.3 THz). As a key challenge for the realization of such a device we have identified the necessity to carry out common mode cancellation and impedance matching at THz frequencies close to the cantilever tip in order to enable sensitive detection of small changes of the tip impedance. We have addressed these challenges by developing an on-chip circuit that can be patterned on the cantilever which comprises a balanced branchline coupler. The coupler functions as an on-chip interferometer and in this manner achieves the required common mode suppression as well as high sensitivity to small impedance changes. To demonstrate the basic functionality of this concept, we have realized a set of devices on substrates and we have characterized their response at THz frequencies. The results can be directly modeled within an analytical model of the circuit. Furthermore, a fabrication technology has been developed that allows for patterning the circuit on a free standing cantilever including the tip. When the released cantilever is landed on different dielectric (Si, SrTiO$_3$) and metallic (Au) samples we observe distinct THz responses which enable us to determine the corresponding capacitive load at the cantilever tip using finite element modelling. Our cantilever removes several critical technological challenges towards scanning impedance microscopy at THz frequencies. \n \n\\section*{Appendix: Analytical Model for the balanced branchline coupler}\n\nIn order to compute the response of the balanced hybrid coupler we describe the signal evolution in the coupler in terms of forward and backward travelling waves in the transmission lines and the reflection coefficents $\\Gamma_\\text{ms}$ and $\\Gamma_\\text{s}$ at the open and shorted transmission line, respectively. Using Kirchoff's rules for the voltage and current at each node of the hybrid, we can construct a system of equations that allows us to determine the voltage measured at the detector line at port 4 upon signal injection at port 1. \n(The full set of equations is provided in the Supplementary material).\nTo describe wave propagation along each transmission line segment with length $L$ and impedance $Z$ we use the frequency dependent wave propagation factor $e^{\\gamma L}$ and\n\n\\begin{align*}\n&\\gamma = \\alpha \\frac{Z}{Z_0} + i\\beta\\\\\n&\\alpha = (1.1f \\times 10^{-9} + 86.9)~\\text{Np\/m} \\\\\n&\\beta = \\frac{2 \\pi}{c} f \\sqrt{\\epsilon_\\text{eff}} \n\\end{align*}\n\nwith $Z_0 = 55.5~\\Omega$, $c = 3\\times 10^8$ m(s)$^{-1}$, $\\epsilon_{eff} = 4.47$ and $\\alpha$ extracted from a direct measurement of a $130~\\mu$m transmission line. For the low impedance lines we have used Z =Z$_l$ = 37 $\\Omega$. The branch lengths of the coupler are $L = 130 \\mu $m. \n\nTo calculate the open terminating capacitance $C_t$ of the open MS line we have used \n\n\\begin{align*}\nC_{t} = G \\frac{\\sqrt{\\epsilon_{eff}}}{c Z_0}, \\\\\nG = \\frac{\\xi_1 \\xi_3 \\xi_5 h}{\\xi_4}\n\\end{align*}\n\ntogether with the following closed form expression:\n\\begin{align*}\n&\\xi_1 = 0.434907\\frac{(\\epsilon_{eff}^{0.81} + 0.26(w\/h)^{0.8544} + 0.236)}{(\\epsilon_{eff}^{0.81} - 0.189(w\/h)^{0.8544} + 0.87)}\\\\\n&\\xi_2 = 1+\\frac{(w\/h)^{0.371}}{2.35\\epsilon_r + 1}\\\\\n&\\xi_3 = 1 + \\frac{0.5274~\\tan^{-1}[0.084(w\/h)^{1.9413\/\\xi_2}]}{\\epsilon_{eff}^{0.9236}}\\\\\n&\\xi_4 = 1+ 0.037\\tan^{-1}[0.067(w\/h)^{1.456}] \\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times (6-5\\exp(0.036(1-\\epsilon_r)))\\\\\n&\\xi_5 = 1-0.218 \\exp(-7.5(w\/h))\n\\end{align*}\n\n\n\n\nfor which we have used the stripline width $w = 3.75~\\mu$m, dielectric thickness $h = 3~\\mu$m, $\\epsilon_{eff} = 4.47$, PECVD SiN$_x$ dielectric constant $\\epsilon_r = 5.9$ \\cite{finkel2017performance}, characteristic impedance $Z_0 = 55.5~\\Omega$ and the vacuum speed of light $ c=3\\times10^{8}$ m\/s. \n\n\n\\section{Acknowledgements}\nWe would like to thank Carmine de Martino and Luca Galatro for help with the THz characterization setup and David Thoen for support during the fabrication process development.\nWe further acknowledge scientific discussions with Andrea Neto, Nuria Llombart, Akira Endo, Jochem Baselmans, Ronald Hesper and Ivan Camara Mayorga. \nThis research has been funded by the European Research Council Advanced Grant No.~339306 (METIQUM).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLow grade gliomas (LGGs) are slowly growing tumours which arise from supporting glial cells in the brain. Although they proliferate slowly, the ultimate behaviour of these tumours is not benign. They are almost invariably incurable due to their diffusive infiltrative nature and often lead to patient death due to malignant transformation, \\emph{i.e.} transformation of the tumour into more aggressive anaplastic form.\n\nTreatment of~LGGs is controversial among clinicians as LGGs usually occur in young and otherwise healthy patients. Besides seizures, patients with LGGs appear neurologically asymptomatic. Thus, life-prolonging treatment should not come at the cost of compromising the quality of life of these otherwise healthy and mostly young patients. Management decisions, whether a~patient with LGG should receive resection, radiation therapy or chemotherapy, are not fully standardized.\nClinicians usually base their decisions about extent of surgical resection, timing of radiotherapy and long-term benefits and risks of chemotherapy on a large number of factors including age, performance status, location of tumour and patient preference. \n\nCurrent standard of care is first to perform maximal resection with minimum impact on the brain's functional areas. Support for such an approach comes from studies which demonstrate that the extent of resection is a~prognostic factor for LGG patients \\cite{Keles,Smith}. Unfortunately, because of the LGGs infiltrative characteristics, surgery alone fails to cure these tumours in most cases as only the tumour bulk can be resected. There is a~trend to use more active treatments and various alternative approaches have been considered. Post-operative radiotherapy could be a~therapeutic option for low-grade gliomas, but it causes long-term neurocognitive toxicity and fails to show a significant improvement in patient survival~\\cite{Karim,Karim1,Shaw}. Therefore radiotherapy is usually deferred and performed routinely only in patients with tumours facing a~high risk of malignant transformation \\cite{Pouratian}. \n\nIn this context, there is an increasing interest in the use of chemotherapeutic agents which could influence tumour evolution and at the same time allow the delay of more aggressive treatments. Currently there are two chemotherapeutic drugs effective for the treatment of glioma patients: temozolomide (TMZ), an oral alkylating agent and procarbazine, lomustine and vincristine (PCV), a~combination of alkylating agents and cell-cycle specific microtubule inhibitor. About 25-50\\% of LGGs show chemotherapeutic responses to treatment with either TMZ~or PCV. PCV has been used from the 1970's and has been suggested as a~clinically relevant option especially for some subtypes of anaplastic gliomas with the 1p\/19q codeletion \\cite{Cairncross, Bent2}. Unfortunately it causes significant myelosuppression, nausea and peripheral neuropathy \\cite{Mason,Buckner}. \n\nTMZ~is a~cell-cycle non-specific prodrug, absorbed with almost 100\\% bioavailability \\cite{Newlands}.~This chemotherapeutic agent can cross the blood-brain barrier and is spontaneously converted in tumour cells to its active metabolite \\cite{Marchesi}. The main cytotoxic action of TMZ occurs at the O$^6-$ position of guanine. During subsequent cycles of replication, the futile mismatch repair system is initiated and O$^6$MeG is incorrectly paired with thymine instead of cytosine, eventually inducing cell death, see \\emph{e.g.} \\cite{Jiricny,Barciszewska}. A~phase III~trial showed the efficacy of TMZ for high-grade gliomas~\\cite{Stupp} and since then it has been used routinely for patients with newly diagnosed glioblastoma (the most malignant type of glioma)~\\cite{Bent3}. Phase II~trials have demonstrated its effectivity against both previously irradiated and unirradiated LGGs \\cite{Kesari,Kaloshi,Pouratian1}. In addition, there are reports of cases where neoadjuvant chemotherapy given to surgically unresectable tumours has allowed subsequent gross total resection \\cite{Blonski,Jo}, which is of great importance especially when the tumour is highly infiltrative or located in~eloquent areas. Large prospective studies with long-term follow-up are under development to verify statistically significant improvement in resections following TMZ~treatment. It has also been reported that TMZ~treatment may lead to an important reduction in seizure frequency in LGG patients \\cite{Koekkoek}. Thus prolonged TMZ~treatment until evidence of resistance is a~clinically interesting option for selected patients as an up-front or adjuvant treatment. Clinical trials are on the way to study the effect of this treatment on survival. \n\nTemozolomide has a~better toxicity profile than PCV, being well tolerated by patients \\cite{Liu,Pace,Pouratian1}. This is due to the fact that, in contrast to procarbazine and lomustine, TMZ does not cause a chemical cross-linking of the DNA strands and it is less toxic to the haematopoietic progenitor cells in the bone marrow \\cite{Agarwala}. Patients treated with TMZ are also seeing a reduced risk of cumulative haematologic toxicity thanks to the rapid elimination of this drug \\cite{Agarwala,Portnow,Newlands}. \n\nThe response of glioma cells to chemotherapy is a~subject of many clinical and biological studies. Recently, it has been shown that in some LGG patients a~metabolic response to TMZ~therapy is observed even 3 months after the end of~the treatment \\cite{Wyss,Guillevin}. Moreover, dynamic volumetric studies have shown that a~treatment-related volume decrease can be observed for many months after the chemotherapy is discontinued \\cite{Peyre,Ricard}. The time to maximum tumour response was reported to be in some cases larger than 2 years \\cite{Chamberlain}.~Other researchers investigated relations between some molecular characteristics of LGGs and response to TMZ~\\cite{Hoang,Kaloshi,Ricard, Ohba}. However, the question of the correct timing of chemotherapy remains unanswered, namely whether it should be given first or when progression has been observed. Another issue to be addressed is~the optimal fractioning of TMZ.\n\nThe typical plan of TMZ~treatment is to give doses of $150$--$200$~mg per m$^2$~of patient body surface once per day for 5 days every 28 consecutive days. The number of such cycles in~clinical practise is usually between 12 and 30 \\cite{Ricard, Ribba,Chamberlain} and it depends on patient-related characteristics and the haematological toxicity observed. There have been many clinical studies on alternative treatment regimens for gliomas. Among others in \\cite{Kesari} patients were treated in cycles with doses of 75 mg\/m$^2$ given daily for 7~weeks followed by four-week breaks. Some trials on dose escalation \\cite{Wick,Taal,Viaccoz} intended to overcome the DNA-repair activity of the enzyme MGMT (O$^6$MeG methyltransferase), which reverses alkylation at the O$^6$ position of guanine \\cite{Everhard,Hegi}. \nHowever, these TMZ~regimes were either not effective or had to be rejected because of a~high toxicity \\cite{Hammond}. Recent studies show that O$^6$MeG is able to trigger not only apoptosis (programmed cell death), but also autophagy (mechanism allowing the degradation and recycling of cellular components) and senescence (state of permanent cell-cycle arrest in~which cells are resistant to apoptotic death) \\cite{Knizhnik}, which may provide an explanation why previous studies failed to improve treatment efficacy. Many clinicians conclude that the chemotherapy fractionation scheme providing the best tumour response and acceptable haematologic toxicity are still to be determined.\n \nMathematical modelling has great potential to help in finding appropriate therapeutic timings and\/or fractionations and in individual patient treatment decision. Even models simple from mathematical point of view can be useful for clinical practice. \n\nIn this paper we describe a~simple mathematical model for LGG growth and response to chemotherapy that fits very well with longitudinal volumetric data of patients diagnosed with LGGs. Interestingly, the model suggests that the response of the tumour to chemotherapy may be related to~its aggressiveness. We also provide an approximate explicit formula for the time of tumour response to chemotherapy. This equation may be helpful to clinicians in selecting patients who will benefit most from early treatment and finding the best personalised therapy.\n\nOur plan in this paper is as follows: in Section 2 we develop the model and describe its parameters. In Section 3 we present the results of model fitting to patient data and suggest the relation between time of response to chemotherapy and patient prognosis. The analytical estimates are shown in Section 4. We discuss the results of our model and therapeutic implications in Section 5.\n\n\\section{Mathematical model} \\label{sec:model}\n\\subsection{The dynamics of tumour cells}\nComputational modelling of glioma growth started 20 years ago \\cite{Murray} and has received strong attention in the last few years (see \\emph{e.g.} \\cite{Unkelbach,Konukoglu,Barazzoul, Bastogne,Kirkby}). \nFor solid tumours mathematical models have considered different chemothe\\-rapy-related factors such as drug diffusion, uptake\/binding, clearance and their effect on cell cycle progression (see \\emph{e.g.} \\cite{Au,Jackson,Tzafriri,Swierniak}). Many mechanistic mathematical models models have been developed to improve the design of chemotherapy regimes (see \\emph{e.g.} \\cite{Gardner} for a~summary). However, very few models have considered chemotherapy of LGGs \\cite{Ribba}.\n\nIn order to keep our description as simple as possible we will build a~continuous macroscopic model assuming that the tumour grows due to net cell division. The simplest choice for the proliferative term is to assume that a~tumour cell number $P(t)$ is governed by a~logistic growth with coefficient $\\rho,$~its inverse giving an estimate of the typical cell doubling times. \n \nTMZ is a small molecule and is easily absorbed in the digestive tract with 100\\% of bioavailability and a half-life of absorption of 7 minutes \\cite{Ostermann}. It should be noted that the intact TMZ molecule easily crosses the blood brain barrier due to its lipophilicity, and is then activated in the brain compartment. It hydrolyses to MTIC, which subsequently undergoes spontaneous breakdown to an inactive metabolite AIC and an active methyldiazonium cation. It is the methyldiazonium ion that causes the damage in patients' DNA, namely it transfers methyl groups to DNA. The most common sites of methylation are the N$^7$ position of guanine (N$^7$-MeG; 60-80\\%) followed by the N$^3$ position of adenine (N$^3$-MeA; 10-20\\%) and the O$^6$ position of guanine (O6-MeG; 5-10\\%) \\cite{Fu}, the last one being responsible for major cytotoxicity.\n\nAs to the TMZ effect on tumour cells, we choose here to formulate a macroscopic model. It is known that TMZ-induced damage leads to cell death long after the end of therapy ~\\cite{Peyre,Bent1,Ricard,Chamberlain}. It has been verified \\emph{in vitro} that the glioma cells death after administration of TMZ is induced most typically in one of the post-treatment cell cycles \\cite{Roos} due to futile mismatch repair cycles (see \\emph{e.g.} \\cite{Marchesi} for a detailed description of this mechanism). This delayed cell death is a key feature of glioma response to chemotherapy. Thus, in line with this biological evidence we will assume that tumour cells treated with chemotherapy die after a~time $k\/\\rho$ of the order of mean $k$~effective tumour doubling times. This type of model has been used successfully to describe the effect of radiotherapy on LGGs \\cite{Victor,Victor2}.\n\nThen, we will complement the equation for the number of functionally alive tumour cells $P(t)$ with an equation for the evolution of the number of cells irreversibly damaged by chemotherapy $D(t)$. The number of cells damaged by the~drug in a~time unit is considered to be proportional to the concentration of the drug $C(t)$ multiplied by the number of proliferating tumour cells with the rate~$\\alpha,$~measuring the influence of TMZ~on cells.\nWe assume that irreversibly damaged tumour cells try to enter mitosis with the same probability as those active, but die after a mean value of $k$ such attempts, which results in the growth rate $\\rho\\left(1-\\frac{P+D}{K}\\right)$ for proliferative cells and the death rate $-\\frac{\\rho}{k}\\left(1-\\frac{P+D}{K}\\right)$ for damaged cells, $K$~being the carrying capacity for both populations $P+D$, which leads to the following set of equations\n\\begin{subequations}\n \\label{ode}\n\t\\begin{eqnarray} \n\t\t\\frac{\\mathrm{d} P}{\\mathrm{d} t} & = & \\rho P\\left(1-\\frac{P+D}{K}\\right) - \\alpha PC, \\label{ode1} \\\\ \n\t\t\\frac{\\mathrm{d} D}{\\mathrm{d} t} & = & -\\frac{\\rho}{k}D\\left(1-\\frac{P+D}{K}\\right) + \\alpha PC, \\label{ode2} \n\t\\end{eqnarray}\nwhere $C(t)$ accounts for the brain chemotherapy concentration to be described in more details later. Taking an average cell volume, we can easily treat the tumour mass $P+D$ as the total tumour volume, which is easier to compare with results obtained from magnetic resonance imaging (MRI), usually used in brain tumour diagnosis and follow-up observation.\n\n\\subsection{Kinetics of chemotherapy drug} \\label{subs:CT}\nThe systemic pharmacokinetic and pharmacodynamic properties of TMZ has been studied in detail in several studies \\cite{Baker,Ostermann}. Baker \\textit{et~al.}\\xspace \\cite{Baker} described the concentrations of TMZ, MTIC and AIC in plasma in detail. Ostermann \\textit{et~al.}\\xspace~\\cite{Ostermann} collected data on TMZ concentration from blood and cerebrospinal fluid (CSF) obtained via lumbar puncture in patients with malignant gliomas. \n\nHowever due to the physiological separation of brain and tumour from both blood and CSF (through blood-brain, blood-tumour, blood-CSF and CSF-tumour barriers) the amount of drug reaching the tumour differs from the amount of drug circulating in blood and CSF \\cite{Shannon,Blakeley}. \nTherefore instead of describing a~complicated mechanism with many unknown parameters based on data collected from blood or CSF, we chose a simpler dynamics based directly on the brain tissue data. Thus we will base our model on data from the study by Portnow \\textit{et~al.}\\xspace~\\cite{Portnow} who examined TMZ concentration in intracerebral microdialysis samples from peritumoural brain interstitium obtained from patients with central nervous system tumours.\n\nWith regard to chemotherapy pharmacokinetics, we will assume, as usual \\cite{Jacqmin,Ribba}, that the concentration of TMZ, $C(t)$, measured in units of days decays exponentially due to the drug clearance with a~constant rate $\\lambda$. It is consistent with the fact that TMZ has linear pharmacokinetics \\cite{Newlands}. Thus, we have to complement Eqs.~\\eqref{ode} with \n\\begin{equation} \\label{ode3}\n\t\t\\frac{\\mathrm{d} C}{\\mathrm{d} t} = -\\lambda C. \n\t\\end{equation}\n\\end{subequations}\nTMZ~reaches a~maximal drug concentration in the brain about two hours after administration~\\cite{Hammond,Portnow,Baker}, what is very short in comparison to the time scale of tumour evolution in the model (of the order of years). Thus, we may treat the whole time of oral drug administration, absorption and transport to the brain as a~discontinuous change in the drug concentration that occurs at given administration times. Such a formulation of the problem enables also the following mathematical analysis and estimations.\n\nChemotherapy will consist of a~sequence of doses $d_1,d_2,\\ldots,d_n$ given at times \n$t_1 < t_2 <\\ldots < t_n$. \nWe assume that initially (at time $t=t_0$, taken to be the start of the tumour observation) the tumour has a~certain mass $P_0=P(t_0)$ and there are neither damaged cells, nor chemotherapy drugs within the tumour, thus \\mbox{$D(t_0) = 0, \\ C(t_0) = 0.$}\nThus for $t\\in (0, t_1)$ solving Eqs.~\\eqref{ode} with initial conditions given above leads to \n\\begin{equation} \\label{solP}\nP(t) = \\frac{ KP_0 \\e^{\\rho t}}{K - P_0 \\left(1-\\e^{\\rho t} \\right)}, \\quad D(t) = 0, \\quad C(t)=0.\n\\end{equation}\nFor $t\\in (t_j, t_{j+1})$, $j=1,2,3,\\ldots,n$ with $n$ being the total number of doses, the values of $P$, $D$ and $C$ change according to Eqs.~\\eqref{ode}, while at $t=t_j$ subsequent doses of the drug are administrated which we model as impulses, so that \n\\begin{equation} \\label{impulses}\n P(t_j)=P(t_j^-), \\quad \n\t D(t_j)=D(t_j^-), \\quad \n\t C(t_j)=C(t_j^-)+C_j, \n\\end{equation}\nwhere $f(t_j^-) = \\lim_{t\\to t_j^-} f(t)$ and $C_{j}$ is the fraction of the dose $d_{j}$ which reaches the tumour tissue, accounting for drug loss during transport to the brain.\n\nThe interval between doses (typically 1 day) will be chosen to be larger than the time of whole dose elimination (reported to be around 7h \\cite{Agarwala}) and the typical damage repair times (of the order of a~few hours \\cite{VanderKogel}), so that one dose will not alter the effect of the next one. \n\nThe asymptotic behaviour of model~\\eqref{ode} under the effect of a~finite number of doses is easy to obtain. When no drug is given for time $t \\ge t_n$ (with $n$ being the index of the last dosis), then \\mbox{$C(t)=C(t_n) \\exp(-\\lambda (t-t_n))\\to 0$} as $t\\to+\\infty$. As $C \\to \\infty$ and $P$ is bounded, then $\\alpha P C \\to 0$ for $t \\to +\\infty.$ As a consequence $D'(t)<0$ for sufficiently large time. \nThen it is easily seen that $D(t)\\to 0$ and $P(t)\\to K$ as $t\\to+\\infty$. This behaviour is not surprising and can be interpreted as patient death due to drug clearance and tumour regrowth. Patient death usually occurs when the tumour reaches a~critical size called the fatal tumour burden considered to be in high-grade glioma models to be around 6 cm in diameter \\cite{Swanson2008,Woodward}.\n\nIt is important to emphasise that model \\eqref{ode} intends to describe the effect of first-line chemotherapy, since after the treatment resistant phenotypes arise leading to the acquisition of drug resistance. Thus a~detailed analysis of second-line chemotherapy would require the introduction of more phenotypes in the model and is beyond the scope of this research.\n\n\\subsection{Parameter estimation} \\label{sec:parameters}\nTo work with Eqs. \\eqref{ode} we need to provide realistic values for the model parameters. \n\nThe saturation coefficient $K$ for LGG growth will be set to the volume of a~sphere of diameter $10$~cm reported to be the maximal mean tumour diameter observed in LGG patients \\cite{Ricard}. In fact, most patients die when the tumour diameter is about 6 cm in size as discussed before \\cite{Swanson2008,Woodward}.\n\nWe can estimate the rate of drug decay $\\lambda$ using values of TMZ~half-life clearance time $t_{1\/2}$. From the definition of $t_{1\/2}$ and assuming exponential decay as in Eqs.\\eqref{ode} we have\n$$\\frac{1}{2}=\\e^{\\textstyle -\\lambda t_{1\/2}}.$$\n\nTo account also for the drug loss during transport to the brain we calculate value $C_{j}$ of the maximal dose $d_{j}$ reaching the tumour as \n\\begin{equation} \\label{dose}\nC_j = \\beta \\cdot d_{j} \\cdot b,\n\\end{equation}\nwhere $\\beta$ is the fraction of TMZ~getting to 1ml of brain interstitial fluid (from a~unit dose) and $b$~is a~surface of a~patient body with $j \\in \\{1,\\ldots, n\\}$ and $n$~being the total number of doses $d$ administered. Then $C_{j}$ can be interpreted as an effective dose per fraction. \n\nThe most typical chemotherapy schedule consists of cycles of 28 days with five TMZ~oral doses on days 1 to 5 followed by a~break of 23~days. Standard dose per day is 150 mg per m$^2$ of patient body surface, which is usually around 1.6~m$^2$ for women and 1.9 m$^2$ for men \\cite{Mosteller} with an average of 1.7~m$^2$~\\cite{Sparreboom}. \nThen in the case of the standard chemotherapy scheme we will fix $d_{j}=d=150$mg\/m$^2$ and $C_{j}=C_0=\\beta \\cdot d \\cdot b.$\n\nThe parameter $\\beta$ can be calculated using the value of maximal TMZ~concentration $C_{max}$ for a~dose of 150 mg\/m$^2$ taken from the literature \\cite{Hammond,Portnow}. Assuming that time to reach peak drug concentration in the brain is negligible (equals $0.85-2$h) in comparison to the time scale of the model, we set the initial drug concentration $C_{0}$ in the moment of its administration to the value $C_{max}.$ \n\nA~summary of the biological parameter values is presented in Table~\\ref{table-parameters}. \n\\begin{table}[ht] \\small \n\t\\begin{center}\n\t\t\\caption{Biological parameters describing TMZ concentration in brain.}\n\t\t\\begin{tabular}{l l l c}\n\t\t\t\\hline\n\t\t\tParameter & Description & Value, references & \\\\ \n\t\t\t\\hline\n\t\t\t$t_{1\/2}$ & TMZ~half-life clearance time & $\\simeq 2$h & \\cite{Hammond} \\\\\n\t\t\t$C_{max}$ & mean peak TMZ~concentration & $0.6 \\mu$g\/ml & \\cite{Portnow} \\\\\n\t\t\t& in brain interstitium & &\\\\\n\t\t\t$\\lambda$ & rate of decay of TMZ~& 0.3466\/h & Calculated \\\\\n\t\t\t& & & from \\cite{Hammond,Portnow} \\\\\n\t\t\t$\\beta$ & fraction of TMZ~getting & $2.1\\cdot 10^{-6}$\/ml (m)& Estimated\\\\ \n\t\t\t& to brain interstitium & $2.5 \\cdot 10^{-6}$\/ml (w)& from \\cite{Hammond,Portnow} \\\\ \n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\label{table-parameters}\n\t\\end{center}\n\\end{table} \n\n\\section{Numerical results}\n\\subsection{Model fitting to patient data} \\label{sec:model fitting}\nTo test if our simple model given by Eqs.~\\eqref{ode} is able to reflect the dynamics of LGG response to chemotherapy, we have used the model to describe volumetric longitudinal data of patients followed at the Bern University Hospital between 1990 and 2013. In this study we selected data on 18 patients who had been treated with TMZ~out of a total number of 82 LGGs patients, see Table~\\ref{tab:patients data}. Radiological glioma growth was quantified by manual measurements of tumour diameters on successive MRIs. The three largest tumour diameters ($D_1,\\ D_2, \\ D_3$) according to three reference orthogonal planes (axial, coronal and sagittal planes) have been measured and tumour volumes have been estimated using the ellipsoidal approximation: $V=(D_1 \\cdot D_2 \\cdot D_3)\/2$, following the standard clinical practice \\cite{Pallud,Mandonnet}. \n\nThe inclusion criteria for patients for the purpose of model fitting in this study included: (i) biopsy\/surgery confirmed LGG (astrocytoma, oligoastrocytoma or oligodendroglioma), (ii) availability of at least 2 MRIs before the onset of TMZ~treatment, (iii) no other treatment given in the period of study and (iv) availability of at least 4 MRIs after TMZ~onset with at least one after the end of the chemotherapy. 7 patients satisfied these criteria. All patients in this group received more than 4~TMZ~cycles and the mean duration of TMZ~treatment was 6.26 months. \n\n\\begin{table}[ht] \\small \n\\begin{center}\n\\caption{Characteristics of patients treated with TMZ}\n\\begin{tabular}{l c c}\n\\hline\nAge at diagnosis, mean (st.~deviation), yr & 47.19 (7.54) \\\\\nSex, M\/F & 14\/4 \\\\\n\\emph{Histology at diagnosis} & \\\\\n\\hspace{0.5cm} Oligodendroglioma & 7 \\\\\n\\hspace{0.5cm} Oligoastrocytoma & 9 \\\\\n\\hspace{0.5cm} Astrocytoma & 1 \\\\\n\\hspace{0.5cm} Unknown & 1 \\\\\n\\emph{Type of surgery} & \\\\\n\\hspace{0.5cm} Biopsy & 10 \\\\\n\\hspace{0.5cm} Resection & 9 \\\\\n\\emph{Radiotherapy} & 8 \\\\\n\\emph{Chemotheraphy (CT)} & 18 \\\\\n\\hspace{0.5cm} Age at CT onset, mean (st.~deviation), yr & 51.8 (8.35) \\\\\n\\hspace{0.5cm} Time from surgery to CT, mean (st.~deviation), yr & 3.7 (4) \\\\ \n\\hspace{0.5cm} Second-line CT & 8 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:patients data}\n\\end{center}\n\\end{table}\n\nThe rate of tumour cell proliferation $\\rho,$~the coefficient $k$ describing the delay in damaged cell death and the parameter of TMZ-cell kill strength $\\alpha$ were considered to be tumour-specific and fitted for each patient. Thus only three parameters are unknown and the others are taken as in Table \\ref{table-parameters}.\n \nTo estimate the parameter $\\rho$ we used patient data before the start of TMZ~administration since it is the only relevant parameter during that time, see Eq.~\\eqref{solP}. The initial tumour volume $P_0$in this equation was fixed to be the volume from first MRI~done after surgery. Then, having obtained the value of $\\rho$, MRI~data after the onset of chemotherapy was used to estimate parameters $\\alpha$~and $k$ in Eqs.~\\eqref{ode}. Model fitting was done using a~weighted least squares method. To simulate Eqs.~\\eqref{ode}, we have used the standard Matlab ODE solver based on the Runge-Kutta 4th-order method.\n\nFigs.~\\ref{fig:proliferacja}, \\ref{fig:tTTP} show both the real tumour volume data obtained from the MRIs (circles) together with the best fit obtained with Eqs.~\\eqref{ode} (solid line). The model dynamics fit the real volumetric tumour evolution well, showing an impressive agreement with a~minimal number of parameters for patients with delayed response to chemotherapy. The minimal value of the fitted proliferation rate for some patients is one order of magnitude smaller than values $(1-5) \\cdot 10^{-3}$ day$^{-1}$ observed in other studies \\cite{Gerin,Victor,Victor3} as in these studies the model for tumour growth also considered a diffusive term. Some of the tumours were relatively large, however no formation of neoangiogenesis or necrotic core was observed. See also Supplementary material for all patients data and estimated parameters.\n\n\\begin{figure}[h!p] \n\\begin{center}\n\\includegraphics[width=0.525\\textwidth]{108} \n\\includegraphics[width=0.525\\textwidth]{159} \n\\includegraphics[width=0.525\\textwidth]{25} \n\\caption{Tumour volume evolution for three selected patients treated with TMZ. The beginning and the end of TMZ~treatment are marked with vertical dashed lines. There are shown the volumes calculated from MRIs (circles) and from the fitted mathematical model (solid lines). The number of TMZ~cycles and the values of parameters were different for each patient.\n(top) Patient treated with 5 TMZ~cycles, $\\alpha=0.971918$ml\/$\\mu$g\/day, $\\rho = 0.001761$\/day, $ \\ k = 0.555867.$\n(center) Patient treated with 11 TMZ~cycles, \\mbox{$\\alpha = 0.279911$ml\/$\\mu$g\/day,} $\\rho = 0.000136$\/day, $ \\ k = 0.025617.$ \n(bottom) Patient treated with 4 TMZ~cycles, $\\alpha=1.387798$ml\/$\\mu$g\/day, \\mbox{$\\rho = 0.002416$\/day,} $ \\ k = 0.272291.$}\n\\label{fig:proliferacja}\n\\end{center}\n\\end{figure}\n\n\\subsection{Tumours with faster response have worse prognosis} \\label{sec:relations}\n\\begin{figure}[h!t]\n\\begin{center}\n\\includegraphics[trim={0.5cm 0cm 1.8cm 0.9cm},clip=true,width=0.45\\textwidth]{fig_rho}\n\\includegraphics[trim={0.5cm 0cm 1.8cm 0.9cm},clip=true,width=0.45\\textwidth]{fig_alpha}\n\\caption{Tumour volume evolution for different values of parameters. Virtual patients were treated with 12 cycles of TMZ~as in the standard fractionation scheme (see Sec.\\ref{sec:parameters}) with $k=0.5.$ (left)~Parameter $\\alpha$ was fixed to value 0.7ml\/$\\mu$g\/day, $\\rho_1=0.004\/$day$, \\ \\rho_2=0.002\/$day. (right) Parameter $\\rho$ was fixed to value 0.0025\/day, \n$\\alpha_1=0.4$ml\/$\\mu$g\/day, $\\alpha_2=0.8$ml\/$\\mu$g\/day. The horizontal dotted lines correspond to tumour sizes equal to the fatal tumour burden.}\n\\label{fig_rho,alpha}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!t]\n\\centering\n\\includegraphics[trim={0.865cm 0cm 1.73cm 0.9cm},clip=true,width=0.49\\textwidth]{alpha,times} \n\\includegraphics[trim={0.865cm 0cm 1.73cm 0.9cm},clip=true,width=0.49\\textwidth]{rho,times}\n\\caption{Characteristic times of tumour response for different proliferation rates $\\rho$ and different levels of TMZ~cell kill strength $\\alpha$.~We considered 12 cycles of TMZ~as in the standard fractionation scheme (see Sec.~\\ref{sec:parameters}) for virtual patients with LGG of initial volume 40 cm$^3$. (left) Results for $\\rho = 0.0008$\/day, $k=0.3$ and $\\alpha \\in [0.1,1]$ml\/$\\mu$g\/day.~(right) Results for $\\alpha=0.8$ml\/$\\mu$g\/day, $k=0.3$ and $\\rho \\in [0.7,8] \\times 10^{-3}$\/day.}\n\\label{fig:relacje}\n\\end{figure}\n\n\\begin{figure}[h!t]\n\\centering\n\\includegraphics[trim={1cm 0.6cm 1.03cm 1.4cm},clip=true,width=0.47\\textwidth]{surfOS} \n\\includegraphics[trim={0.63cm 0.6cm 0.95cm 1.4cm},clip=true,width=0.47\\textwidth]{surfTRP,GD}\n\\caption{Characteristic times of tumour response for different proliferation rates $\\rho$ and different levels of TMZ~cell kill strength $\\alpha$.~We considered 12 cycles of TMZ~as in the standard fractionation scheme (see Sec.~\\ref{sec:parameters}) with $k=0.3.$\nValues of time to radiological progression (left), growth delay and overall survival (right) are shown for virtual patients with LGG of an initial volume of 40~cm$^3$.\n}\n\\label{fig:surf}\n\\end{figure}\n\n\\begin{figure}[h!t]\n\\centering \n\\includegraphics[trim={0.83cm 0cm 1.8cm 1.02cm},clip=true,width=0.5\\textwidth]{TRP,OS}\n\\caption{Time to radiological response and overall survival of LGGs patients treated with TMZ. Patients were divided into two groups: in the first, denoted by circles, patients were treated with TMZ~for less than 10 months (average: 5.72, st.~deviation: 3.8), in the second group, denoted by triangles, patients were treated with TMZ~for more than 10 months (average: 13.65, st.~deviation: 3.14).}\n\\label{fig:patients-PFS,OS}\n\\end{figure}\nWe have also studied how the tumour response depends on parameters fitted, see Figs. \\ref{fig:relacje}, \\ref{fig:surf}. \nWe will denote by ``time to radiological progression'' (TRP) the time when the tumour attains its minimum volume after the chemotherapy onset and starts regrowing. We will refer to ``growth delay'' as the time for which the tumour volume equals the initial one when regrowing after the therapy, see~\\cite{Victor}. We will refer to ``early response'' when TRP is attained shortly after the end of chemotherapy and ``no response'' when there was no decrease in tumour volume detectable by radiologist. In case of frequent MRIs these times can be easily obtained from model simulations and compared with the values obtained from patient's MRI~volumetry. It can be estimated with an error being the time between two subsequent MRIs. We have also computed the overall survival (OS) as the time until reaching the fatal tumour burden defined in Sec.\\ref{subs:CT}. We have considered only the cases of virtual tumours whose volumes decreased below the volume at TMZ onset. Note that for the purpose of analysis of response to TMZ, OS is computed for virtual tumours responding to TMZ and without any other treatment in the following course of disease, thus in general could be overestimated and thus should not be compared to the values of real-patients overall survival. \n\nAfter performing many simulations for different initial values and chemotherapy schemes, we conclude that both larger proliferation rates $\\rho$ and smaller TMZ~cell kill strengths $\\alpha$, lead to an earlier response to TMZ~treatment. A~more systematic study is shown in Fig.~\\ref{fig_rho,alpha} for two specific parameter sets, providing representative examples. Virtual patients who responded earlier to TMZ~(had smaller TRP) had a~faster regrowth and reached the fatal tumour burden earlier. Thus, a shorter TRP is an indicator of worse prognosis. \n\nAre these model features also present in the patient's data? Fig.~\\ref{fig:patients-PFS,OS} shows how the overall survival rate correlates with the time to radiological progression. For the purpose of this analysis we also included patients with only one MRI before treatment with TMZ. We excluded (i) two patients in which only two MRIs were available after the end of chemotherapy showing no tumour regrowth and (ii) one patient treated with TMZ~for only 1.5 month showing no response. For each patient TRP was estimated as the time to the MRI in which tumour volume was the smallest after the onset of treatment. Due to the differing duration of TMZ treatments we divided patients into a group receiving less than 10 TMZ cycles and those receiving 10 or more cycles.\n\nThe Spearman rank correlation coefficient between TRP and OS equals 1 for data of patients treated with TMZ~for less than 10 months and 0.9047619 for those treated with TMZ~for longer time. The exact Spearman coefficient test significance levels equal 0.008333 and 0.002282 for right-tailed tests for group of patients treated with less and more than 10 cycles of TMZ, respectively. This result indicates a~positive correlation between TRP and OS. The significance levels were calculated using R. \nData on overall survival is right-censored, however the results suggest that the early regrowth of the tumour after chemotherapy is related to its aggressiveness. Despite therapies used after progression, those tumours that responded faster to TMZ~treatment progressed faster, suggesting either larger proliferation potential and\/or smaller TMZ~cell kill strength. \n\n\\section{Results (II): Analytical estimates of tumour response} \n\n\\subsection{Survival fraction}\nUp to now our analysis has been based on numerical simulations of Eqs. \\eqref{ode}. We can calculate the fraction of tumour cells eliminated by a~single dose of chemotherapy, which in the context of radiotherapy is usually referred to as \\emph{survival fraction}. To do so, we will assume that the time of drug absorption, distribution and elimination from the human body is much shorter than the doubling time of the tumour cell population, what is true for LGGs. Therefore, focusing on short-term effects of the drug we may neglect the term describing tumour proliferation. Consequently the size of damaged cells population remains zero and we consider instead the~simplified model\n\\begin{subequations}\n\\begin{eqnarray}\n\t\t\\frac{d P}{d t} & = - \\alpha P C, \\\\ \n\t\t\\frac{d C}{d t} & = -\\lambda C.\n\\end{eqnarray}\n\t\\end{subequations}\nFor time before the second drug administration $t_1\\le t0$, thus showing a radiologically visible decrease in total volume. Thus, in our approach, in which only first-line chemotherapy is described and resistant cells do not arise, tumour progression will occur after the end of chemotherapy ($s_{\\textrm{TRP}} \\gg s_{n}$) provided tumour growth is slow as happens in the case of LGGs. Thus we will try to obtain explicit formulae approximating $x$ and $y$~for $s \\gg s_{n}.$\n\nWe will consider tumours with initial sizes significantly smaller than the carrying capacity, then the Eq. \\eqref{ode_scaled} takes the simpler form\n\\begin{subequations}\\label{ode_final}\n\\begin{eqnarray} \n\t\t\\frac{d x}{d s} & = & x - xz, \\\\ \n\t\t\\frac{d y}{d s} & = & -\\frac{1}{k}y + xz, \\\\ \n\t\t\\frac{d z}{d s} & = & -\\mu z,\n\t\\end{eqnarray}\n\\end{subequations}\nwith initial conditions: $ x(0)=x_0, \\ y(0)=0, \\ z(0)=z_0.$ Then for rescaled time TRP we have $x'(s_{\\textrm{TRP}}) + y'(s_{\\textrm{TRP}})=0,$ therefore \n\\begin{equation} \\label{formula s_ttp}\nx(s_{\\textrm{TRP}})=\\frac{1}{k}y(s_{\\textrm{TRP}}).\n\\end{equation}\nEqs.~\\eqref{ode_final} are a~set of ordinary differential equations with impulses, the functions $x$ and $y$ being continuous, and $z$~being discontinuous at times $s_{j}$ for $j \\in \\{2,\\ldots,n\\}$ as \n\\begin{equation}\n z(s_j)=z(s_j^-)+z_0.\n \\end{equation}\nSince dose clearance time is about two hours we may assume that each dose is cleared in one day, then, in the rescaled units\n\\mbox{$z((s_{j}+\\rho)^{-}) \\approx 0$} for \\mbox{$j \\in \\{1,\\ldots,n\\}.$} Therefore, we can approximate\n\\begin{equation} \\label{z approximation}\nz(s) \\approx \n\\left\\{\n\t\\begin{aligned}\n\t\t& z_0 \\e^{\\textstyle -\\mu(s-s_{j})} && s \\in (s_{j}, s_{j}+\\rho), \\\\ \n\t\t& 0 && \\mbox{for other $s$},\n\t\\end{aligned}\n\t\\right.\n\\end{equation} \nwhere $j=\\displaystyle \\operatorname*{arg\\,max}_{i \\in \\{1,\\ldots, n\\}} \\left\\{ s_{i} \\leq s \\right\\}.$\nLet us define\n\\begin{subequations}\n\\begin{align}\n& w(s) =\\int_{0}^{s} z(t) \\mathrm{d} t, \\\\\n& w_0 =w(\\rho)=\\int_{0}^{\\rho} z(t) \\mathrm{d} t= \\frac{z_0}{\\mu}\\left(1- \\e^{\\textstyle -\\mu \\rho} \\right).\n\\end{align}\n\\end{subequations}\nWe should emphasise that for $s> s_2,$ $w(s)\\neq z_0\\left(1- \\e^{\\textstyle -\\mu s} \\right)\/\\mu$ due to the administration of the next drug dose.\nFurthermore, from the definition of function $z$ we have \n\\begin{equation} \\label{periodicity}\n\\begin{aligned}\nw(s) & = \\int_{0}^{s_j} z(t) \\mathrm{d} t + \\int_{s_{j}}^{s} z(t) \\mathrm{d} t = w(s_{j}) + \\int_{0}^{s-s_{j}} z(t) \\mathrm{d} t \n\\approx (j-1)w_{0}+ w(s-s_{j}) \\\\\n& \\approx \\left\\{ \\begin{array}{ll}\n(j-1)w_{0}+ \\frac{z_0}{\\mu}\\left(1- \\e^{\\textstyle -\\mu (s-s_{j})} \\right) & s-s_{j} \\leq \\rho,\\\\\njw_0 & \\textrm{otherwise,}\n\\end{array} \\right.\n\\end{aligned}\n\\end{equation}\nwhere $j=\\displaystyle \\operatorname*{arg\\,max}_{i \\in \\{1,\\ldots, n\\}} \\left\\{ s \\geq s_{i} \\right\\}.$\nHence for $s>s_{n}+\\rho$ the formulae for rescaled proliferating and the damaged part of tumour take the~form\n\\begin{subequations}\\label{xy}\n\\begin{eqnarray} \\label{x}\nx(s) &= & x_0 \\e^{\\textstyle s-w(s)}=x_0 \\e ^{\\textstyle s - n w_0 }, \\\\\n\\label{y}\ny(s) & = & \\int_{0}^{s} \\e^{\\textstyle -\\frac{s-t}{k}} x(t) z(t) \\mathrm{d} t. \n\\end{eqnarray}\n\\end{subequations}\nWe look for $s_{\\textrm{TRP}}$ which fulfils condition \\eqref{formula s_ttp}. Using Eqs. \\eqref{xy} we have \n\\begin{equation}\nk\\e^{ \\textstyle \\tilde{k}s_{\\textrm{TRP}} -n w_0} = \\int_{0}^{s_{\\textrm{TRP}}} \\e^{\\textstyle \\tilde{k}t - w(t) } z(t) \\mathrm{d} t,\n\\end{equation}\n\\begin{equation} \\label{sTTP1}\ns_{\\textrm{TRP}}=\\frac{1}{\\tilde{k}}\\Bigg[ nw_0 + \\ln \\left( \\frac{1}{k} \\int_{0}^{s_{\\textrm{TRP}}} \\e^{\\textstyle \\tilde{k}t - w(t) } z(t) \\mathrm{d} t \\right) \\Bigg],\n\\end{equation}\nwhere $\\tilde{k}=1+1\/k$. \nAs a~result of approximations \\eqref{z approximation} and \\eqref{periodicity} we conclude that for \\mbox{$s\\geq s_{n}+ \\rho$} \n\\begin{align} \\label{rhs}\n& \\displaystyle \\int_{0}^{s} \\e^{\\textstyle \\tilde{k}t - w(t) } z(t) \\mathrm{d} t =\n\\displaystyle \\sum_{j=1}^{n} z_0 \\displaystyle \\int_{s_{j}}^{s_{j}+\\rho} \\e^{\\textstyle \\tilde{k}t - w(t)} \\e^{\\textstyle -\\mu(t-s_{j})} \\mathrm{d} t \n\\nonumber \\\\\n & = \\displaystyle z_0 \\sum_{j=1}^{n} \\e^{\\textstyle -(j-1)w_{0} +\\tilde{k}s_{j}} \\displaystyle \\int_{s_{j}}^{s_{j}+\\rho} \\e^{\\textstyle \n(\\tilde{k} -\\mu)(t-s_{j}) + \\frac{\\textstyle z_0}{\\textstyle \\mu}\\left(\\e^{\\textstyle -\\mu (t-s_{j})} -1 \\right)} \\mathrm{d} t \n\\nonumber \\\\\n& = z_0 \\left( \\sum_{j=1}^{n} \\e^{\\textstyle -(j-1)w_{0} +\\tilde{k}s_{j}} \\right) \\int_{0}^{\\rho}\n\\e^{\\textstyle \n(\\tilde{k} -\\mu)t + \\frac{\\textstyle z_0}{\\textstyle \\mu}\\left( \\e^{\\textstyle -\\mu t}-1\\right)} \\mathrm{d} t. \n\\end{align} \nUsing Taylor expansion of an exponential function for $t<1\/\\mu$ we approximate \n\\[\n\t\\e^{-\\mu t}-1 \\approx \\begin{cases}\n\t\t\t-\\mu t & 0\\le t<1\/\\mu, \\\\\n\t\t\t-1 & t\\ge 1\/\\mu,\n\t\t\\end{cases}\n\\]\nand obtain\n\\begin{flalign} \\label{integral}\n& \\displaystyle \\int_{0}^{\\rho}\n\\e^{\\textstyle (\\tilde{k} -\\mu)t + \\frac{\\textstyle z_0}{\\textstyle \\mu}\\left( \\e^{\\textstyle -\\mu t}-1\\right)} \\mathrm{d} t \\approx\n\\displaystyle \\int_{0}^{\\frac{1}{\\mu}} \\e^{\\textstyle (\\tilde{k} -\\mu)t - z_0 t} \\mathrm{d} t \n+ \\int_{\\frac{1}{\\mu}}^{\\rho} \\e^{\\textstyle (\\tilde{k} -\\mu)t - \\frac{z_0}{\\mu}} \\mathrm{d} t \\nonumber \\\\\n&= \\frac{1}{\\tilde{k}-\\mu -z_{0}}\n\\left( \\e^{\\textstyle \\frac{\\tilde{k}-\\mu-z_0}{\\mu}} -1 \\right) +\n\\frac{1}{\\tilde{k}-\\mu} \\left( \\e^{\\textstyle (\\tilde{k}-\\mu)\\rho -\\frac{z_0}{\\mu}}-\\e^{\\textstyle\\frac{\\tilde{k}-\\mu-z_0}{\\mu}} \\right) \\nonumber \\\\\n&= \\frac{1}{\\left(\\tilde{k}-\\mu -z_{0}\\right)\\left(\\tilde{k}-\\mu\\right)} \\Bigg[ z_0 \\e^{\\textstyle \\frac{\\tilde{k}-\\mu-z_0}{\\mu}} -\\tilde{k} +\\mu + \n\\left(\\tilde{k} -\\mu -z_0\\right) \\e^{\\textstyle \\left(\\tilde{k}-\\mu\\right)\\rho - \\frac{z_0}{\\mu}} \n\\Bigg].\n\\end{flalign}\nTo compute the sum term in Eq.~\\eqref{rhs} we need the relation between dose indexes $j$~and the times of their administration $s_{j}.$ Taking into consideration assumptions from Sec. \\ref{sec:estimates} and Eqs.~(\\ref{dosingtimes}-\\ref{dosingindexes}) we get\n$ s_{j}= \\big[(i-1)r+ mT\\big]\\rho, $\nwhere $m\\in \\{0, n\/p -1\\} $ is the~number of completed chemotherapy cycles before dose $s_{j},$ $j \\in \\{1,\\ldots, n\\}$ and $i \\in \\{1, \\ldots, p\\}$ is an index of dose within the~TMZ~cycle.\nAs a~result we obtain\n\\begin{multline} \n\\sum_{j=1}^{n} \\exp \\left(-(j-1)w_{0} +\\tilde{k}s_{j}\\right) \\\\\n= \\sum_{m=0}^{ n\/p-1} \\sum_{i=1}^{p}\n\\exp \\left( m \\left(-pw_{0} +\\tilde{k}\\rho T \\right)+ (i-1)(-w_{0} +\\tilde{k}\\rho r) \\right)\n \\\\\n = \\frac{1-\\e^{\\textstyle \\left(-p w_{0} +\\tilde{k}\\rho T \\right) \\frac{n}{p} }}{1-\\e^{\\textstyle -pw_{0} + \\tilde{k}\\rho T}} \n\\cdot \\frac{1-\\e^{\\textstyle \\left(-w_{0} +\\tilde{k}\\rho r \\right)p}}{1-\\e^{\\textstyle -w_{0} +\\tilde{k}\\rho r}},\n \\end{multline}\nwhere $n$ is the total number of doses and $n\/p$ is the total number of chemotherapy cycles as previously stated. \nThen using Eqs. \\eqref{sTTP1}, \\eqref{rhs} and \\eqref{integral} we get \n\\begin{align}\n& s_{\\textrm{TRP}} = \n\\frac{n w_0 }{\\tilde{k}}\n+ \\frac{1}{\\tilde{k}}\\ln \n\\left\\{ z_0 \\e^{\\textstyle \\frac{\\tilde{k}-\\mu-z_0}{\\mu}} -\\tilde{k} +\\mu + \n\\left(\\tilde{k} -\\mu -z_0\\right) \\e^{\\textstyle \\left(\\tilde{k}-\\mu\\right)\\rho - \\frac{ z_0}{\\mu}} \n\\right\\} \\nonumber \\\\\n& + \\frac{1}{\\tilde{k}}\\ln \\left\\{\n\\frac{\\textstyle z_0\\left( 1-\\e^{\\textstyle \\left(-pw_{0} +\\tilde{k}\\rho T \\right) \\frac{n}{p} }\\right) \n\\left(1-\\e^{\\textstyle \\left(-w_{0} +\\tilde{k}\\rho r\\right)p}\\right)}\n{\\textstyle k\\left( 1-\\e^{\\textstyle -pw_{0} + \\tilde{k}\\rho T} \\right) \n\\left(1-\\e^{\\textstyle -w_{0} +\\tilde{k}\\rho r}\\right) \n\\left(\\tilde{k}-\\mu -z_{0} \\right)\\left(\\tilde{k}-\\mu\\right)} \\right\\} .\n\\end{align}\nIn terms of the initial time scale, the time to tumour progression can be estimated as \n$t_{\\textrm{TRP}}=s_{\\textrm{TRP}}\/ \\rho,$ giving the final result\n\\begin{align} \\label{formula}\n& t_{\\textrm{TRP}} = \n\\frac{n w_0 }{\\tilde{k}\\rho}\n+ \\frac{1}{\\tilde{k}\\rho}\\ln \n\\left\\{ z_0 \\e^{\\textstyle \\frac{\\tilde{k}-\\mu-z_0}{\\mu}} -\\tilde{k} +\\mu + \n\\left(\\tilde{k} -\\mu -z_0\\right) \\e^{\\textstyle \\left(\\tilde{k}-\\mu\\right)\\rho - \\frac{ z_0}{\\mu}} \n\\right\\} \\nonumber \\\\\n& + \\frac{1}{\\tilde{k}\\rho}\\ln \\left\\{\n\\frac{\\textstyle z_0\\left( 1-\\e^{\\textstyle \\left(-pw_{0} +\\tilde{k}\\rho T \\right) \\frac{n}{p} }\\right) \n\\left(1-\\e^{\\textstyle \\left(-w_{0} +\\tilde{k}\\rho r\\right)p}\\right)}\n{\\textstyle k\\left( 1-\\e^{\\textstyle -pw_{0} + \\tilde{k}\\rho T} \\right) \n\\left(1-\\e^{\\textstyle -w_{0} +\\tilde{k}\\rho r}\\right) \n\\left(\\tilde{k}-\\mu -z_{0} \\right)\\left(\\tilde{k}-\\mu\\right)} \\right\\}.\n\\end{align}\nEq. \\eqref{formula} gives the time to radiological progression as a~function of parameters with relevant biological and\/or therapeutical meaning.\nSince TRP is a~metric of practical relevance, it is very interesting that it is possible to estimate its value analytically. \n\n\\subsection{Validation}\nEq. \\eqref{formula} has been obtained via a~number of approximations and thus it is relevant to compare its predictions with the results of the original equation \\eqref{ode} and real patient data.\n\nTo do the latter we first fix the treatment parameters to match those routinely used for TMZ therapeutic schedules. Since TMZ~is given on 5 consecutive days in cycles consisting of 28 days, we get $p=5$, $T=28$, $r=1$. Taking dose per fraction to be 150~mg\/m$^2$ and the other parameters as in Sec.~\\ref{sec:estimates} we can then estimate the time of progression for the individual patients studied previously (accounting for the number of cycles received by each patient). \n\nFig.~\\ref{fig:tTTP}~shows how well formula \\eqref{formula} estimates the response to chemotherapy for three patients chosen from our database. The task of comparing simulation results with real patients MRI data requires a lot of caution. In particular, we need to take into account the limitations of calculations of tumour volume using the method of three largest diameters. The method is only an approximation of real tumour volume and its accuracy is limited by slice thickness, changes in head position \\cite{Schmitt} or even by perception of medical doctor who calculate these diameters. However in this case one can conclude that we have obtained a satisfactory result in fit. In the future we hope that MRI data will be analysed through automatic segmentation, \\textit{e.g.}~with algorithm suggested by Porz \\textit{et~al.}\\xspace \\cite{Porz} and the real tumour volume will be calculated more accurately. \n\nFig.~\\ref{fig:diff} presents relative differences between TRP from the estimated formula and simulations for different sets of parameters, suggesting a very good approximation.\n\n\\begin{figure}[h!p]\n\\centering\n\\includegraphics[width=0.525\\textwidth]{10_TRP}\n\\includegraphics[width=0.525\\textwidth]{57_TRP}\n\\includegraphics[width=0.525\\textwidth]{151_TRP}\n\\caption{Tumour volume evolution for three patients treated with TMZ. \nVertical dashed lines mark the start and the end of TMZ~treatment. \nCircles denote the volumes obtained from MRIs and solid lines the results of the best fit using Eqs.~\\eqref{ode}. \n(top) Woman treated with 6~TMZ cycles, $\\alpha=0.199094$ml\/$\\mu$g\/day, $\\rho = 0.00022\/$day, $k = 0.075644.$\n(center) Man treated with 11~TMZ~cycles, $\\alpha = 0.17367$ml\/$\\mu$g\/day, $\\rho = 0.000338\/$day, $k = 0.019279.$ \n(bottom) Man treated with 4~TMZ~cycles, $\\alpha=0.236439$ml\/$\\mu$g\/day, $\\rho = 0.000701\/$day, $k = 0.257806.$\nThe times to radiological progression computed using Eq.~\\eqref{formula} are marked with vertical dashed-dotted lines, showing a very good agreement with the data and the simulations of Eqs.~\\eqref{ode}. The relative differences between $t_{\\textrm{TRP}}$ calculated from Eq.~\\eqref{formula} and from Eqs.~\\eqref{ode} were respectively $0.042043, \\ 0.041671$ and $0.037481$ years, for the selected patients. \n}\n\\label{fig:tTTP}\n\\end{figure}\n\n\\begin{figure}[h!p]\n\t\\begin{center}\n\t\t\\includegraphics[trim={0.6cm 0.1cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth]{diff_k_rho0004alpha4}\n\t\t\\includegraphics[trim={0.6cm 0.1cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth]{diff_k_rho0008alpha8}\n\t\t\\ \\\\ \\ \\\\\t\t\n\t\t\\includegraphics[trim={0.6cm 0.1cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth, height=0.2\\textheight]{diff_rho_k3alpha8}\t\t\n\t\t\\includegraphics[trim={0.6cm 0.1cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth, height=0.2\\textheight]{diff_rho_k6alpha4}\n\t\t\\ \\\\ \\ \\\\\t\n\t\t\\includegraphics[trim={0.6cm 0cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth, height=0.2\\textheight]{diff_alpha_k6rho0008}\n\t\t\\includegraphics[trim={0.6cm 0cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth, height=0.2\\textheight]{diff_alpha_k3rho0004}\n\t\t\\caption{Relative percentage difference between time to radiological progression estimated from simulations of model~\\eqref{ode} and formula \\eqref{formula}. We considered 12 cycles of TMZ~as in the standard fractionation scheme (see Sec.~\\ref{sec:parameters}) for virtual patients with LGG having an initial volume of 40 cm$^3$. Results for $\\rho = 0.0004$\/day, $\\alpha=0.4$ml\/$\\mu$g\/day, $k \\in [0.02, 1]$ (left) and $\\rho = 0.0008$\/day, $\\alpha=0.8$ml\/$\\mu$g\/day, $k \\in [0.02, 1].$ (center) Results for $\\alpha=0.8$ml\/$\\mu$g\/day, $k=0.3$ for $\\rho \\in [0.2,8] \\times 10^{-3}$\/day (left) and $\\alpha=0.8$ml\/$\\mu$g\/day, $k=0.3$ for $\\rho \\in [0.2,3] \\times 10^{-3}$\/day. (bottom) Results for $\\rho = 0.0008$\/day, $k=0.6$ and $\\alpha \\in [0.3,1.5]$ml\/$\\mu$g\/day (left) and $\\rho = 0.0004$\/day, $k=0.3$ and $\\alpha \\in [0.15,1.5]$ml\/$\\mu$g\/day (right).\n\t\t}\n\t\t\\label{fig:diff}\n\t\\end{center}\n\\end{figure} \n\n\\subsection{The study of tumour response for other chemotherapy protocols}\nWe have also verified that Eq.~\\eqref{formula} provides a good approximation of TRP for model \\eqref{ode} for other fractionation schemes. Fig.~\\ref{fig:tTTP1} shows some examples. \n\n\\begin{figure}[h!t]\n\\begin{center}\n\\includegraphics[trim={0.8cm 0cm 1.8cm 0.9cm},clip=true,width=0.46\\textwidth]{TRP2}\n\\includegraphics[trim={0.8cm 0cm 1.8cm 0.9cm},clip=true,width=0.46\\textwidth]{TRP5}\n\\caption{Tumour volume evolution for virtual patients simulated from Eqs.~\\eqref{ode}. Times to radiological progression estimated from Eq.~\\eqref{formula} are marked with vertical dashed-dotted lines. Values of parameters were $k =0.5$, $\\alpha = 0.4$ml\/$\\mu$g\/day and $\\rho = 0.005\/$day. The start and the end of TMZ~treatment are marked with vertical dashed lines. (left) 9 TMZ~cycles of 34 days with doses given every 2 days for a~total of 10 doses per cycle. The dose per fraction was $d$=100~mg\/m$^2.$ (right) 18~TMZ~cycles of 77 days with doses given every 7 days for a~total of 10 doses per cycle. The dose per fraction was $d$=50 mg\/m$^2.$ Relative differences between times free of progression calculated from Eq.~\\eqref{formula} and estimated from simulations were 0.026756 and 0.025303 years, respectively.}\n\\label{fig:tTTP1}\n\\end{center}\n\\end{figure} \n\n\\section{Discussion and therapeutic implications}\n\nDue to the observed clinical significance of TMZ there is an increasing interest in studying its characteristics. Up to now there have been a great deal of relevant research into the pharmacokinetic\/pharmacodynamic properties of TMZ \\cite{Baker,Hammond,Ostermann,Portnow}, its specific mechanism of action \\cite{Marchesi,Roos,Barciszewska,Agarwala} and modelling its concentration dynamics \\emph{in vitro} and \\emph{in vivo} \\cite{Zhou,Rosso,Ballesta}. However there are fewer mathematical studies of patient response to TMZ in LGGs. \nHere we intended to construct a mathematical model which would enable understanding of delayed response to chemotherapy observed in LGGs without using an excessive number of unknown parameters. Cells were assumed to grow logistically, chemotherapy drug kinetics and its effect on glioma cells was based on TMZ concentration in brain tissue \\cite{Portnow} and clinical observations \\cite{Bent1,Ricard,Chamberlain}. Note that even the authors of the very complicated model \\cite{Ballesta}, constructed for the purpose of describing pharmacokinetics and pharmacodynamis of TMZ, validated their model for human cerebral tumours on the basis of data of Portnow \\textit{et~al.}\\xspace \\cite{Portnow}.\nIt is remarkable that a~simple model such as the one presented here with essentially only three unknown parameters ($\\alpha, \\rho, k$) is able to describe the response of real patients to a~variable number of cycles of TMZ.\n\nThe model also shows a~correlation between a~short time to radiological progression and a~poor virtual patient outcome. We may conclude that time to radiological progression can be useful as a~measure of tumour aggressiveness due to its dependence on tumour-specific parameters: proliferation rate $\\rho$ and TMZ~cell kill strength $\\alpha$ (see Figs.~\\ref{fig:relacje},\\ref{fig:surf}). \nOur data on patients treated with first-line TMZ~suggests likewise that despite other therapies used in the follow-up, patients who \nhad shorter estimated TRP had worse prognosis. Such observation has been made for radiotherapy \\cite{Pallud1, Ducray}, but so far no similar analysis of response to TMZ~has been done. The velocity of tumour decrease after radiotherapy (or equivalently time of progression-free survival) is strongly associated with the risk of rapid progression and poor overall survival. Here we suggest a similar result for the response to chemotherapy, namely that short time to radiological progression results in shorter overall survival. \n\nThis outcome makes us think of the possibility of using chemo\\-therapy to probe tumours, hence providing estimates of tumour-specific parameters $\\rho$ and $\\alpha.$ We could apply a small number of cycles of TMZ~causing minimal toxicity and monitor with MRI~the tumour response to chemotherapy. In order to assure the reasonable measurement error, we would need at least two measurements before and three after TMZ onset. We predict that the time horizon would be of around 2 years from the time of the first MRI. We believe it could be feasible as even up to now there were cases when MRI was performed three times a year. Based on our database there will be no progression at this time horizon. Such a procedure can be used as a~novel way to assess tumour aggressiveness. Our mathematical model suggests that tumour which attains its minimal volume early after a short course of TMZ~treatment (has shorter TRP)~may be more aggressive, therefore in such a~case the remaining TMZ~dose has to be finished as soon as possible and other therapeutic options (further surgery if feasible or radiotherapy) should be considered. Such a concept can be supported also by the\\emph{in vitro} results of Roos \\textit{et~al.}\\xspace \\cite{Roos}, who shown that higher proliferation rates accelerate apoptosis after TMZ treatment, thus in terms of our idea, shorter TRP.\n\nThis idea resembles that described in \\cite{Victor}, but with chemotherapy instead of radiotherapy.~Although the modelling principles are similar, from the clinical point of view the use of TMZ~is a much more interesting as a way to probe a tumour than the use of radiotherapy as its side-effects are long-term and non-reversible. On the other hand, TMZ~has significantly lower and largely reversible side-effects. Moreover, radiotherapy is well known to induce changes in the MRI~images due to inflammation which may distort the analysis of the tumour response. Finally, TMZ~is easily managed because it is administered orally.\n\n\\section{Conclusions}\nWe have build a~model which is simple from the mathematical point of view, but which incorporates the basic biological features of LGG growth and the response to chemotherapy. \n\nThe model is able to describe response to TMZ with a minimal number of parameters and suggests that tumours having a shorter time to radiological response after TMZ~treatment~may be more aggressive in terms of proliferation potential.~We plan to reassess this observation using a~larger data set, if possible.~In this case we would like also to verify whether the survival curves obtained can be fitted for a~cohort of virtual patients, as done by Kirkby \\textit{et~al.}\\xspace \\cite{Kirkby2007}. \n\nMoreover, we propose a~paradigm for probing tumour with TMZ~which could be used in clinical practice. We have also found an equation giving the time free of progression as a~function of the relevant biological and therapeutic parameters. In future studies it may be helpful in designing treatment schedules giving the longest TRP possible with the additional condition for the toxicity level.\n\nIn order to address other clinically relevant issues and the biological perspective several improvements to the present model will be implemented in the near future. First it may be appropriate to include more biological details such as the potential existence of the so-called cancer initiating cells or cancer stem cells. Also incorporating different phenotypes may be relevant to describe the process of acquiring resistances to TMZ.\n\nIt would also be interesting to find the minimal doses and minimal frequency of therapy such that the solution (namely tumour mass) stays below a~given threshold. In principle, survival could be improved and chemoresistance deferred using metronomic fractionations, \\emph{i.e.} schedules consisting of many, equally spaced and generally low doses of chemotherapeutic drugs without extended rest periods (see \\textit{e.g.}~\\cite{Benzekry,Andre}).\n\nWe hope that optimized cancer treatment protocols on the basis of models such as the one presented in this paper may become in the future a~standard element of personalised medicine.\n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{}{}{0pt}{\\Large\\scshape\\bfseries\\filcenter}\n\\tableofcontents\n\\titleformat\\section{}{}{0pt}{\\Large\\scshape\\bfseries\\filcenter\\thesection{} - }\n\n\\section{Introduction}\\label{intro}\n\nThe aim of this paper is to derive an error estimate for approximate solutions of the compressible barotropic Navier-Stokes equations obtained by a discretisation scheme.\nThese equations are posed on the time-space domain $Q_T=(0,T)\\times\\Omega$, where $\\Omega$ is a bounded polyhedral domain of $\\R^d$, $d=2,3$ and $T>0$, and read:\n\\begin{subequations}\n\t \\begin{align}\n\t\t& \\partial_t \\vr + \\dv ( \\vr \\bu) = 0, \\label{cont2} \\\\\n\t\t& \\partial_t (\\vr\\bu) +{\\rm div}(\\vr\\bu\\otimes\\bu) - \\mu \\Delta \\bu -(\\mu+\\lambda) \\nabla \\dv \\bu+ \\nabla_x p(\\vr) = \\bm{0}, \\label{mov2}\n\t\\end{align}\n\t\\label{pbcont}\n\\end{subequations}\nsupplemented with the initial conditions\n\\begin{equation}\\label{ci1}\n\t \\vr(0,x) = \\vr_0 (x),~\\vr\\bu(0,x) =\\vr_0\\bu_0,\n\\end{equation}\nwhere $\\varrho_0$ and $\\bu_0$ are given functions from $\\Omega$ to $\\R_+$ and $\\R^d$ respectively, and boundary conditions\n\\begin{equation}\\label{ci2}\n\t\\bu_{|(0,T)\\times\\partial \\Omega} =0.\n\\end{equation}\nIn the above equations, the unknown functions are the scalar density field $\\vr(t,x)\\ge 0$ and vector velocity field $\\bu=(u_1,\\ldots,u_d)(t,x)$, where $t \\in (0,T)$ denotes the time and $x\\in \\Omega$ is the space variable.\nThe viscosity coefficients $\\mu$ and $\\lambda$ are such that\n\\begin{equation}\\label{visc}\n\t\\mu > 0,~ \\lambda + \\frac{2}{d}\\mu \\ge 0.\n\\end{equation}\nThe pressure $p$ is a given by an equation of state, that is a function of density which satisfies\n\\begin{equation}\\label{hypp}\n\tp \\in C([0,\\infty)) \\cap C^1(0,\\infty),\\;p(0)=0,\\; p'(\\vr)>0.\n\\end{equation}\nIn addition to (\\ref{hypp}), in the error analysis, we shall need to prescribe the asymptotic behavior of the pressure at large densities\n\\begin{equation}\\label{pressure1}\n\t\\lim_{\\vr\\to \\infty}\\frac {p'(\\vr)}{\\vr^{\\gamma-1}}=p_\\infty>0\\quad\\mbox{with some } \\gamma\\ge 1;\n\\end{equation}\nfurthermore, if $\\gamma <2$ in \\eqref{pressure1}, we need the additional condition (for small densities):\n\\begin{equation}\\label{pressure2}\n\\liminf_{\\vr\\to 0}\\frac {p'(\\vr)}{\\vr^{\\alpha+1}}=p_0>0\\quad\\mbox{with some }\\alpha\\le 0.\n\\end{equation}\n\nThe main underlying idea of this paper is to derive the error estimates for approximate solutions of problem \\eqref{pbcont}{--(\\ref{ci2})} obtained by time and space discretization by using the discrete version of the {\\em relative energy method} { introduced on the continuous level in \\cite{FeJiNo,FENOSU, FENO7}. In spite of the fact that the relative energy method looks at the first glance pretty much similar to the widely used {\\em relative entropy method} (and both approaches translate the same thermodynamic stability conditions), they are very different in appearance and formulation and may provide different results.}\nThe notions of relative entropy and relative entropy inequality were first introduced by Dafermos \\cite{Daf4} in the context of systems of conservation laws and in particular for the compressible Euler equations.\nThe relative energy functional was suggested and successfully used for the investigation of the stability of weak solutions to the equations of viscous compressible and heat conducting fluids in \\cite{FENO7}.\nIn contrast with the relative entropy of Dafermos, { for the viscous and heat conducting fluids}, the relative energy approach is able to provide the structural stability of weak solutions, while the relative entropy approach fails in this case.\n\\\\ \\\\\nBoth functionals coincide { (modulo a change of variables)} in the case of (viscous) compressible flows in the barotropic regime. The relative energy functional and the intrinsic version of the relative energy inequality have been recently employed to obtain several stability results for the weak solutions to these equations, including the weak strong uniqueness principle, see \\cite{FeJiNo,FENOSU}.\nNote that particular versions of the relative entropy inequality with particular specific test functions had been previously derived in the context of low Mach number limits, see e.g. \\cite{MASS,WanJia}.\n\\\\ \\\\\nThe { discrete} version of the Dafermos relative entropy was employed in the non viscous case to derive an error estimate for the numerical approximation to a hyperbolic system of conservation laws and, in particular, to the compressible Euler equations \\cite{CancesMathisSeguin2014Relative}.\nIn this latter paper, the authors assume an $L^\\infty$ bound for the discrete solution, which is uniform with respect to the size of the space and time disretization (usually called stability hypothesis), that is not provided by the discrete equations. { The same method with the same severe hypotheses have been used in\n\\cite{YOVAN} to treat the compressible Navier-Stokes equations}.\nThe error analysis in the present paper relies on the theoretical background introduced in \\cite{FeJiNo} and yields an {\\it unconditional result}; in particular, {\\it we do not need any assumed bound} on the solution to get the error estimate.\n\\\\ \\\\\nThe mathematical analysis of numerical schemes for the discretization of the steady and\/or non steady compressible Navier-Stokes and\/or compressible Stokes equations has been the object of some recent works.\nThe convergence of the discrete solutions to the weak solutions of the compressible stationary Stokes was shown for a finite volume-- non conforming P1 finite element \\cite{GHL2009iso,EGHL2010isen,eym-10-convII} and for the wellknown MAC scheme which was introduced in \\cite{har-65-num} and is widely used in computational fluid dynamics {\\ (see e.g. \\cite{Sun-2014})}.\nThe unsteady Stokes problem was also discretized by some other discretization schemes on a reformulation of the problem, which were proven to be convergent \\cite{KK2010Stokes,KK2011Stokes,KK2012Stokes}.\n{ The unsteady barotropic Navier-Stokes equations was recently { investigated} in \\cite{KARPER} in the case $\\gamma>3$ (there is a real difficulty in the realistic case $\\gamma \\le 3$ arising from the treatment of the non linear convective term).}\nHowever, in these works, the rate of convergence is not provided; in fact, to the best of our knowledge, no error analysis has yet been performed for any of the numerical schemes that have been designed for the compressible Navier-Stokes equations, in spite of its great importance for the numerical analysis of the equations and for the mathematical simulations of compressible fluid flows.\nWe present here a general technique to obtain an error analysis and apply it to one of the available numerical schemes.\nTo the best of our knowledge, this is the first result of this type in the mathematical literature on the subject.\n\\\\ \\\\\nTo achieve the goal, we systematically use the relative energy method on the discrete level.\nFrom this point of view, this paper is as valuable for the introduced methodology as for the result itself.\nHere, we apply the method to the scheme of \\cite{KARPER}.\n{ In spite of the fact that this latter scheme is not used in practice (see e.g. \\cite{KHL2013baro} for a related schemes used in industrial codes), we begin the error analysis with the scheme \\cite{KARPER}} because of its readily available convergence proof.\nIn fact, we aim to use this approach to investigate the numerical errors of less academic numerical schemes, such as the finite volume -- non conforming P1 finite element \\cite{GHL2010drift,KHL2013baro,GGHL2008baro,GHKLL2011allspeed} or the MAC scheme \\cite{bab-11-dis,her-14-inc}.\n\\\\ \\\\\nThe paper is organized as follows.\nAfter recalling the fundamental setting of the problem and the relative energy inequality in the continuous case in Section \\ref{1}, we proceed in Section \\ref{2} to the discretization: we introduce the discrete functional spaces and the definition of the numerical scheme, { and state the main result of the paper, that is the error estimate formulated} in Theorem \\ref{Main}.\nThe remaining sections are devoted to the proof of Theorem \\ref{Main}:\n\\begin{list}{$\\bullet$}{}\n\\item\n In Section \\ref{4} we recall the existence theorem for the numerical scheme (Lemma \\ref{Theorem1}) and derive estimates provided by the scheme. \n\\item In Section \\ref{5}, we derive the discrete intrinsic version of the relative energy inequality for the solutions of the numerical scheme (see Theorem \\ref{Theorem4}).\n\\item The relative energy inequality is transformed to a more convenient form in Section \\ref{6}, see Lemma \\ref{refrelenergy}.\n\\item Finally, in Section \\ref{7}, we investigate the form of the discrete relative energy inequality with the test function { being a strong} solution to the original problem. This investigation is formulated in Lemma \\ref{strongentropy} and finally leads to a Gronwall type estimate formulated in Lemma \\ref{Gronwall}. The latter yields the error estimates and finishes the proof of the main result.\n \\end{list}\n Fundamental properties of the discrete functional spaces\nneeded throughout the paper are reported in Appendix (Section \\ref{3}). Some of them (especially those referring to the $L^p$ setting, $p\\neq 2$ that are not currently available in the mathematical literature) are proved. Section \\ref{3} is therefore of the independent interest.\n\n\\section{The continuous problem}\\label{1}\nThe aim of this section is to recall some fundamental notions and results.\nWe begin by the definition of weak solutions to problem \\eqref{pbcont}-- \\eqref{ci2}.\n\n\\begin{df}[Weak solutions]\\label{ws}\n{ Let $\\varrho_0 : \\Omega \\to [0, +\\infty) $ and $\\bu_0 :\\Omega \\to \\R^d$ with finite energy $E_0=\\int_\\Omega (\\frac{1}{2} \\varrho_0 |\\bu_0|^2 + {{H}}(\\vr_0)) \\dx$ and finite mass $0d\/(d-1)$, see Lions \\cite{LI4} for \"large\" values of $\\gamma$, Feireisl and coauthors \\cite{FNP} for $\\gamma>d\/(d-1)$.\n\n\\medskip\n\nLet us now introduce the notion of relative energy.\nWe first introduce the function\n\\begin{equation}\\label{E}\n\t\\begin{array}{lll}\n\t\t&E: &[0,\\infty)\\times(0,\\infty)\\to \\R,\\\\\n\t\t& & (\\vr,r) \\mapsto E(\\vr|r)=H(\\vr)-H'(r)(\\vr-r) -H(r),\n\t\\end{array}\n\\end{equation}\nwhere $H$ is defined by \\eqref{H}.\nDue to the monotonicity hypothesis in (\\ref{hypp}), $H$ is strictly convex on $[0,\\infty)$, and therefore\n\\[\n\tE(\\vr|r)\\ge 0\\quad\\mbox{and}\\quad E(\\vr|r)=0\\;\\Leftrightarrow\\;\\vr=r.\n\\]\nIn order to measure a ``distance'' between a weak solution $(\\vr,\\bu)$ of the compressible Navier-Stokes system and any other state $(r,\\bU)$ of the fluid , we introduce the relative energy functional, defined by\n\\begin{equation}\\label{ent}\n\t{\\cal{E}}(\\vr,\\bu\\Big|r,\\bU) = \\int_\\Omega \\Big(\\frac{1}{2} \\vr| \\bu -\\bU|^2 + E(\\vr\\,|\\,r)\\Big) \\dx.\n\\end{equation}\nIt was proved recently in \\cite{FeJiNo} that, provided assumption \\eqref{hypp} holds, any weak solution satisfies the following so-called relative energy inequality\n\\begin{equation}\\label{p5}\n\\begin{aligned}\n\t&{\\mathcal E} \\left( \\vr, \\bu \\Big| r, \\bU \\right) (\\tau) -{\\mathcal E} \\left( \\vr, \\bu \\Big| r, \\bU \\right)(0)+ \\int_0^\\tau \\intO{ \\Big( \\mu|\\nabla(\\bu-\\bU)|^2 +(\\mu+\\lambda)|\\dv(\\bu-\\bU)|^2 \\Big) } \\dt \\\\\n\t&\\phantom{{\\mathcal E} \\left( \\vr, \\bu \\right)} \\le \\int_0^\\tau \\intO{ \\Big( \\mu \\nabla\\bU:\\nabla(\\bU-\\bu) +(\\mu+\\lambda)\\dv \\bU \\dv(\\bU-\\bu) \\Big) } \\dt \\\\\n\t & \\phantom{{\\mathcal E} \\left( \\vr, \\bu \\right) ) \\le } + \\int_0^\\tau \\intO{ \\vr \\partial_t \\bU\\cdot (\\bU - \\bu )}\\dt + \\int_0^\\tau\\intO{\\vr\\bu {\\cdot} \\nabla \\bU \\cdot(\\bU - \\bu )}\\dt \\\\\n\t & \\phantom{{\\mathcal E} \\left( \\vr, \\bu \\right) \\le } -\\int_0^\\tau \\intO{p(\\vr)\\dv \\bU} \\dt + \\int_0^\\tau \\intO{ (r - \\vr) \\partial_t H'(r)}\\dt -\\int_0^\\tau \\intO{\\vr \\nabla H'(r) \\cdot\\bu }\\dt\n\\end{aligned}\n\\end{equation}\nfor a.a. $\\tau\\in (0,T)$, and for any pair of test functions\n\\[\nr \\in C^1 ([0,T] \\times \\Ov{\\Omega}),\\ r > 0,\\ \\bU \\in C^1([0,T] \\times \\Ov{\\Omega};\\R^3 ),\\ \\bU|_{\\partial \\Omega} = 0.\n\\]\n\n\nThe stability of strong solutions in the class of weak solutions is stated in the following proposition.\n\\begin{Proposition}[Estimate on the relative energy]\\label{proposition}\n Let $\\Omega$ be a Lipschitz domain. Assume that the viscosity coefficients satisfy assumptions (\\ref{visc}), that the pressure $p$ is a twice continuously differentiable function on $(0,\\infty)$ satisfying (\\ref{hypp}) and (\\ref{pressure1}), and that $(\\vr,\\bu)$ is a weak solution to problem \\eqref{pbcont}--\\eqref{ci2} emanating from initial data $(\\vr_0\\ge 0,\\bu_0)$, with finite energy $E_0$ and finite mass $M_0{\\ =\\int_\\Omega\\vr_0{\\rm d} x}>0$.\nLet $(r,\\bU)$ in the class\n\\begin{equation} \\label{r,U}\n\t\\left\\{\\begin{array}{l}\n\t\t \\displaystyle r\\in C^1([0,T]\\times\\overline\\Omega),\\; 0<\\underline r =\\min_{(t,x)\\in\\overline Q_T}r(t,x)\\le r(t,x)\\le \\overline r= \\max_{(t,x)\\in\\overline Q_T}r(t,x), \\\\ ~ \\\\\n \t\t \\bU\\in C^1([0,T]\\times \\overline\\Omega;\\R^3),\\; \\bU|_{(0,T)\\times\\partial\\Omega}=0\n\t\\end{array}\\right.\n \\end{equation}\nbe a (strong) solution of problem \\eqref{pbcont} emanating from the initial data $(r_0,\\bU_0)$.\nThen there exists\n\\[\nc=c(T,\\Omega, M_0, E_0, \\underline r, \\overline r, |p'|_{C^1([\\underline r,\\overline r])}, \\|(\\nabla r, \\partial_t r, \\bU, \\nabla\\bU, \\partial_t \\bU) \\|_{L^\\infty(Q_T;\\Rm^{19})} )>0\n\\]\nsuch that for almost all $t\\in (0,T)$,\n \\begin{equation}\\label{cerrorestimate}\n {\\cal E}(\\vr,\\bu\\Big|r,\\bU)(t)\\le c{\\cal E}(\\vr_0,\\bu_0\\Big|r_0,\\bU_0).\n\\end{equation}\n\\eP\nThis estimate (implying among others the weak-strong uniqueness) was proved in \\cite{FeJiNo} (see also \\cite{FENOSU}) for pressure laws (\\ref{pressure1}) with $\\gamma>d\/(d-1)$.\nIt remains valid under weaker hypothesis on the pressure, such as (\\ref{pressure1}) with $\\gamma\\ge 1$; this can be proved using ideas introduced in \\cite{BEFENO} and \\cite{MANO}.\n\n\n\n\n\n\n\\section{The numerical scheme}\\label{2}\n\\subsection{Partition of the domain}\\label{3.1}\nWe suppose that $\\Omega$ is a bounded domain of $\\R^d$, polygonal if $d=2$ and polyhedral if $d=3$.\nLet ${\\cal{T}}$ be a decomposition of the domain $\\Omega$ in { tetrahedra}, which we call hereafter a triangulation of $\\Omega$, regardless of the space dimension. By ${\\cal{E}}(K)$, we denote the set of the edges ($d=2$) or faces ($d=3$) $\\sigma$ of the element $K \\in {\\cal{T}}$ called hereafter faces, regardless of the dimension.\nThe set of all faces of the mesh is denoted by ${\\cal{E}}$; the set of faces included in the boundary $\\partial\\Omega$ of $\\Omega$ is denoted by ${\\cal{E}}_{\\extt}$ and the set of internal faces (i.e ${\\cal{E}} \\setminus {\\cal{E}}_{\\extt} $) is denoted by ${\\cal{E}}_{\\intt}$.\nThe triangulation ${\\cal{T}}$ is assumed to be regular in the usual sense of the finite element literature (see e.g. \\cite{cia-91-bas}), and in particular, ${\\cal{T}}$ satisfies the following properties:\n\\begin{itemize}\n\t\\item[$\\bullet$] $ \\overline{\\Omega} = \\cup_{K \\in {\\cal{T}}} \\overline{K} $;\n\t\\item[$\\bullet$] if $(K,L) \\in {\\cal T}^2$, then $ \\overline{K} \\cap \\overline{L} = \\emptyset $ or $\\overline{K} \\cap \\overline{L} $ is a vertex or $\\overline{K} \\cap \\overline{L} $ is a common face of $K$ and $L$; in the latter case it is denoted by $K|L$.\n\\end{itemize}\nFor each internal face of the mesh $\\sigma=K|L$, $\\bn_{\\sigma, K}$ stands for the normal vector of $\\sigma$, oriented from $K$ to $L$ (so that $\\bn_{\\sigma, K}=-\\bn_{\\sigma,L}$).\nWe denote by $ |K|$ and $ |\\sigma|$ the ($d$ and $d-1$ dimensional) Lebesgue measure of the { tetrahedron} $K$ and of the face $\\sigma$ respectively, and by $h_K$ and $h_\\sigma$ the diameter of $K$ and $\\sigma$ respectively.\nWe measure the regularity of the mesh thanks to the parameter $\\theta$ defined by\n{\n\\begin{equation}\\label{reg}\n\t{\\ \\theta = \\inf \\{ \\frac{\\xi_K}{h_K}, K \\in {\\cal{T}} \\} }\n\\end{equation}\n}\nwhere $\\xi_K$ stands for the diameter of the largest ball included in $K$.\nLast but not least we denote by $h$ the maximal size of the mesh,\n\\begin{equation}\\label{maxh}\n {\\\th=\\max_{K \\in {\\cal{T}}} h_K.}\n \\end{equation}\nThe triangulation $\\mathcal T$ is said to be regular if it satisfies\n\\begin{equation}\\label{reg1-}\n \t{\\ \\theta\\ge\\theta_0>0.}\n\\end{equation}\n\n\\subsection{Discrete {\\ function} spaces}\nLet $\\mathcal T$ be a mesh of $\\Omega$.\nWe denote by $L_h({ \\Omega})$ the space of piecewise constant functions on the cells of the mesh;\nthe space $L_h{ (\\Omega)}$ is the approximation space for the pressure and density.\nFor $1\\le p<\\infty$, the mapping\n\\[\n\tq \\mapsto \\|q\\|_{L_h^p(\\Omega)}= \\|q\\|_{L^p(\\Omega)}=\\Big(\\sum_{K\\in{\\cal T}}|K| |q_K|^p\\Big)^{1\/p}\n\\]\nis a norm on $L_h(\\Omega)$.\nWe also introduce spaces of non-negative and positive functions:\n\\[\n L_h^+{ (\\Omega)} = \\{q \\in L_h{ (\\Omega)}, ~q_K \\ge 0, ~\\forall K \\in \\T \\},\\quad L_h^{++}{ (\\Omega)} = \\{q \\in L_h{ (\\Omega)}, ~q_K > 0, ~\\forall K \\in \\T \\}.\n\\]\nThe approximation space for the velocity field is the space $\\bW_h(\\Omega)=V_h(\\Omega;{\\R^d})$, where $V_h(\\Omega)$ is the non conforming piecewise linear finite element space \\cite{cro-73-con,ern-04-the} defined by:.\n\\begin{multline}\n\t V_h(\\Omega) = \\{ v \\in L^2(\\Omega), ~\\forall K \\in {\\cal{T}}, ~v_{|K} \\in \\mathbb{P}_1(K),\\\\\n\t \\forall \\sigma \\in {\\cal{E}}_{\\intt} ,\\; \\sigma=K|L,\\; \\int_{\\sigma}v_{|K} \\dS=\\int_{\\sigma}v_{|L}\\dS,\\quad\n\t\\forall \\sigma \\in {\\cal{E}}_{\\extt},\\; \\int_\\sigma v \\dS=0 \\},\n \\end{multline}\nwhere $\\mathbb{P}_1(K)$ denotes the space of affine functions on $K$ and $\\dS$ the integration with respect to the $(d-1)$-dimensional Lebesgue measure { on the face $\\sigma$}.\nEach element $v\\in V_h(\\Omega)$ can be written in the form\n\\begin{equation}\\label{CRP}\n\tv{ (x)}=\\sum_{\\sigma\\in{\\cal E}_{\\rm int}}v_\\sigma\\varphi_\\sigma{ (x)},\\quad x\\in \\Omega\n\\end{equation}\nwhere the set $\\{\\varphi_\\sigma\\}_ {\\sigma\\in{\\cal E}_{\\rm int}}\\subset V_h(\\Omega)$ is the classical basis determined by\n\\begin{equation}\\label{CRB}\n\\forall (\\sigma,\\sigma')\\in {\\cal E}^2_{\\rm int},\\;\\int_{\\sigma'}\\varphi_\\sigma \\dS=\\delta_{\\sigma,\\sigma'},\\quad\\forall\\sigma'\\in {\\cal E}_{\\rm ext},\\; \\int_{\\sigma'}\\varphi_\\sigma \\dS=0\n\\end{equation}\n{ and $\\{v_\\sigma\\}_{\\sigma\\in {\\cal E}_{\\rm int}}\\subset R$ is the set of degrees of freedom relative to $v$. }\nNotice that $V_h(\\Omega)$ approximates the functions with zero traces in the sense that for all elements in $V_h(\\Omega)$, $v_\\sigma=0$ provided $\\sigma\\in {\\cal E}_{\\rm ext}$.\nSince only the continuity of the integral over each face of the mesh is imposed, the functions in $V_h(\\Omega)$ may be discontinuous through each face; the discretization is thus nonconforming in $ W^{1,p}(\\Omega;\\R^d)$, $1\\le p\\le \\infty$.\nFinally, we notice that for any $1\\le p<\\infty$ the expression\n\\[\n\t|v|_{V_h^p(\\Omega)}=\\Big(\\sum_{K\\in {\\cal T}}\\|\\nabla v\\|_{L^p(K;\\Rm^d)}^p\\Big)^{1\/p}\n\\]\nis a norm on $V_h(\\Omega)$ and we denote by $V^p_h{ (\\Omega)}$ the space $V_h{ (\\Omega)}$ endowed with this norm.\n\n\n\nWe finish this section by introducing some notations.\nFor a function $v$ in $L^1(\\Omega)$, we set\n\\begin{equation}\n \tv_K = \\frac 1{|K|}\\int_K v \\dx \\textrm{ for } K \\in {\\cal T} \\textrm{ and } \\hat v{ (x)}= \\sum_{K\\in {\\cal T}} v_K 1_K{ (x)},\\; { x\\in \\Omega}\n\t\\label{vhat}\n\\end{equation}\nso that $\\hat v \\in L_h(\\Omega)$. { Here and in what follows, $1_K$ is the characteristic function of $K$}.\n\n{ If} $v \\in W^{1,p}(\\Omega)$, we set\n\\begin{equation}\n\tv_\\sigma=\\frac 1{|\\sigma|}\\int_\\sigma v{\\rm d} S\\textrm{ for } \\sigma\\in {\\cal E}.\n\\label{vtilde}\n\\end{equation}\nFinally, if $v \\in W^{1,p}_0(\\Omega)$, we set\n\\begin{equation}\n\n v_h{ (x)}=\\sum_{\\sigma\\in {\\cal E_{\\rm int}}} v_\\sigma \\varphi_\\sigma{ (x)},\\; { x\\in \\Omega.}\n\t\\label{vh}\n\\end{equation}\nso that $v_h \\in V_h(\\Omega)$.\n{ In accordance with the above notation, for $v\\in W^{1,p}_0(\\Omega)$, the symbol $\\hat v_{h}$ means $\\widehat{v_h}(x)=\\sum_{\\sigma\\in {\\cal E}_{\\rm int}}v_\\sigma\\hat\\phi_\\sigma(x)$, and the symbol $v_{h,K}=\\frac 1{|K|}\\int_K v_h(x){\\rm d}x$ and the symbol $\\hat v_{h,\\sigma}^{\\rm up}= [\\widehat{(v_h)}]_\\sigma^{\\rm up}$.}\n\n\n\n\n\n\\subsection{Discrete equations}\nLet us consider a partition $0=t_0 0\\\\ q_L\\; \\mbox{if } \\bu_\\sigma\\cdot\\bn_{\\sigma, K}{ \\le} 0,\n \\end{cases}\n\\end{equation}\nso that\n\\[\n\\sum_{\\sigma\\in{\\cal E}(K)} q_\\sigma^{\\rm up}{\\bu}_\\sigma\\cdot{\\vc n}_{\\sigma,K}=\\stik \\Big(q_K [{\\bu}_\\sigma\\cdot{\\vc n}_{\\sigma,K}]^+ - q_L [{\\bu_\\sigma}\\cdot{\\vc n}_{\\sigma,K}]^-\\Big),\n\\]\nwhere $a^+ = \\max(a,0)$, $a^- =- \\min(a,0)$.\n\n\n\nLet us then consider the following numerical scheme \\cite{KARPER}:\n\n\\vspace{2mm}\n\n{\\it Given $(\\vr^0,\\bu^0) \\in L_h^+(\\Omega) \\times \\bW_h(\\Omega) $\n{find}} $ (\\vr^n)_{1\\le n \\le N}\\subset (L_h(\\Omega))^N, (\\bu^n)_{1\\le n \\le N} \\subset (\\bW_h(\\Omega))^N $ {\\it such that for all} $ n=1,...,N $\n\\vspace{2mm}\n\\begin{subequations}\\label{scheme}\n\\begin{align}\n \\label{dcont}\n\t&|K| \\frac{ \\vr_{K}^n - \\vr_{K}^{n-1}}{\\deltat} + \\sum_{\\sigma \\in {\\cal E}(K)} |\\sigma| \\vr_\\sigma^{n,{\\rm up}}[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}]=0,\\quad\\forall K \\in {\\cal{T}}, \\\\\n\t& \\sum_{K\\in{\\cal T}}\\frac{|K|}{\\deltat} \\Big({\\vr^n_K{{\\bu}}^n_{K}- \\vr^{n-1}_K{{\\bu}}^{n-1}_{K}} \\Big)\\cdot \\bv_K+ \\sum_{K\\in {\\cal T}}\\sum_{\\sigma \\in {\\cal E}(K)} |\\sigma|\\vr^{n,{\\rm up}}_\\sigma {{ \\hat\\bu}}_{\\sigma}^{n,{\\rm up}}[\\bu^n_\\sigma\\cdot \\bn_{\\sigma,K}]\\cdot \\bv_K \\nonumber \\\\\n\t&\\vspace{-.3cm}\\qquad \\qquad - \\sum_{K\\in{\\cal T}}p(\\vr^n_K)\\sum_{\\sigma \\in {\\cal E}(K)} |\\sigma|\\bv_\\sigma\\cdot {\\vc n}_{\\sigma,K}+\\mu\\sum_{K\\in{\\cal T}}\\int_K \\nabla\\bu^n : \\nabla\\bv \\ \\dx \\label{dmom}\\\\ \\vspace{-.3cm}\n\t&\\qquad \\qquad \\qquad \\qquad\\qquad + ( \\mu +\\lambda)\\sum_{K\\in{\\cal T}}\\int_K{\\rm div}\\bu^n{\\rm div}\\bv\\, \\dx =0,\n\t\\; \\forall \\bv \\in {\\bW_h(\\Omega) }. \\nonumber\n\\end{align}\n\\end{subequations}\nNote that the boundary condition $\\bu^n_{\\sigma}=0\\quad\\mbox{if $\\sigma\\in {\\cal E}_{\\rm ext}$}$ is ensured by the definition of the space $V_h(\\Omega)$.\nNote also that if $\\sigma \\in {\\cal E}_{\\rm int}$, $\\sigma = K|L$, one has, following \\eqref{vhat} and \\eqref{upwind1},\n\\[\n{{ \\hat\\bu}}_{\\sigma}^{n,{\\rm up}}=u^n_K=\\frac 1 {|K]} \\int_K \\bu^n(x) \\dx \\textrm{ if } \\bu^n_\\sigma\\cdot \\bn_{\\sigma,K}>0 \\textrm{ and }\n{{ \\hat\\bu}}_{\\sigma}^{n,{\\rm up}}=u^n_L=\\frac 1 {|L]} \\int_L \\bu^n(x) \\dx \\textrm{ if } \\bu^n_\\sigma\\cdot { \\bn_{\\sigma,K}< 0}.\n\\]\n\nIt is well known that any solution $(\\vr^n)_{1\\le n \\le N}\\subset (L_h(\\Omega))^N$ satisfies $\\vr^n >0$ thanks to the upwind choice in \\eqref{dcont} { (see e.g. \\cite{GGHL2008baro,KARPER})}.\nFurthermore, summing \\eqref{dcont} over $K \\in \\mesh$ immediately yields the total conservation of mass, which reads:\n\\begin{equation}\\label{masscons}\n\t\\forall n=1,...N,\\quad \\int_\\Omega \\vr^n \\dx = \\int_\\Omega \\vr^0 \\dx.\n\\end{equation}\n\nWe finally state in this section the existence result, which can be proved by a topological degree argument,\n\\cite{GGHL2008baro,KARPER}.\n\\begin{Proposition}[Existence]\\label{Theorem1}\n\tLet $(\\vr^0,\\bu^0){ \\in L_h^{++}}(\\Omega) \\times \\bW_h(\\Omega)$.\n\tUnder assumptions \\eqref{visc} and \\eqref{hypp}, Problem \\eqref{scheme} admits at least one solution\n\t\\[\n\t\t(\\vr^n)_{1\\le n \\le N}\\in [{ L_h^{++}}(\\Omega)]^N, (\\bu^n)_{1\\le n \\le N} \\in [\\bW_h(\\Omega)]^N.\n\t\\]\n\\end{Proposition}\n\n\n\n\\subsection{Main result: error estimate}\n\nLet $(r,\\bU) : [0,T]\\times\\overline\\Omega\\mapsto (0,\\infty)\\times \\R^3$ be $C^2$ functions such that $\\bU=\\bzero$ on $\\partial \\Omega$.\nLet $(\\vr,\\bu)$ be a solution of the discrete problem \\eqref{scheme}.\nInspired by (\\ref{ent}), we introduce the discrete relative energy functional\n\\begin{align}\\label{dent}\n\t{\\cal E}(\\vr^n,\\bu^n\\Big|r^n,\\bU^n) &=\\int_{\\Omega}\\Big(\\frac 12\\vr^n|\\hat\\bu^n-\\hat{\\bU}_h^n|^2\n\n\t+ E(\\vr^n|\\hat r^n)\\Big)\\dx \\\\\n\t&=\\sum_{K\\in{\\cal T}}|K|\\Big(\\frac 12\\vr_K|\\bu_K^n-\\bU^n_{h,K}|^2+ E(\\vr_K^n|r_K^n)\\Big), \\nonumber\n\\end{align}\nwhere\n\\begin{equation}\\label{notation2-}\nr^n(x)=r(t_n,x),\\;\\bU^n(x)=\\bU(t_n,x),\\;{ n=0,\\ldots,N,}\n\\end{equation}\n $(\\vr^n,\\bu^n)$ is defined in (\\ref{notation0}), and $E$ is defined by \\eqref{E}.\nLet us finally introduce the notations\n\\[\nM_0=\\sum_{K\\in{\\cal K}}|K|\\vr_K^0,\\mbox{ and } E_0=\\sum_{K\\in{\\cal K}}|K|\\Big(\\frac 12\\vr_K^0|\\bu_K^0|^2 + H(\\vr_K^0)\\Big).\n\\]\n\nNow, we are ready to state the main result of this paper.\nFor the sake of clarity, we shall state the theorem and perform the proofs only in the most interesting three dimensional case.\nThe modifications to be done for the two dimensional case, which is in fact more simple, are mostly due to the different Sobolev embedings, and are left to the interested reader.\n\n\n\\begin{Theorem}[Error estimate]\\label{Main}\nLet $\\theta_0 > 0$ and ${\\cal{T}} $ be a { regular} triangulation of a bounded polyhedral domain $ \\Omega\\subset\\R^3 $ { introduced in Section \\ref{3.1}} such that $ \\theta \\ge \\theta_0 $, where $ \\theta $ is defined in (\\ref{reg}).\nLet $p$ be a twice continuously differentiable function satisfying assumptions (\\ref{hypp}), (\\ref{pressure1}) with $\\gamma\\ge 3\/2$, and the additional assumption (\\ref{pressure2}) in the case $\\gamma<2$.\nLet the viscosity coefficients satisfy assumptions (\\ref{visc}).\nSuppose that $(\\vr^0,\\bu^0) \\in L_h^{+}(\\Omega) \\times \\bW_h(\\Omega)$ and that $(\\vr^n)_{1\\le n \\le N}\\subset [L_h^{+}(\\Omega)]^N$, $(\\bu^n)_{1\\le n \\le N} \\subset [\\bW_h(\\Omega)]^N$ is a solution of the discrete problem \\eqref{scheme}.\n Let $(r,\\bU)$ in the class\n\\begin{subequations}\\label{dr,U}\n\\begin{align}\n& r\\in C^2([0,T]\\times\\overline\\Omega),\\quad 0<\\underline r:=\\min_{(t,x)\\in \\overline Q_T}\\le r(t,x)\\le\\overline r:= \\max_{(t,x)\\in \\overline Q_T}r(t,x), \\label{dr}\\\\\n & \\bU\\in C^2([0,T]\\times\\overline\\Omega;\\R^3), \\;\\bU|_{\\partial\\Omega}=0 \\label{U}\n\\end{align}\n \\end{subequations}\n be a (strong) solution of problem \\eqref{pbcont}.\n Then there exists\n\\begin{multline*}\n c=c\\Big(T,|\\Omega|, {\\rm diam}(\\Omega),\\theta_0,\\gamma, M_0, E_0, \\underline r,\\overline r,\\\\ |p'|_{C^1([\\underline r,\\overline r])},\n \\|(\\nabla r, \\partial_t r, \\partial_t\\nabla r, \\partial^2_t r, \\bU, \\nabla\\bU, { \\nabla^2\\bU}, \\partial_t\\bU, { \\partial_t^2\\bU,} \\partial_t\\nabla\\bU)\\|_{L^\\infty(Q_T;{ \\Rm^{68}})}\\Big) \\in (0,+\\infty)\n\\end{multline*}\n (independent of $h$, $\\deltat$) such that for any $m=1,\\ldots,N,$\n \\begin{equation}\\label{errorestimate}\n {\\cal E}(\\vr^{ m},\\bu^{ m}\\Big|r^{ m},\\bU^{ m}){ +\\deltat\\sum_{n=1}^m\\sum_{{K}\\in {\\cal T}}\\int_K|\\Grad(u^n-\\vc U^n_h)|^2{\\rm d x}}\n \\le c\\Big({\\cal E}(\\vr^0,\\bu^0\\Big|r^0,\\bU^0)+ h^{A}\n +\\sqrt{\\deltat}\\Big),\n \\end{equation}\nwhere\n\\begin{equation}\\label{defAmain}\n A = \\begin{cases}\n\t\t\\frac {2\\gamma-3}\\gamma\\quad\\mbox{if }\\gamma\\in (3\/2,2],\\\\\n\t\t1\/2\\quad\\mbox{if }\\gamma> 2.\n \\end{cases}\n\\end{equation}\n\\end{Theorem}\n\n{ Starting from this point, unlike in Section \\ref{intro}, here and hereafter, the symbol ${\\cal E}$ refers always to the {\\it discrete} relative energy functional defined in (\\ref{dent}).}\n\n\n\\begin{Remark}\\label{rem1}{\\rm ~}\\\\\n{\nAssumptions (\\ref{dr,U}) on the regularity of the strong solution $(r,\\vc U)$ in Theorem \\ref{Main} may be slightly relaxed: It is enough to suppose\n$$\n(r,\\vc U)\\in C^1([0,T]\\times\\overline\\Omega;\\R^4),\\;\\nabla^2 \\vc U\\in C([0,T]\\times\\overline\\Omega;\\R^3),\\;\n0<\\inf_{(t,x)\\in \\overline Q_T} r(t,x),\n$$\n$$\n\\partial_t^2r\\in L^1(0,T;L^{\\gamma'}(\\Omega)),\\;\\partial_t\\nabla r\\in L^2(0,T; L^{6\\gamma\/(5\\gamma-6)}(\\Omega;\\R^3)),\\;\n(\\partial_t^2\\vc U,\\partial_t\\nabla\\vc U)\\in L^2(0,T; L^{6\/5}(\\Omega;\\R^{12})).\n$$\nThe constant in the error estimate depends on $\\underline r$ and the norms of $r$ and $\\vc U$ in these spaces.\nThis improvement is at the price of more technicalities in estimates of several residual terms, namely in estimates (\\ref{R2.1}--\\ref{R2.2}), (\\ref{R5.2}), (\\ref{R6.4}), (\\ref{cR2.1}), (\\ref{cR2.3}--\\ref{cR2.4})\nand (\\ref{cP1}).}\n\\end{Remark}\n\\begin{Remark}\\label{rmq-thmp}{\\rm ~ }\\\\\n\\vspace{-3mm}\n\\begin{enumerate}\n\\item\nTheorem \\ref{Main} holds also for two dimensional bounded { polygonal} domains under the assumption that $\\gamma\\ge1$.\nAssumption (\\ref{pressure2}) on the asymptotic behavior of pressure near $0$ is no more necessary in this case.\nThe value of $A$ in the error estimate (\\ref{errorestimate}) is\n\\[\nA =\\left\\{\n\\begin{array}{c}\n\\frac {2\\gamma-2}\\gamma\\quad\\mbox{if $\\gamma\\in (1,2]$},\\\\\n1\\quad\\mbox{if $\\gamma> 2$}.\n\\end{array}\n\\right.\n\\]\n{\n\\item Suppose that the discrete initial data $(\\vr^0,\\vc u^0)$ coincide with the projection $(\\hat r^0,\\hat {\\vc U}_{h}^0)$ of the initial data determining the strong solution. Then formula (\\ref{errorestimate}) provides in terms of classical Lebesgue spaces the following bounds:\n $$\n \\|\\vr^m-r^m\\|^2_{L^2(\\Omega\\cap\\{\\underline r\/2\\le\\vr^m\\le 2\\overline r\\})}+\n \\|\\hat{\\vc u}^m- {\\vc U}^m\\|^2_{L^2(\\Omega\\cap\\{\\underline r\/2\\le\\vr^m\\le 2\\overline r\\})}\\le\n c\\Big(h^{A}\n +\\sqrt{\\deltat}\\Big)\n $$\n for the \"essential part\" of the solution (where the numerical density remains bounded from above and from below outside zero), and\n $$\n |\\{\\vr^m\\le\\underline r\/2\\}|+ |\\{\\vr^m\\ge 2\\overline r\\}| +\\|\\vr^m\\|^\\gamma_{L^\\gamma(\\Omega\\cap \\{\\vr^m\\ge 2\\overline r\\})}+\\|\\vr^m|\\hat{\\vc u}^m-{\\vc U}^m|^2\\|_{L^1(\\Omega\\cap\\{\\vr^m\\ge 2\\overline r\\})}\\le c\\Big(h^{A}\n +\\sqrt{\\deltat}\\Big)\n $$\n for the \"residual part\" of the solution, where the numerical density can be \"close\" to zero or infinity.\n (In the above formula, for $B\\subset\\Omega$, $|B|$ denotes the Lebesgue measure of $B$.)\n\n Moreover, in the particular case of $p(\\vr)=\\vr^2$ (that however represents a non physical situation)\n $E(\\vr| r)=(\\vr- r)^2$ and the error estimate (\\ref{errorestimate}) reads\n $$\n \\|\\vr^m- r^m\\|^2_{L^2(\\Omega)}+\n \\|\\vr^m|\\hat{\\vc u}^m- {\\vc U}^m|^2\\|_{L^1(\\Omega)}\\le\n c\\Big(\\sqrt h\n +\\sqrt{\\deltat}\\Big)\n $$\n}\n\n\\item Theorem \\ref{Main} can be viewed as a discrete version of Proposition \\ref{proposition}.\nIt is to be noticed that the assumptions on the constitutive law for pressure guaranteeing the error estimates for the scheme \\eqref{scheme} are somewhat stronger ($\\gamma\\ge 3\/2$) than the assumptions needed for the stability in the continuous case ($\\gamma\\ge 1$).\nThe threshold value $\\gamma=3\/2$ is however in accordance with the existence theory of weak solutions.\nThe assumptions on the regularity of the strong solution to be compared with the discrete solution in the scheme are slightly stronger than those needed to establish the stability estimates in the continuous case.\n\n\\item If $d=3$, we notice that the assumptions on the pressure (as function of the density) in Theorem \\ref{Main} are compatible with the isentropic case $p(\\vr)=\\vr^\\gamma$ for all values $\\gamma\\ge 3\/2$.\n\n \\item { The scheme \\cite{KARPER} contains in addition artificial stabilizing terms both in the continuity and momentum equations. These terms are necessary for the convergence proof in \\cite{KARPER} even for the large values of $\\gamma$. It is to be noticed that the error estimate in Theorem \\ref{Main} is formulated for the numerical scheme without these stabilizing terms. Of course similar error estimate is a fortiori valid also for the scheme with the stabilizing terms, however, this issue is not discussed in the present paper.}\n\n\\end{enumerate}\n\\end{Remark}\n\n{ The rest of the paper is devoted to the proof of Theorem \\ref{Main}. For the sake of simplicity, and in order to simplify notation, we present the proof for the uniformly regular mesh meaning that there exist positive numbers $c_i=c_i(\\theta_0)$ such that\n\\begin{equation}\\label{reg1}\nc_1h_K\\le h\\le c_2 h_\\sigma\\le c_3 h_K,\\quad c_1|K|\\le |\\sigma| h\\le c_2|\\sigma|h_K\\le c_3|\\sigma|h_\\sigma\\le c_4 |K|\n\\end{equation}\nfor any $K\\in {\\cal T}$ and any $\\sigma\\in {\\cal E}$.\nThe necessary (small) modifications needed to accommodate the regular mesh satisfying only (\\ref{reg1-}) are straightforward. Even with this simplification the proof is quite involved, and some details have to be necessarily omitted to keep its length within reasonable bounds. The reader can eventually find them in the extended version of this paper available on ArXiv \\cite{GHMNArxive}.\n}\n\n\\section{Mesh independent estimates}\\label{4}\n\nWe start by a remark on the notation.\nFrom now on, the letter $c$ denotes positive numbers that may tacitly depend on $T$, $|\\Omega|$, ${\\rm diam}(\\Omega)$, $\\gamma$, $\\alpha$, $\\theta_0$, $\\lambda$ and $\\mu$, and on other parameters;\nthe dependency on these other parameters (if any) is always explicitly indicated in the arguments of these numbers.\nThese numbers can take different values even in the same formula.\nThey are always independent of the size of the discretisation $\\deltat$ and $h$.\n\n\n\\subsection{Energy { Identity}}\nOur analysis starts with an energy inequality, which is crucial both in the convergence analysis and in the error analysis.\nWe recall this energy estimate which is already given in \\cite{KARPER}, along with its proof for the sake of completeness.\n\n\\begin{lm}\\label{Theorem3}\n Let $(\\vr^0,\\bu^0) \\in L_h^{+}(\\Omega) \\times \\bW_h(\\Omega)$ and suppose that $(\\vr^n)_{1\\le n \\le N}\\in [L_h^{+}(\\Omega)]^N$, $(\\bu^n)_{1\\le n \\le N} \\in [\\bW_h(\\Omega)]^N$ is a solution of the discrete problem \\eqref{scheme} with the pressure $p$ satisfying condition (\\ref{hypp}).\nThen there exist\n\\begin{align*}\n&\\overline \\vr^n_{\\sigma}\\in [\\min(\\vr^n_K,\\vr^n_L), \\max(\\vr^n_K,\\vr^n_L)],\\;\\sigma=K|L\\in {\\cal E}_{\\rm int},\\; n=1,\\ldots,N \\\\\n&\\overline\\vr_K^{n-1,n}\\in [\\min(\\vr^{n-1}_K,\\vr^n_K), \\max(\\vr^{n-1}_K,\\vr^n_K)],\\; K\\in {\\cal T},\\; n=1,\\ldots,N\n\\end{align*}\nsuch that\n\\begin{multline}\n\\sum_{K\\in {\\cal T}}{|K|}\\Big(\\frac 12\\vr^m_K|{\\bu}^m_K|^2\n +H(\\vr_K^m)\\Big)\n-\\sum_{K\\in {\\cal T}}|K|\\Big(\\frac 12\\vr^{0}_K|{\\bu}^{0}_K|^2\n+H(\\vr_K^{0})\\Big)\n\\\\\n+\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K|\\Grad\\bu^n|^2 \\dx+( \\mu +\\lambda)\\int_K|{\\rm div}\\bu^n|^2 \\dx\\Big)\n\\\\\n+ [D^{m,|\\Delta\\bu|}_{\\rm time}]+ [D^{m,|\\Delta\\vr|}_{\\rm time}]+ [D^{m,|\\Delta\\bu|}_{\\rm space}] + [D^{m, |\\Delta\\vr|}_{\\rm space}]= 0,\n\\label{denergyinequality}\n\\end{multline}\nfor all $m=1,\\ldots,N$,\nwhere\n\\begin{subequations}\\label{upwinddissipation}\n\t\\begin{align}\n\t\t& [D^{m,|\\Delta\\bu|}_{\\rm time}]=\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{|K|}\\vr_K^{n-1}\\frac {|{\\bu}_K^n-{\\bu}_K^{n-1}|^2} 2\n\t\t\\label{upwinddissipation_1}\\\\\n\t\t&[D^{m,|\\Delta\\vc \\vr|}_{\\rm time}]=\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}|K|H''(\\overline\\vr_K^{n-1,n})\\frac {|{\\vr}_K^n-{\\vr }_K^{n-1}|^2} 2,\n\t\t\\label{upwinddissipation_2}\\\\\n\t\t&[D^{m,|\\Delta\\bu|}_{\\rm space}]=\\deltat\\sum_{n=1}^m\\sum_{\\sigma=K|L\\in{\\cal E}_{\\rm int}}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\frac{({\\bu}^n_K-{{\\bu}}^n_L)^2} 2\\;|{\\bu}^n_\\sigma\\cdot\\bn_{\\sigma,K}|,\n\t\t\\label{upwinddissipation_3}\\\\\n\t\t&[D^{m, |\\Delta\\vr|}_{\\rm space}]=\\deltat\\sum_{n=1}^m\\sum_{\\sigma=K|L \\in{\\cal E}_{\\rm int}}|\\sigma|H''(\\overline \\vr^n_{\\sigma})\\frac{(\\vr^n_K-\\vr^n_L)^2}2 \\;|{\\bu}^n_\\sigma\\cdot{\\vc n}_{\\sigma,K}|.\n\t\t\\label{upwinddissipation_4}\n\\end{align}\n \\end{subequations}\n\\end{lm}\n\n\\begin{proof}\nMimicking the formal derivation of the total energy conservation in the continuous case we take as test function $\\bv = \\bu^n$ in the discrete momentum equation (\\ref{dmom})$^n$ and obtain\n\\begin{equation}\\label{denergy1}\n\t I_1+ I_2+I_3+I_4 = 0,\n\\end{equation}\nwhere\n\\begin{align*}\n& { I_1=\\sum_{K\\in {\\cal T}}\\frac {|K|}\\deltat(\\vr_k^n\\bu_K^n-\\vr_K^{n-1}\\bu_K^{n-1})\\cdot\\bu_K^n},\n \t&&{ I_2}= \\sum_{K\\in {\\cal T}}\\stik |\\sigma|\\vr^{n,{\\rm up}}_\\sigma {{ \\hat\\bu}}_{\\sigma}^{n,{\\rm up}}\\cdot\\bu^n_K\\,[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}],\n \\\\\n &{ I_3} = -\\sum_{K\\in {\\cal T}}\\stik |\\sigma|p(\\vr^n_K)[\\bu^n_{\\sigma}\\cdot{\\vc n}_{\\sigma,K}], &&\n\t{ I_4= \\sum_{K\\in{\\cal T}}\\int_K\\Big(\\mu\\nabla\\bu^n:\\nabla\\bu^n +( \\mu +\\lambda){\\rm div}{\\bu^n}{\\rm div}{{\\bu}^n}\\Big) \\dx.}\n\\end{align*}\nNext, we multiply the continuity equation (\\ref{dcont})$^n_K$ by $\\frac 12|\\bu^n_K|^2$ and sum over all\n$K\\in {\\cal T}$. We get\n\\begin{equation}\\label{denergy2}\nI_5+I_6 = 0\n\\end{equation}\n\\[\n\\mbox{with }I_5 = -\\sum_{K\\in {\\cal T}}\\frac 12\\frac{|K|}{\\deltat} ({ \\vr_{K}^n - \\vr_{K}^{n-1}})|\\bu^n_K|^2 \\mbox{ and } I_6=-\\sum_{K\\in {\\cal T}}\\stik\\frac 12 |\\sigma| \\vr_\\sigma^{n,\\rm up}[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}]|\\bu^n_K|^2.\n\\]\nFinally, we multiply the continuity equation (\\ref{dcont})$^n_K$ by $H'(\\vr_K^n)$ and sum over all $K\\in {\\cal T}$.\nWe obtain\n\\begin{equation}\\label{denergy3}\nI_7+I_8 =0,\n\\end{equation}\n\\[\n\\mbox{with }I_7=\\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat} ({ \\vr_{K}^n - \\vr_{K}^{n-1}}) H'(\\vr^n_K)\n \\mbox{ and } I_8=\\sum_{K\\in {\\cal T}}\\stik |\\sigma| \\vr_\\sigma^{n,\\rm up}[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}] H'(\\vr^n_K).\n\\]\n\nWe now sum formulas (\\ref{denergy1})--(\\ref{denergy3}) in several steps.\n\n\\vspace{2mm}\n\n{\\bf Step 1:} {\\it Term $I_1+ I_7$.}\nWe verify by a direct calculation that\n\\[\nI_1= \\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}\\Big(\\frac 12 \\vr^n_K|\\bu^n_K|^2-\\frac 12 \\vr^{n-1}_K|\\bu^{n-1}_K|^2\\Big)\n+\\sum_{{ K\\in{\\cal T}}}\\frac{|K|}{\\deltat}\\vr_K^{n-1}\\frac {|{\\bu}_K^n-{\\bu}_K^{n-1}|^2} 2.\n\\]\nIn order to transform the term $I_7$, we employ the Taylor formula\n\\[\nH'(\\vr_K^n)\\Big(\\vr^n_K-\\vr^{n-1}_K\\Big)= H(\\vr_K^n)-H(\\vr_K^{n-1}) +\\frac 12 H''(\\overline\\vr_K^{n-1,n})(\\vr_K^n-\\vr_K^{n-1})^2,\n\\]\nwhere $\\overline\\vr_K^{n-1,n}\\in [\\min(\\vr_K^{n-1},\\vr_K^{n}), \\max(\\vr_K^{n-1},\\vr_K^{n})]$.\nConsequently,\n\\begin{multline}\\label{denergy4}\nI_1+I_7=\\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}\\Big(\\frac 12 \\vr^n_K|\\bu^n_K|^2-\\frac 12 \\vr^{n-1}_K|\\bu^{n-1}_K|^2\\Big)\n+ \\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}\\Big(H(\\vr_K^n)-H(\\vr_K^{n-1})\\Big)\n\\\\+\\sum_{K\\in{\\cal T}}\\frac{|K|}{\\deltat}\\vr_K^{n-1}\\frac {|{\\bu}_K^n-{\\bu}_K^{n-1}|^2} 2\n+\\sum_{K\\in{\\cal T}}\\frac{|K|}{\\deltat}H''(\\overline\\vr_K^{n-1,n})\\frac {|{\\vr}_K^n-{\\vr }_K^{n-1}|^2} 2.\n\\end{multline}\n{\\bf Step 2:} {\\it Term $I_2+I_6$}.\nThe contribution of the face $\\sigma=K|L$ to the sum $I_2+I_6$ reads, by virtue of (\\ref{upwind1}),\n\\begin{multline*}\n|\\sigma|\\, [\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}]^+\\,\\vr_K \\Big(|\\bu_K^n|^2 -\\bu^n_K\\cdot\\bu_L^n -\n\\frac 12|\\bu^n_K|^2 + \\frac 12|\\bu^n_L|^2\\Big) \\\\\n+ |\\sigma|\\,[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,L}]^+\\,\\vr_L \\Big(|\\bu_L^n|^2 -\\bu^n_K\\cdot\\bu_L^n -\n\\frac 12|\\bu^n_L|^2 + \\frac 12|\\bu^n_K|^2\\Big).\n \\end{multline*}\nConsequently,\n\\begin{equation}\\label{denergy5}\nI_2+I_6=\\sum_{\\sigma=K|L\\in{\\cal E}_{\\rm int}} |\\sigma| |\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}|\\vr_{\\sigma}^{n,{\\rm up}}\n\\frac{(\\bu^n_K-\\bu^n_L)^2} 2.\n\\end{equation}\n{\\bf Step 3:} {\\it Term $I_3+I_8$.}\nWe have\n\\begin{multline*}\nI_8=\\sum_{K\\in {\\cal T}}\\stik|\\sigma|\\,[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}]\\,\\Big( H'(\\vr^n_K)( \\vr_\\sigma^{n,\\rm up}-\\vr^n_K)+ H(\\vr^n_K)\\Big)\n\\\\+ \\sum_{K\\in {\\cal T}}\\stik|\\sigma|\\,[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}]\\,\\Big( \\vr^n_K H'(\\vr^n_K) -H(\\vr^n_K)\\Big).\n \\end{multline*}\nRecalling (\\ref{upwind1}), we may write the contribution of the face $\\sigma=K|L$ to the first sum in $I_8$; it reads\n\\begin{multline*}\n|\\sigma|\\,[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}]^+\\,\\Big(H(\\vr^n_K)- H'(\\vr^n_L)(\\vr^n_K-\\vr^n_L)-H(\\vr^n_L)\\Big)\n\\\\+|\\sigma|\\,[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,L}]^+\\,\\Big(H(\\vr^n_L)- H'(\\vr^n_K)(\\vr^n_L-\\vr^n_K)-H(\\vr^n_K)\\Big).\n \\end{multline*}\nRecalling that $rH'(r)-H(r)=p(r)$, we get, employing the Taylor formula\n\\[\nI_3+I_8=\\sum_{\\sigma=K|L\\in {\\cal E}_{\\rm int}}\\,|{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}| H''(\\overline \\vr^n_{\\sigma})\\frac{(\\vr^n_K-\\vr^n_L\\Big)^2} 2\n\\]\nwith some $\\overline \\vr^n_{\\sigma}\\in [\\min(\\vr^n_K,\\vr^n_L), \\max(\\vr^n_K,\\vr^n_L)]$.\n\n\\vspace{2mm}\n\n{\\bf Step 4:} {\\it Conclusion}\n\nCollecting the results of Steps 1-3 we arrive at\n\\begin{multline}\\label{denergyinequality1}\n\\sum_{K\\in {\\cal T}}\\frac 12\\frac{|K|}{\\deltat}\\Big(\\vr^n_K|{\\bu}^n_K|^2-\\vr^{n-1}_K|{\\bu}^{n-1}_K|^2\\Big)\n+ \\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}\\Big(H(\\vr_K^n)-H(\\vr_K^{n-1})\\Big) +\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K|\\Grad\\bu^n|^2 \\dx\n\\\\+( \\mu+\\lambda)\\int_K|{\\rm div}\\bu^n|^2 \\dx\\Big)\n+\\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}\\vr_K^{n-1}\\frac {|{\\bu}_K^n-{\\bu}_K^{n-1}|^2} 2\n+\\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}H''(\\overline\\vr_K^{n-1,n})\\frac {|{\\vr}_K^n-{\\vr }_K^{n-1}|^2} 2\n\\\\+ \\sum_{\\substack{\\sigma\\in{\\cal E}_{\\rm int} \\\\ \\sigma=K|L}}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\frac{({\\bu}^n_K-{{\\bu}}^n_L)^2} 2\\;|{\\bu}^n_\\sigma\\cdot\\bn_{\\sigma,K}|\n+\\sum_{\\substack{\\sigma\\in{\\cal E}_{\\rm int} \\\\ \\sigma=K|L}}|\\sigma|H''(\\overline \\vr^n_{\\sigma})\\frac{(\\vr^n_K-\\vr^n_L\\Big)^2} 2 \\;|{\\bu}^n_\\sigma\\cdot{\\vc n}_{\\sigma,K}|= 0.\n\\end{multline}\nAt this stage, we get the statement of Lemma \\ref{Theorem3} by multiplying (\\ref{denergyinequality1})$^{n}$ by $\\deltat$ and summing from $n=1$ to $n=m$. Lemma \\ref{Theorem3} is proved.\n\\end{proof}\n\n\\subsection{Estimates}\n\n\n\nWe have the following corollary of Lemma \\ref{Theorem3}.\n\\begin{cor}\\label{Corollary1}\n\\begin{description}\n\\item{(1)}\nUnder assumptions of Lemma \\ref{Theorem3}, there exists $c=c(M_0,E_0)>0$ (independent of $h$ and $\\deltat$) such that\n\\begin{equation}\\label{est0}\n|\\bu|_{L^2(0,T;V^2_h(\\Omega;\\Rm^3)}\\le c\n\\end{equation}\n\\begin{equation}\\label{est1}\n\\|\\bu\\|_{L^2(0,T;L^6(\\Omega;\\Rm^3))}\\le c\n\\end{equation}\n\\begin{equation}\\label{est2}\n\\|\\vr\\hat{\\bu}^2\\|_{L^\\infty(0,T;L^1(\\Omega))}\\le c.\n\\end{equation}\n \\item{(2)} If in addition the pressure satisfies assumption (\\ref{pressure1}) then\n\\begin{equation}\\label{est3}\n\\|\\vr\\|_{L^\\infty(0,T;L^\\gamma(\\Omega))}\\le c\n\\end{equation}\n\\item{(3)} If the pair $(r,\\bU)$ belongs to the class (\\ref{dr,U}) there exists $c=c(M_0,E_0,\\underline r,\\overline r, \\|\\bU, \\nabla \\bU\\|_{L^\\infty(Q_T;\\Rm^{12})})>0$\nsuch that for all $n=1,\\ldots,N$,\n\\begin{equation}\\label{est4}\n{\\cal E}(\\vr^n,\\bu^n|r^n,\\bU^n)\\le c,\n\\end{equation}\nwhere the discrete relative energy ${\\cal E}$ is defined in (\\ref{dent}).\n\\end{description}\n\\end{cor}\n\\begin{proof}\nRecall that\n\\[|\\bu|^2_{L^2(0,T;V^2_h(\\Omega;\\Rm^3)} = \\deltat \\sum_{n=1}^N\\sum_{K\\in {\\cal T}}\\int_K|\\Grad\\bu^n|^2 \\dx;\\]\nthe estimate (\\ref{est0}) follows from (\\ref{denergyinequality}).\nThe estimate (\\ref{est1}) holds due to imbedding (\\ref{sob1}) in Lemma \\ref{Lemma2+} and bound (\\ref{est0}).\nThe estimate (\\ref{est2}) is just a short transcription of the bound for the kinetic energy in (\\ref{denergyinequality}).\n\n{ We prove estimate (\\ref{est3}). First, we deduce from (\\ref{hypp}) and the definition (\\ref{H}) of $H$ that $0\\le -H(\\vr)\\le c_1$ with some $c_1>0$, provided $0<\\vr \\le 1$ and $H(\\vr)> 0$ if $\\vr>1$. This fact in combination with the bound for $\\int_\\Omega H(\\vr){\\rm d}x$ derived in (\\ref{denergyinequality}) yields\n \\begin{equation}\\label{H+}\n \\int_\\Omega |H(\\vr)|{\\rm d}x\\le c<\\infty.\n\\end{equation}\nSecond, relations (\\ref{hypp}--\\ref{pressure2}) imply that there are $\\overline\\vr>1$ and $0<\\underline p<\\overline p<\\infty$ such that\n$$\n\\left\\{\n\\begin{array}{c}\n\\vr^{\\alpha}p_0\/2\\le\\frac {p(\\vr)}{\\vr^2}\\mbox{if $0<\\vr<1\/\\overline\\vr$},\\\\\n\\underline p\\le \\frac {p(\\vr)}{\\vr^2}\\le\\overline p\\;\\mbox{if $1\/\\overline\\vr\\le\\vr\\le \\overline\\vr$},\\\\\n\\vr^{\\gamma-2} {p_\\infty}\/2\\le \\frac {p(\\vr)}{\\vr^2}\\;\\mbox{if $\\vr>2\\overline\\vr$}\n\\end{array}\n\\right\\}.\n$$\nUsing these bounds and the definition (\\ref{H}) of $H$ we verify that\n$$\n\\vr^\\gamma\\le c (|H(\\vr)|+ \\vr +1)\n$$\nwith a convenient positive constant $c$. Now, bound (\\ref{est3}) follows readily\nfrom the boundedness of\n$\\int_\\Omega\\vr^m{\\rm d}x\\equiv\\sum_{K\\in {\\cal T}}|K|\\vr_K^m$ and $\\int_\\Omega H(\\vr^m){\\rm d}x\\equiv\\sum_{K\\in {\\cal T}}{|K|} H(\\vr_K^m)$ established in (\\ref{masscons}) and\n(\\ref{denergyinequality}).\n}\n\n Finally, to get (\\ref{est4}), we have employed (\\ref{E}), (\\ref{dent}), (\\ref{masscons}), (\\ref{H+})\nto estimate $\\int_\\Omega E(\\vr^n|\\hat r^n)\\dx$ and (\\ref{est2}), (\\ref{L2-3}), (\\ref{L1-3}) to evaluate $\\sum_{K\\in {\\cal T}}\\int_K\\vr_K^n|\\bU_{h,K}^n-\\bu_K^n|^2\\dx$.\n\\end{proof}\n\nThe following estimates are obtained thanks to the numerical diffusion due to the upwinding, as is classical in the framework of hyperbolic conservation laws, see e.g. \\cite{egh-book}.\n\n\\begin{lm}[Dissipation estimates on the density]\\label{Lemma5}\nLet $(\\vr^0,\\bu^0) \\in L_h^{+}(\\Omega) \\times \\bW_h(\\Omega)$.\nSuppose that $(\\vr^n)_{1\\le n \\le N}\\subset [L_h^{+}(\\Omega)]^N$, $(\\bu^n)_{1\\le n \\le N} \\subset [\\bW_h]^N(\\Omega)$ is a solution of problem \\eqref{scheme}.\nFinally assume that the pressure satisfies hypotheses (\\ref{hypp}) and\n(\\ref{pressure1}).\nThen we have:\n\\begin{description}\n\\item {(1)} If $\\gamma\\ge 2$ then there exists $c=c(\\gamma,\\theta_0, E_0)>0$ such that\n\\begin{equation}\\label{dissipative2}\n\\deltat \\sum_{n=1}^N\\sum_{\\sigma=K|L \\in{\\cal E}_{\\rm int}}|\\sigma|\\frac{(\\vr^n_K-\\vr^n_L)^2}{{\\rm max}(\\vr^n_K,\\vr^n_L)} \\;|{\\bu}^n_\\sigma\\cdot{\\vc n}_{\\sigma,K}|\\le c.\n\\end{equation}\n\\item{(2)} If $\\gamma\\in [1, 2)$ and the pressure satisfies additionally assumption (\\ref{pressure2})\nthen there exists $c=c(M_0, E_0)>0$ such that\n\\begin{multline}\\label{dissipative1}\n\\deltat\\sum_{n=1}^N \\sum_{\\sigma=K|L \\in{\\cal E}_{\\rm int}}|\\sigma|\\frac{(\\vr^n_K-\\vr^n_L)^2}{[{\\rm max}(\\vr^n_K,\\vr^n_L)]^{2-\\gamma}} 1_{\\{\\overline\\vr^n_\\sigma\\ge 1\\}} \\;|{\\bu}^n_\\sigma\\cdot{\\vc n}_{\\sigma,K}|\n\\\\+\n\\deltat \\sum_{n=1}^N\\sum_{\\sigma=K|L \\in{\\cal E}_{\\rm int}}|\\sigma|(\\vr^n_K-\\vr^n_L)^2 1_{\\{\\overline\\vr^n_\\sigma <1\\}} \\;|\\bu^n_\\sigma\\cdot{\\vc n}_{\\sigma,K}|\\le c,\n\\end{multline}\nwhere the numbers $\\overline\\vr^n_\\sigma$ are defined in Lemma \\ref{Theorem3}.\n\\end{description}\n\\end{lm}\n\\begin{proof} We start by proving the simpler statement \\textit{ (2)}.\nTaking into account the continuity of the pressure, we deduce from assumptions (\\ref{pressure1}) and (\\ref{pressure2}) that there exist numbers $\\overline p_0>0$, $\\overline p_\\infty>0$ such that\n\\[\nH''(s)\\ge \\begin{cases}\n \\frac{\\overline p_\\infty}{s^{2-\\gamma}},\\quad\\mbox{if } s\\ge 1,\n\\\\ \\overline p_0 s^\\alpha\\ge\n\\overline p_0, \\quad\\mbox{if }s< 1,.\n \\end{cases}\n\\]\nwhence, splitting the sum in the definition of the term $[D^{N,\\Delta\\vr}_{\\rm space}]$ (see (\\ref{upwinddissipation_4})) into two sums, where $(\\sigma, n)$ satisfies $\\overline\\vr_\\sigma^n\\ge 1$ for the first one and $\\overline\\vr_\\sigma^n< 1$ for the second, we obtain the desired result.\n\nLet us now turn to the proof of statement \\textit{(1)}.\nMultiplying the discrete continuity equation (\\ref{dcont})$_K^n$ by $\\ln \\vr_K^n$ and summing over $K\\in {\\cal T}$, we get\n\\begin{equation*}\n\\sth |K| \\frac{ \\vr_{K}^n - \\vr_{K}^{n-1}}{\\deltat}\\ln\\vr_K^n + \\sth \\sum_{\\sigma\\in {\\cal E}(K),\\sigma=K|L}(\\ln\\vr_K^n)\\vr_\\sigma^{n,{\\rm up}} \\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}=0.\n\\end{equation*}\nBy virtue of the convexity of the function $\\vr\\mapsto \\vr\\ln\\vr-\\vr$ on the positive real line, and due to the Taylor formula, we have\n\\[\n\\vr_K^n\\ln \\vr_K^n-\\vr_K^{n-1}\\ln \\vr_K^{n-1} -( \\vr_K^n-\\vr_K^{n-1})\\le \\ln \\vr_K^n (\\vr_K^n-\\vr_K^{n-1});\n\\]\nwhence, thanks to the mass conservation \\eqref{masscons} and the definition of $\\vr_\\sigma^{{\\rm up}}$, we arrive at\n\\begin{multline*}\n\\sth |K| \\frac{ \\vr_{K}^n\\ln\\vr_K^n- \\vr_{K}^{n-1}\\ln\\vr_K^{n-1}}{\\deltat}\n+ \\stkl |\\sigma| \\vr_K^n [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,K}]^+ \\Big( \\ln\\vr_K^n-\\ln\\vr_L^n\\Big)\n\\\\+ \\stkl |\\sigma| \\vr_L^n [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,L}]^+ \\Big( \\ln\\vr_L^n-\\ln\\vr_K^n\\Big)\\le 0,\n\\end{multline*}\nor equivalently\n\\begin{multline}\\label{est5}\n\\deltat\\stkl |\\sigma| [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,K}]^+ \\Big( \\vr_K^n(\\ln\\vr_K^n-\\ln\\vr_L^n) -(\\vr_K^n-\\vr_L^n)\\Big)\n+\\deltat\\stkl |\\sigma| [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,L}]^+ \\Big( \\vr_L^n ( \\ln\\vr_L^n-\\ln\\vr_K^n)-(\\vr_L^n-\\vr_K^n)\\Big)\\le\n\\\\-\\sth |K| \\Big({ \\vr_{K}^n\\ln\\vr_K^n- \\vr_{K}^{n-1}\\ln\\vr_K^{n-1}}\\Big)\n +\\deltat\\stkl |\\sigma| \\Big([\\bu_\\sigma^n \\cdot \\bn_{\\sigma,K}]^+ (\\vr_L^n-\\vr_K^n))+ [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,L}]^+ (\\vr_K^n-\\vr_L^n))\\Big).\n\\end{multline}\nFrom \\cite[Lemma C.5]{FettahGallouet2013Stokes}, we know that if $\\varphi$ and $\\psi$ are functions in $C^1((0,\\infty);\\R)$ such that $s\\psi'(s)=\\varphi'(s)$ for all $ s \\in (0,\\infty)$, then for any $(a,b)\\in (0,\\infty)^2$ there exits $c \\in [a,b]$ such that\n\\[\n\t\t(\\psi(b)-\\psi(a))b -(\\varphi(b)-\\varphi(a)) = \\frac{1}{2}(b-a)^2 \\psi'(c).\n\\]\nApplying this result with $\\psi(s)=\\ln s$, $\\varphi(s)=s$ we obtain that the left hand side of \\eqref{est5} is greater or equal to\n\\[\n\\deltat\\stkl |\\sigma| \\Big([\\bu_\\sigma^n \\cdot \\bn_{\\sigma,K}]^+ + [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,L}]^+\\Big)\n\\frac{(\\vr^n_K-\\vr^n_L)^2}{\\max(\\vr^n_K,\\vr^n_L)}.\n\\]\nOn the other hand, the first term at the right hand side is bounded from above by $\\|\\vr^n\\|_{L^\\gamma(\\Omega)}^\\gamma$.\n Finally the second term at the right hand side is equal to\n\\[\n-\\deltat\\sum_{K\\in {\\cal T}}\\int_K\\vr_K^n{\\rm div}\\bu^n {\\le \\deltat\\sum_{K\\in {\\cal T}}\\|\\vr_K\\|_{L^2(K)}\n\\|{\\rm div}\\vc u^n\\|_{L^2(K)},}\n\\]\nwhence bounded from above by $\\deltat\\|\\bu^n\\|_{V_h^2(\\Omega;\\Rm^3)}\\|\\vr^n\\|_{L^2(\\Omega)}$, { where we have used the H\\\"older inequality and the definition of the $V_h^2(\\Omega)$-norm}. The statement \\textit{(1)} of Lemma \\ref{Lemma5} now follows from the estimates of Corollary \\ref{Corollary1}.\n\n\\end{proof}\n\n\n\\section{Exact relative energy inequality for the discrete problem}\\label{5}\n\nThe goal of this section is to prove the discrete version of the relative energy inequality.\n\n\\begin{Theorem}\\label{Theorem4}\n\nSuppose that $\\Omega\\subset \\R^3$ is a polyhedral domain and ${\\cal T}$ its regular triangulation { introduced in Section \\ref{3.1}}. Let $p$ satisfy hypotheses (\\ref{hypp}) and the viscosity coefficient $\\mu$, $\\lambda$ obey (\\ref{visc}).\nLet $(\\vr^0,\\bu^0) \\in L_h^{+}(\\Omega) \\times \\bW_h(\\Omega)$ and suppose that $(\\vr^n)_{1\\le n \\le N}\\in [L_h^{+}(\\Omega)]^N$, $(\\bu^n)_{1\\le n \\le N} \\in [\\bW_h(\\Omega)]^N$ is a solution of the discrete problem \\eqref{scheme}.\nThen there holds for all $m=1,\\ldots,N$,\n\\begin{equation}\\label{drelativeenergy}\n\\begin{aligned}\n&\\sum_{K\\in {\\cal T}}\\frac 12{|K|}\\Big(\\vr^{ m}_K|{\\bu}^m_K-{\\bU}^m_{h,K}|^2-\\vr^{0}_K|{\\bu}^{0}_K-{\\bU}^{0}_{h,K}|^2\\Big)\n+ \\sum_{K\\in {\\cal T}}{|K|}\\Big(E(\\vr_K^m| r_K^m)-E(\\vr_K^{0}| r_K^{0})\\Big)\n\\\\ &\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad+\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K|\\Grad(\\bu^n-\\bU^n_h)|^2 \\dx+( \\mu +\\lambda)\\int_K|{\\rm div}(\\bu^n-\\bU^n_h)|^2 \\dx\\Big)\n\\\\&\\phantom{\\sum_{K\\in {\\cal T}}} \\le\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K\\Grad\\bU^n_h:\\Grad(\\bU^n_h-\\bu^n) \\dx+(\\mu +\\lambda)\\int_K{\\rm div}\\bU^n_h{\\rm div}(\\bU^n_h-\\bu^n) \\dx\\Big)\n\\\\&\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad+\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}{|K|}\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big(\\frac{{\\bU}_{h,K}^{n-1} + {\\bU}_{h,K}^{n} }2 -\n\\bu_K^{n-1}\\Big)\n\\\\\n&\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad-\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\stik|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\Big(\\frac{\\bU^n_{h,K}+\\bU^n_{h,L}}2-\n\\hat{\\bu}_{\\sigma}^{n,{\\rm up}}\\Big)\\cdot{\\bU}^n_{h,K} [\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}]\n\\\\\n&\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad-\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\stik |\\sigma|p(\\vr^n_K)[{\\bU}_{h,\\sigma}^{n}\\cdot\\bn_{\\sigma,K}]\n\\\\&\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad+\n\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat} ( r^n_K-\\vr^n_K)\\Big(H'( r^n_K)-H'( r^{n-1}_K)\\Big)\n\\\\\n&\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad+\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\stik |\\sigma|\\vr_\\sigma^{n,{\\rm up}}H'( r_K^{ n-1})[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}],\n\\end{aligned}\n\\end{equation}\nfor any $ 00\n\\]\n(where $\\overline r=\\max_{(t,x)\\in \\overline {Q_T}} r(t,x)$, $\\underline r=\\min_{(t,x)\\in \\overline {Q_T}} r(t,x)$),\nsuch that for all $m=1,\\ldots,N$, we have:\n\\begin{equation}\\label{relativeenergy-}\n\\begin{aligned}\n\t&{\\cal E}(\\vr^m,\\bu^m\\Big| r^m, \\bU^m)- {\\cal E}(\\vr^0,\\bu^0\\Big|r(0),\\bU(0))\n\t\\\\& \\qquad \\qquad \\qquad +\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K|\\Grad(\\bu^n-\\bU^n_h)|^2 \\dx+( \\mu +\\lambda)\\int_K|{\\rm div}(\\bu^n-\\bU^n_h)|^2 \\dx\\Big)\n\t\\\\& \\qquad \\le \\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K\\Grad\\bU^n_h:\\Grad(\\bU^n_h-\\bu^n) \\dx+(\\mu +\\lambda)\\int_K{\\rm div}\\bU^n_h{\\rm div}(\\bU^n_h-\\bu^n) \\dx\\Big)\n\t\\\\& \\qquad+\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K}^{n} -\\bu_K^{n}\\Big)\n\t\\\\ &\\qquad { + \\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\n\\Big(\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}-\\hat{\\bu}^{n,{\\rm up}}_{\\sigma}\\Big)\\cdot\\Big(\\bU^n_\\sigma-{\\bU}^n_{h,K}\\Big) \\hat\\bU_{h,\\sigma}^{n,{\\rm up}}\\cdot\\bn_{\\sigma,K}}\n\t\\\\& \\qquad-\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}} \\int_K p(\\vr_K^n)\\dv\\bU^n\\dx\n\t+\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_K ( r^n_K-\\vr^n_K)\\frac{p'(r_K^n)}{r_K^n} [\\partial_t r]^n\\dx\n\t\\\\ &\\qquad -\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\int_K\\frac{\\vr^n_K}{r^n_K} p'(r^n_K) { \\bu^n}\\cdot\\nabla r^n\\dx\t\n\\, { + R^m_{h,\\deltat}} { + G^m}\n\\end{aligned}\n\\end{equation}\nfor any pair $(r,\\bU)$ belonging to the class (\\ref{dr,U}), where\n\\begin{equation}\\label{A1}\n{ |G^m|\\le c\\,\\deltat\\sum_{n=1}^m{\\cal E}(\\vr^n,\\bu^n\\Big| r^n, U^n),}\\;\\;\n{ |R^m_{h,\\deltat}|\\le c (\\sqrt{\\deltat} + h^A)},\\quad \\mbox{ and } A=\\left\\{\\begin{array}{c}\n{ \\frac{2\\gamma-3}{\\gamma}}\\;\\mbox{if $\\gamma\\in [3\/2,2)$}\n\\\\\n1\/2 \\;\\mbox{ if { $\\gamma\\ge 2$}},\n\\end{array}\n\\right.\n\\end{equation}\n { and where we have used notation (\\ref{notation2-}) for $r^n$, $\\bU^n$ and (\\ref{vhat}--\\ref{vtilde}) for $\\vc U^n_h$, $\\vc U^n_{h,K}$, $r^n_K$, $\\vc u^n_{\\sigma}$.}\n\\end{lm}\n\n\n\n\\begin{proof}\n\\label{6.0}\nThe right hand side of the relative energy inequality (\\ref{drelativeenergy}) is a sum $\\sum_{i=1}^6T_i$, where\n\\begin{align*}\n& T_1=\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K\\Grad\\bU^n_h:\\Grad(\\bU^n_h-\\bu^n) \\dx+(\\mu +\\lambda)\\int_K{\\rm div}\\bU^n_h{\\rm div}(\\bU^n_h-\\bu^n) \\dx\\Big),\n\\\\ & T_2=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}{|K|}\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big(\\frac{{\\bU}_{h,K}^{n-1} + {\\bU}_{h,K}^{n} }2 -\n\\bu_K^{n-1}\\Big),\n\\\\ & T_3=-\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\Big(\\frac{\\bU^n_{h,K}+\\bU^n_{h,L}}2-\n\\hat {\\bu}_{\\sigma}^{n,{\\rm up}}\\Big)\\cdot{\\bU}^n_{h,K} [\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}],\n\\\\ & T_4 = -\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|p(\\vr_K)[{\\bU}_{h,\\sigma}^{n}\\cdot\\bn_{\\sigma,K}],\n\\\\ & T_5=\n\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}|K| ( r^n_K-\\vr^n_K)\\frac{H'( r_K^{n})-H'(r^{n-1}_K)}{\\deltat},\n\\\\& T_6=\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}H'( r_K^{ n-1})[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}].\n \\end{align*}\n\nThe term $T_1$ will be kept as it is; all the other terms $T_i$ will be transformed to a more convenient form, as described in the following steps.\n\n\\vspace{2mm}\\noindent\n{\\bf Step 1:} {\\it Term $T_2$.}\n\\label{6.1}\nWe have\n\\[\nT_2=T_{2,1}+ R_{2,1},\\mbox{ with }T_{2,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K\n}^{n-1} -\n\\bu_K^{n-1}\\Big),\n\\mbox{ and }R_{2,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}R^{n,K}_{2,1},\n\\]\nwhere\n$$\n R_{2,1}^{n,K}=\n\\frac {|K|}2\n\\vr_K^{n-1}\\frac{({\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1})^2}{\\deltat}={ \\frac {|K|}2\n\\vr_K^{n-1}\\frac{([{\\bU}^{n}-{\\bU}^{n-1}]_{h,K})^2}{\\deltat}}.\n$$\nWe may write by virtue of the first order Taylor formula applied to function $t\\mapsto \\vc U(t,x)$,\n$$\n\\Big|\\frac{[{\\bU}^{n}-{\\bU}^{n-1}]_{h,K}}{\\deltat}\\Big|=\n\\Big|\\frac 1{|K|}\\int_K\\Big[\\frac 1\\deltat \\Big[\\int_{t_{n-1}}^{t_n} \\partial_t\\vc U(z,x) {\\rm d} z\\Big]_h\\Big]{\\rm d}x\\Big|\n$$\n$$\n=\\Big|\\frac 1{|K|}\\int_K\\Big[\\frac 1\\deltat \\int_{t_{n-1}}^{t_n} [\\partial_t\\vc U(z) \\Big]_h(x){\\rm d} z\\Big]{\\rm d}x\\Big|\\le \\|[\\partial_t\\vc U ]_h\\|_{L^\\infty(0,T;L^\\infty(\\Omega;\\Rm^3))}\\le \\|\\partial_t\\vc U \\Big\\|_{L^\\infty(0,T;L^\\infty(\\Omega;\\Rm^3))},\n$$\nwhere we have used the property (\\ref{ddd}) of the projection onto the space $V_h{ (\\Omega)}$.\nTherefore, thanks to the mass conservation \\eqref{masscons}, we finally get\n\\begin{equation}\\label{R2.1}\n|R^{n,K}_{2,1}|\\le\\frac {M_0} 2|K|\\deltat\\|\\partial_t\\bU\\|^2_{L^\\infty(0,T; L^{\\infty}(\\Omega;\\Rm^3))}.\n\\end{equation}\n\nLet us now decompose the term $T_{2,1}$ as\n\\begin{equation}\\label{T2}\nT_{2,1}=T_{2,2}+R_{2,2},\\mbox{ with }T_{2,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K\n}^{n} - \\bu_K^{n}\\Big),\n \\mbox{ and } R_{2,2}=\\deltat\\sum_{n=1}^m R_{2,2}^n,\n\\end{equation}\nwhere $\\displaystyle\n\tR^n_{2,2}=\\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K}^{n-1} - \\bU_{h,K}^{n}\\Big)\n \t- \\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bu}_{K}^{n-1} \t\t -\\bu_{K}^{n}\\Big).$\n\nBy the same token as above, we may estimate the residual term as follows\n\\[\n|R^n_{2,2}|\\le \\deltat\\; c M_0 \\|\\partial_t\\bU\\|^2_{L^\\infty(0,T;W^{1,\\infty}(\\Omega;\\Rm^3)}+\nc M_0^{1\/2}\\Big(\\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}|{\\bu}_{K\n}^{n-1} -\n\\bu_{K}^{n}|^2\\Big)^{1\/2} \\|\\partial_t\\bU\\|_{L^\\infty(0,T;L^{\\infty}(\\Omega;\\Rm^3))},\n\\]\nwhere we have used the H\\\"older inequality to treat the second term;\nwhence, by virtue of estimate (\\ref{upwinddissipation_1}),\n\\begin{equation}\\label{R2.2}\n|R_{2,2}|\\le \\sqrt{\\deltat} \\,c(M_0, E_0, \\|(\\partial_t\\bU, \\partial_t\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{12})}).\n\\end{equation}\n\n\n\n\n\\vspace{2mm}\\noindent\n{\\bf Step 2:} {\\it Term $T_3$.}\n\\label{6.2}\nEmploying the definition (\\ref{upwind1}) of upwind quantities, we easily establish that\n\\begin{align*}\n& T_3= T_{3,1} + R_{3,1}, \\\\\n& \\mbox{with }T_{3,1}= \\deltat\\sum_{n=1}^m \\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E(K)}}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\Big(\\hat {\\bu}_{\\sigma}^{n,{\\rm up}}-\n\\hat {\\bU}^{n, {\\rm up}}_{h,\\sigma}\\Big)\\cdot{\\bU}^n_{h,K} \\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}, \\quad\nR_{3,1}=\n\\deltat\\sum_{n=1}^m { \\sum_{\\sigma \\in {\\cal E}_{\\rm int}}R_{3,1}^{n,\\sigma}}, \\\\\n&\\mbox{and }{ R_{3,1}^{n,\\sigma}}= |\\sigma|\\vr_K^n \\frac{|\\bU_{h,K}^n-\\bU_{h,L}^n|^2}2\\,[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}]^+\n+ |\\sigma|\\vr_L^n \\frac{|\\bU_{h,L}^n-\\bU_{h,K}^n|^2}2\\,[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,L}]^+, \\; \\forall \\sigma=K|L \\in {\\cal E}_{\\rm int}.\n\\end{align*}\n{ Writing\n$$\n\\bU^n_{h,K}-\\bU^n_{h,L}= \\bU^n_{h,K}-\\bU^n_{h,\\sigma}+\\bU^n_{h,\\sigma}-\\bU^n_{h,L},\\; \\sigma=K|L\\in {\\cal E}_{\\rm int},\n$$\n}\nemploying estimates (\\ref{L2-1}) and (\\ref{L1-3})$_{s=1}$ and the continuity of the mean value $\\bU^n_\\sigma{ =\\bU_{h,\\sigma}^n}$ of $\\bU^n_h$ over faces $\\sigma$, we infer by using the Taylor formula applied to function\n$x\\mapsto U^n(x)$,\n\\[\n\t{ |R_{3,1}^{n,\\sigma}|}\\le h^2\\;c \\|\\nabla\\bU\\|^2_{L^\\infty (Q_T;\\Rm^9)} |\\sigma|(\\vr^n_K+\\vr^n_L) |\\bu^n_\\sigma|, \\; \\forall \\sigma=K|L \\in {\\cal E}_{\\rm int},\n\\]\nwhence\n\\begin{equation}\\label{R3.1}\n\\begin{aligned}\n |R_{3,1}| & \\le h\\;c \\|\\nabla\\bU\\|^2_{L^\\infty (Q_T;\\Rm^9)}\\Big(\\sum_{K\\in{\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)} h|\\sigma|(\\vr^n_K+\\vr^n_L)^{6\/5}\\Big)^{5\/6}\n\\Big[\\deltat \\sum_{n=1}^m\\Big(\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma||\\bu^n_\\sigma|^6\\Big)^{1\/3}\\Big]^{1\/2}\n\\\\ &\\le h \\; c(M_0,E_0,\\|\\nabla\\bU\\|_{L^\\infty(Q_T;\\Rm^{9})}),\n\\end{aligned}\n\\end{equation}\nprovided $\\gamma\\ge 6\/5$,\nthanks to the discrete H\\\"older inequality, the equivalence relation (\\ref{reg1}), the equivalence of norms (\\ref{norms1}) and energy bounds listed in Corollary \\ref{Corollary1}.\n\nEvidently, for each face $\\sigma=K|L\\in {\\cal E}_{\\rm int}$,\n$\n\\bu_{\\sigma}^n\\cdot\\bn_{\\sigma,K}+\\bu_\\sigma^n\\cdot{\\vc n}_{\\sigma,L}=0;$ whence, finally\n\\begin{equation}\\label{T3.1}\nT_{3,1}= \\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\Big(\\hat\n{\\bu}_{\\sigma}^{n,{\\rm up}}-\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}\\Big)\\cdot\\Big({\\bU}^n_{h,K}-\\bU^n_\\sigma\\Big) \\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}\n\\end{equation}\n{ Let us now decompose the term $ T_{3,1}$ as\n\\begin{equation*}\n \\begin{aligned}\n&T_{3,1}= T_{3,2}+ R_{3,2}, \\mbox{ with } { R_{3,2}=\\deltat\\sum_{n=1}^mR^{n}_{3,2}}, \\\\\n&T_{3,2}= \\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\n\\Big(\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}-\\hat{\\bu}^{n,{\\rm up}}_{\\sigma}\\Big)\\cdot\\Big(\\bU^n_\\sigma-{\\bU}^n_{h,K}\\Big) \\hat\\bu_\\sigma^{n,{\\rm up}}\\cdot\\bn_{\\sigma,K}, \\mbox{ and }\\\\\n&{ R^{n}_{3,2}}=\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|\\vr_\\sigma^{n,{\\rm up}}\n\\Big(\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}-\\hat{\\bu}^{n,{\\rm up}}_{\\sigma}\\Big)\\cdot\\Big(\\bU^n_\\sigma-{\\bU}^n_{h,K}\\Big) \\Big(\\bu_{\\sigma}^n-\\hat\\bu_\\sigma^{n,{\\rm up}}\\Big)\\cdot\\bn_{\\sigma,K}.\n \\end{aligned}\n\\end{equation*}\nBy virtue of { discrete} H\\\"older's inequality and the { first order Taylor formula applied to function $x\\mapsto \\vc U^n(x)$ in order to evaluate the difference $\\bU^n_\\sigma-{\\bU}^n_{h,K}$}, we get\n\\begin{equation*}\n \\begin{aligned}\n\t{ |R^{n}_{3,2}|} & \\le\n\tc \\|\\nabla\\vc U\\|_{L^\\infty(Q_T;\\Rm^9)}\\Big(\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma|\n\\vr_\\sigma^{n,{\\rm up}}\\Big|\\hat\\bu_\\sigma^{n,{\\rm up}}-\\hat\\bU_{h,\\sigma}^{n,{\\rm up}}\\Big|^2\\Big)^{1\/2}\\\\\n& \\times\n\\Big(\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma|\n|\\vr_\\sigma^{n,{\\rm up}}|^{\\gamma_0}\\Big)^{1\/(2{\\gamma_0})}\\Big(\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma|\n\\Big|\\bu_\\sigma^{n}-\\hat\\bu_\\sigma^{n,{\\rm up}}\\Big|^q\\Big)^{1\/q},\n \\end{aligned}\n\\end{equation*}\nwhere $\\frac 12+\\frac 1{2\\gamma_0}+\\frac 1q=1$, $\\gamma_0={\\rm min}\\{\\gamma, 2\\}$ and $\\gamma\\ge 3\/2$. For the sum in the last term of the above product, we have\n$$\n\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma|\n\\Big|\\bu_\\sigma^{n}-\\hat\\bu_\\sigma^{n,{\\rm up}}\\Big|^q\\le c \\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma|\n|\\bu_\\sigma^{n}-\\bu_K^{n}|^q\n$$\n$$\n\\le c \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\n\\Big(\\|\\bu_\\sigma^{n}-\\bu^n\\|_{L^q(K;\\Rm^3)}^q+\n\\sum_{K\\in {\\cal T}}\n\\|\\bu^n-\\bu_K^{n}\\|_{L^q(K;\\Rm^3)}^q\\Big)\n\\le c h^{\\frac {2\\gamma_0-3}{2\\gamma_0}q}|\\bu^n|_{V^2_h(\\Omega;\\Rm^3)}^q,\n$$\nwhere we have used the definition (\\ref{upwind1}), the Minkowski inequality and the interpolation inequalities\n(\\ref{L2+-1}--\\ref{L2+-2}). Now we can go back to the estimate of $R_{3,2}^n$ taking into account the upper bounds\n(\\ref{est0}), (\\ref{est3}--\\ref{est4}), in order to get\n \\begin{equation}\\label{R3.2}\n |R_{3,2}|\\le h^A\\;c (M_0,E_0,\\|\\nabla\\vc U\\|_{L^\\infty(Q_T;\\Rm^9)})\n \\end{equation}\n provided $\\gamma\\ge 3\/2$, where $A$ is given in (\\ref{A1}).\n\n Finally, we rewrite term $T_{3,2}$ as\n \\begin{equation}\\label{T3}\n \\begin{aligned}\n&T_{3,2}= T_{3,3}+ R_{3,3}, \\mbox{ with } { R_{3,3}=\\deltat\\sum_{n=1}^mR^{n}_{3,3}}, \\\\\n&T_{3,3}= \\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\n\\Big(\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}-\\hat{\\bu}^{n,{\\rm up}}_{\\sigma}\\Big)\\cdot\\Big(\\bU^n_\\sigma-{\\bU}^n_{h,K}\\Big) \\hat\\bU_{h,\\sigma}^{n,{\\rm up}}\\cdot\\bn_{\\sigma,K}, \\mbox{ and }\\\\\n&{ R^{n}_{3,3}}=\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|\\vr_\\sigma^{n,{\\rm up}}\n\\Big(\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}-\\hat{\\bu}^{n,{\\rm up}}_{\\sigma}\\Big)\\cdot\\Big(\\bU^n_\\sigma-{\\bU}^n_{h,K}\\Big) \\Big(\\hat\\bu_\\sigma^{n,{\\rm up}}-\\hat \\bU_{h,\\sigma}^{n,{\\rm up}}\\Big)\\cdot\\bn_{\\sigma,K};\n \\end{aligned}\n\\end{equation}\nwhence\n\\begin{equation}\\label{R3.3}\n|R_{3,3}|\\le c(\\|\\nabla\\bU\\|_{L^\\infty(Q_T,\\Rm^9)})\\; \\deltat\\sum_{n=1}^m{\\cal E}(\\vr^n,\\vc u^n\\,|\\, r^n,\\bU^n).\n\\end{equation}\n\n }\n\n\n\n\\vspace{2mm}\\noindent\n{\\bf Step 3:} {\\it Term $T_4$.}\n\\label{6.3}\nUsing the Stokes formula and the property (\\ref{L1-1}) in Lemma \\ref{Lemma1}, we easily see that\n \\begin{equation}\\label{T4}\n T_4=-\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{{\\rm \\int_K}} p(\\vr_K^n)\\dv\\bU^n\\dx.\n \\end{equation}\n\n\\vspace{2mm}\\noindent\n{\\bf Step 4:} {\\it Term $T_5$.}\n \\label{6.4}\nUsing the Taylor formula, we get\n\\[\nH'(r_K^n)-H'(r_K^{n-1})=H''(r_K^{n})(r_K^n-r_K^{n-1}) -\\frac 12H'''(\\overline r_K^n)(r_K^n-r_K^{n-1})^2,\n\\]\nwhere $\\overline r_K^n\\in[\\min(r_K^{n-1},r_K^n), \\max(r_K^{n-1},r_K^n)]$;\nwe infer\n\\begin{equation*}\n\\begin{aligned}\n\t& T_5= T_{5,1}+ R_{5,1},\\mbox{ with } T_{5,1}=\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}|K| ( r^n_K-\\vr^n_K)\\frac{p'(r_K^n)}{r_K^n} \\frac{r_K^n-r_K^{n-1}}{\\deltat}, \\, R_{5,1}=\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}} R_{5,1}^{n,K}, \\mbox{ and } \\\\\n\t& R_{5,1}^{n,K}= \\frac 12|K|H'''(\\overline r_K^n)\\frac{(r_K^n-r_K^{n-1})^2}{\\deltat}(\\vr_K^n-r_K^n).\n \\end{aligned}\n\\end{equation*}\nConsequently, by the { first order Taylor formula applied to function $t\\mapsto r(t,x)$ on the interval $(t_{n-1}, t_n)$} and thanks to the mass conservation \\eqref{masscons}\n\\begin{equation}\\label{R5.1}\n\t|R_{5,1}|\\le \\deltat \\;c(M_0,\\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r]},\\|\\partial_t r\\|_{L^\\infty(Q_T)}),\n\\end{equation}\nwhere $\\underline r$, $\\overline r$ are defined in (\\ref{dr,U}).\n\n\\vspace{2mm}\nLet us now decompose $T_{5,1}$ as follows:\n\\begin{equation}\\label{T5}\n\\begin{aligned}\n\t& T_{5,1}=T_{5,2}+ R_{5,2}, \\mbox{ with }T_{5,2}=\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{ \\int_K} ( r^n_K-\\vr^n_K)\\frac{p'(r_K^n)}{r_K^n} [\\partial_t r]^n { {\\rm d}x}, \\, R_{5,2}=\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}} R_{5,2}^{n,K}, \\mbox{ and} \\\\\n\t& R_{5,2}^{n,K}= {\\int_K} ( r^n_K-\\vr^n_K)\\frac{p'(r_K^n)}{r_K^n}\\Big(\\frac{r_K^n-r_K^{n-1}}{\\deltat} -[\\partial_t r]^n\\Big){ {\\rm d} x}. \\end{aligned}\n\\end{equation}\nIn accordance with (\\ref{notation2-}), here and in the sequel, $[\\partial_t r]^n(x)=\\partial_t r(t_n,x)$.\n{ We write using twice the Taylor formula in the integral form and the Fubini theorem,\n$$\n|R_{5,2}^{n,K}|= \\frac 1\\deltat\\Big|{p'(r^n_K)}{r^n_K} (\\vr^n_K-r^n_K)\\int_K\\int^{t_n}_{t_{n-1}}\\int_s^{t_n}\\partial_t ^2 r(z){\\rm d}z{\\rm d}s {\\rm d}x\\Big|\n$$\n$$\n\\le \\frac {p'(r^n_K)}{r^n_K}\\int^{t_n}_{t_{n-1}}\\int_K|\\vr^n_K-r^n_K|\\Big|\\partial_t ^2 r(z)\\Big|{\\rm d}x{\\rm d}z{\\rm d}s\n$$\n$$\n\\le \\frac{p'(r^n_K)}{r^n_K}\n\\|\\vr^n-\\hat r^n\\|_{L^{\\gamma}(K)}\\int^{t_n}_{t_{n-1}}\\|\\partial_t^2 r(z)\\|_{L^{\\gamma'}(K)}{\\rm d z}{\\rm d s}.\n$$\nTherefore, by virtue of Corollary \\ref{Corollary1}, we have estimate\n\\begin{equation}\\label{R5.2}\n\t|R_{5,2}|\\le \\deltat\\; c(M_0, E_0,\\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r]},\\|\\partial^2_t r\\|_{L^1(0,T; L^{\\gamma'}(\\Omega)}).\n\\end{equation}\n}\n\n\n\n\\vspace{2mm}\\noindent\n{\\bf Step 5:} {\\it Term $T_6$.}\nUsing the same argumentation as in formula (\\ref{T3.1}), we may write\n\\begin{equation}\n \\begin{aligned}\n\t &T_6=T_{6,1} + R_{6,1},\\quad R_{6,1}=\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}R_{6,1}^{n,\\sigma,K}, \\mbox{ with} \\\\\n\t&T_{6,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|\\vr_K^{n}\\Big( H'( r_K^{ {n-1}})-H'(r_\\sigma^{ {n-1}})\\Big)\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}, \\mbox{ and} \\\\\n\t& R_{6,1}^{n,\\sigma,K}=|\\sigma|\\Big(\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n}\\Big)\\Big( H'( r_K^{ {n-1}})-H'(r_\\sigma^{ {n-1}})\\Big)\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}, { \\mbox{ for } \\sigma=K|L.}\n \\end{aligned}\n\\end{equation}\nWe estimate this term separately for $\\gamma\\le 2$ and $\\gamma>2$.\nIf $\\gamma\\le 2$, motivated by Lemma \\ref{Lemma5}, we may write\n\\begin{multline}\\label{R6.1a}\n\t |R_{6,1} ^{n,\\sigma,K}| \\le \\sqrt h\\, \\|\\nabla H'(r)\\|_{L^\\infty(Q_T;\\Rm^3)} |\\sigma| \\\\\n\t\\times \\Big( \\frac { |\\vr_\\sigma^{n,{\\rm up}} - \\vr_K^{n} |} {\\max(\\vr_K,\\vr_L)^{(2-\\gamma)\/2}} \\,\\sqrt{|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}}| 1_{\\overline\\vr_\\sigma^n\\ge1} \\sqrt h(\\vr^n_K+\\vr^n_L)^{(2-\\gamma)\/2} \\sqrt{|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}}|\n\t\\\\ + |\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n}|\\,\\sqrt{|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}|} 1_{\\overline\\vr_\\sigma^n<1}\n\\sqrt h \\sqrt{|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}|} \\Big),\n\\end{multline}\nwhere we again use the { first order Taylor formula applied to function $H'$ between endpoints $r_K^{n-1}$,\n$r_\\sigma^{n-1}$,} and where the numbers $\\overline\\vr_\\sigma^n$ are defined in Lemma \\ref{Theorem3}.\nConsequently, an application of the H\\\"older and Young inequalities yields\n\\begin{equation}\\label{R6.1b}\n \\begin{aligned}\n |R_{6,1}|\n\t& \\le \\sqrt h\\, { c} \\|\\nabla H'(r)\\|_{L^\\infty(Q_T;\\Rm^3)}\n\t\t\\deltat \\sum_{n=1}^m\\Big[\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma| \\frac {(\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n})^2}{\\max(\\vr_K,\\vr_L)^{(2-\\gamma)}}\\,|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}| 1_{\\overline\\vr_\\sigma^n\\ge1}\\Big)^{1\/2}\n\t\t\\\\ & \\hspace{6cm} \\times \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h\\vr_K^{2-\\gamma} \\,|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}|\\Big)^{1\/2}\n\t\t\\\\ & \\quad + \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|h (\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n})^2\\,|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}| 1_{\\overline\\vr_\\sigma^n<1}\\Big)^{1\/2}\n \t\t \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h \\,|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}|\\Big)^{1\/2}\\Big]\n\t\\\\ & \\le \\sqrt h\\, { c} \\|\\nabla H'(r)\\|_{L^\\infty(Q_T;\\Rm^3)} \\deltat \\sum_{n=1}^m\\Big[\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in \t{\\cal E}(K)}|\\sigma| \\frac {(\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n})^2} {\\max(\\vr_K,\\vr_L)^{(2-\\gamma)}} \\, |\\bu_\\sigma^n \\cdot \\bn_{\\sigma,K}| 1_{\\overline\\vr_\\sigma^n\\ge1}\n\t\t\\\\ & \\qquad \\qquad + \\Big(\\sum_{K\\in {\\cal T}} |K|\\vr_K^{6(2-\\gamma)\/5}\\Big)^{5\/6} \\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/6}\n\t\t\\\\ & \\qquad \\qquad + \\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|h (\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n})^2\\,|\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}| 1_{\\overline\\vr_K^n<1}+ |\\Omega|^{5\/6} \\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/6}\\Big]\n\\end{aligned}\n\\end{equation}\nWe deduce employing the discrete H\\\"older inequality\n{\n$$\n\\deltat \\sum_{n=1}^m\\Big(\\sum_{K\\in {\\cal T}} |K|\\vr_K^{6(2-\\gamma)\/5}\\Big)^{5\/6} \\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/6}\n$$\n$$\n\\le \\Big[\\deltat \\sum_{n=1}^m \\Big(\\sum_{K\\in {\\cal T}} |K|\\vr_K^{6(2-\\gamma)\/5}\\Big)^{5\/3} \\Big]^{1\/2}\\Big[\\deltat \\sum_{n=1}^m\\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/3}\\Big]^{1\/2},\n$$\nand\n$$\n\\deltat\\sum_{n=1}^m\\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/6}\\le \\sqrt T\n\\Big[k\\sum_{n=1}^m\\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/3}\\Big]^{1\/2}\n$$\n}\nComing back to (\\ref{R6.1b}) we deduce that\n\\begin{equation}\\label{R6.1}\n|R_{6,1}|\n\\le \\sqrt h \\; c(M_0,E_0,\\underline r,\\overline r, |p'|_{C([\\underline r,\\overline r])}, \\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)})\n\\end{equation}\nprovided $\\gamma\\ge 12\/11$, where we use estimate (\\ref{dissipative1}), estimates (\\ref{est1}), (\\ref{est3}) of Corollary \\ref{Corollary1} and equivalence relation (\\ref{norms1}).\nIn the case $\\gamma>2$, the same final bound may be obtained by a similar argument, replacing the estimate (\\ref{dissipative1}) by (\\ref{dissipative2}).\n\n\\vspace{2mm}\nLet us now decompose the term $T_{6,1}$ as\n{\n\\begin{equation*}\n\\begin{aligned}\n\t&T_{6,1}=T_{6,2}+ R_{6,2}, \\mbox{ with } T_{6,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}\n|\\sigma|\\vr^n_K H''(r_K^{ n-1})(r_K^{ n-1}-r_\\sigma^{ n-1}) [\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}],\n \\\\ & R_{6,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal K}}\\sum_{\\sigma\\in {\\cal E}(K) }\nR_{6,2}^{n,\\sigma,K},\\;\n\\mbox{and }\\\\ & R_{6,2}^{n,\\sigma,K}=|\\sigma|\\vr^n_K \\Big(H'(r^{n-1}_K)-H'(r^{n-1}_\\sigma)-H''(r^{n-1}_K)(r^{n-1}_K-r^{n-1}_\\sigma)\\Big)\n[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}]\n \\end{aligned}\n\\end{equation*}\n}\nTherefore, by virtue of the { second order Taylor formula applied to function H'}, H\\\"older's inequality, (\\ref{sob1}), (\\ref{norms1}),\nand (\\ref{est0}), (\\ref{est3}) in Corollary \\ref{Corollary1}, we have, provided\n$\\gamma\\ge 6\/5$,\n{\n\\begin{align}\n| R_{6,2}| &\\le h c \\Big(|H''|_{C([\\underline r,\\overline r])}+|H'''|_{C([\\underline r,\\overline r])}\\Big)\\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)} \\|\\vr\\|_{L^\\infty(0,T;L^\\gamma(\\Omega))} \\|\\bu\\|_{L^2(0,T;V_h^2(\\Omega;\\Rm^3))}\n\\nonumber \\\\\n\\label{R6.2}\n& \\le h\\;\nc(M_0,E_0,\\underline r, \\overline r, |p'|_{C^1([\\underline r,\\overline r])}, \\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)} ),\n\\end{align}\n}\nwhere in the first line we have used notation (\\ref{notation1}).\n\n\\vspace{2mm}\n\nLet us now deal with the term $T_{6,2}$.\nNoting that $ \\displaystyle\n\\int_K\\nabla r^{ n-1} \\dx = \\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|(r_\\sigma^{ n-1}-r_K^{ n-1})\\bn_{\\sigma,K},$ we may write\n\\begin{multline*}\n\\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|\\vr^n_K H''(r_K^{ n-1})(r_K^{ n-1}-r_\\sigma^{ n-1}) [\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}]\n\\\\= -\\int_K\\vr^n_K H''(r_K^{ n-1}) \\bu^n_K\\cdot\\nabla r^{ n-1}\\dx +\n\\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|\\vr^n_K H''(r_K^{ n-1})(r_K^{ n-1}-r_\\sigma^{ n-1}) (\\bu^n_\\sigma-\\bu^n_K)\\cdot\\bn_{\\sigma,K}.\n\\end{multline*}\nConsequently, $T_{6,2}= T_{6,3}+ R_{6,3},$ with\n$$\n\\begin{aligned}\n& T_{6,3}=\n -\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\int_K{\\vr^n_K} H''(r_K^{ n-1}) \\bu^n \\cdot\\nabla r^{ n-1}\\dx, \\\\\n& R_{6,3} = \\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}} \\int_K\\vr^n_K H''(r_K^{ n-1})(\\bu^n- \\bu^n_K)\\cdot\\nabla r^{ n-1}\\dx\n \\\\ & \\hspace{3cm}+\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|\\vr^n_K H''(r_K^{ n-1})(r_K^{ n-1}-r_\\sigma^{ n-1})\n(\\bu^n_\\sigma-\\bu^n_K)\\cdot\\bn_{\\sigma,K},\n\\end{aligned}\n$$\n{\n$$\n|R_{6,3}|\\le c\\,\\|H''(r)\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)}\\Big[ \\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\|\\vr^n_K\\|_{L^{\\gamma_0}(K)}\\|\\bu^n- \\bu^n_K\\|_{L^{\\gamma_0'}(K;\\Rm^3)}+\n$$\n$$\n\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)} \\|\\vr^n_K\\|_{L^{\\gamma_0}(K)}\\|\\bu_\\sigma^n- \\bu^n_K\\|_{L^{\\gamma_0'}(K;\\Rm^3)}\\Big],\\quad \\gamma_0=\\min\\{\\gamma,2\\},\n$$\nwhere we have used the H\\\"older inequality, and also the Taylor formula applied to function $x\\mapsto r(t_{n-1},x)$\ntogether with equivalence relation (\\ref{reg1}) yielding $|\\sigma|h\\le |K|$, to treat the second term.\nConsequently, by virtue of H\\\"older's inequality, interpolation inequality (\\ref{interpol1}) (to estimate\n$\\|\\vc u^n-\\vc u^n_K\\|_{L^{\\gamma_0'}(K;\\Rm^3)}$ by $h^{(5\\gamma_0-6)\/(2\\gamma_0)}\\|\\Grad\\vc u^n \\|_{L^2(K;\\Rm^9)}$, $\\gamma_0=\\min\\{\\gamma,2\\}$)\nin the first term, and by the the H\\\"older inequality\n and (\\ref{interpol1}--\\ref{interpol2}) (to estimate $\\|\\vc u_\\sigma^n-\\vc u^n_K\\|_{L^{\\gamma'}(K;\\Rm^3)}$ by $h^{(5\\gamma_0-6)\/(2\\gamma_0)}\\|\\Grad\\vc u^n \\|_{L^2(K;\\Rm^9)}$) in the second term, we get\n$$\n|R_{6,3}|\\le c\\,h^{(5\\gamma_0-6)\/(2\\gamma_0)}\\,\\|H''(r)\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)} \\deltat\\sum_{n=1}^m\\Big(\\sum_{K \\in {\\cal T}}\\|\\vr^n_K\\|^2_{L^{\\gamma_0}(K)}\\Big)^{1\/2}\\Big(\\sum_{K \\in {\\cal T}}\\|\\Grad\\bu^n\\|^2_{L^{2}(K;\\Rm^9)}\\Big)^{1\/2}\n$$\n$$\n\\le\nc\\,h^{(5\\gamma_0-6)\/(2\\gamma_0)}\\,\\|H''(r)\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)} \\deltat\\sum_{n=1}^m\\Big(\\sum_{K \\in {\\cal T}}\\|\\vr^n_K\\|^{\\gamma_0}_{L^{\\gamma_0}(K)}\\Big)^{1\/\\gamma_0}\\Big(\\sum_{K \\in {\\cal T}}\\|\\Grad\\bu^n\\|^2_{L^{2}(K;\\Rm^9)}\\Big)^{1\/2},\n$$\nprovided $\\gamma\\ge 6\/5$,\nwhere we have used the discrete H\\\"older inequality and the algebraic inequality (\\ref{dod1*}).\n}\nNow it remains to use (\\ref{est0}), (\\ref{est3}) in Corollary \\ref{Corollary1} in order to get\n\\begin{equation}\\label{R6.3}\n|R_{6,3}|\\le h^A \\;c(M_0,E_0,\\underline r, \\overline r, |p'|_{C^1([\\underline r,\\overline r])} \\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)} ),\n\\end{equation}\nwhere $A$ is defined in \\eqref{A1}.\n\n{\nFinally we write\n $T_{6,3}= T_{6,4}+ R_{6,4},$ with\n\\begin{equation}\\label{T6}\n\\begin{aligned}\n& T_{6,4}=\n -\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\int_K{\\vr^n_K}\\frac{ p'(r_K^{n})}{r_K^n} \\bu^n\\cdot\\nabla r^{ n}\\dx, \\\\\n& R_{6,4} = \\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}} \\int_K\\vr^n_K \\Big(H''(r_K^{ n})\\nabla r^{n}-\nH''(r_K^{n-1})\\nabla r^{n-1}\\Big)\\cdot\\bu^n\\dx,\n\\end{aligned}\n\\end{equation}\n{ where by the same token as in (\\ref{R5.2}),\n\\begin{equation}\\label{R6.4}\n|R_{6,4}|\\le \\deltat\\; c (M_0,E_0, \\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r])}, \\|\\nabla r, \\partial_t r\\|_{L^\\infty(Q_T;\\Rm^4)}, \\|\\partial_t\\nabla r\\|_{L^2(0,T; L^{{6\\gamma}\/{(5\\gamma-6)}}(\\Omega;\\Rm^3))}).\n\\end{equation}\n}\n}\n\n\n\\vspace{2mm}\nWe are now in position to conclude the proof of Lemma \\ref{6.6}: we obtain the inequality \\eqref{relativeenergy-} by gathering the principal terms (\\ref{T2}), (\\ref{T3}), (\\ref{T4}), (\\ref{T5}), (\\ref{T6}) and the residual terms estimated in (\\ref{R2.1}), (\\ref{R2.2}), (\\ref{R3.1}), (\\ref{R3.2}), (\\ref{R3.3}), (\\ref{R5.1}), (\\ref{R5.2}), (\\ref{R6.1a}), \\eqref{R6.1b}, (\\ref{R6.2}), (\\ref{R6.3}), { (\\ref{R6.4}) at} the right hand side $\\sum_{i=1}^6 T_i$ of the discrete relative energy inequality (\\ref{drelativeenergy}).\n\n\\end{proof}\n\n\\section{ A discrete identity satisfied by the strong solution}\\label{7}\n{ This section is devoted to the proof of a discrete identity satisfied by any strong solution. This identity\nis stated in Lemma \\ref{strongentropy} below. It will be used in combination with the approximate relative energy inequality stated in Lemma \\ref{refrelenergy} to deduce the convenient form of the relative energy inequality verified by any function being a strong solution to the compressible Navier-Stokes system. This last step is performed in the next section.}\n\n\n\\begin{lm}[A discrete identity for strong solutions]\\label{strongentropy}\nSuppose that $\\Omega\\subset \\R^3$ is a bounded polyhedral domain and ${\\cal T}$ a regular triangulation { introduced in Section \\ref{3.1}}.\nLet the pressure $p$ be a $C^2(0,\\infty)$ function satisfying hypotheses (\\ref{hypp}) and (\\ref{pressure1}) with\n$\\gamma\\ge 3\/2$. Let $(r,\\bU)$ belong to the class (\\ref{dr,U}) satisfy equation \\eqref{pbcont} with the viscosity coefficients $\\mu$, $\\lambda$ obeying (\\ref{visc}).\n\nLet $(\\vr^0,\\bu^0) \\in L_h^{+}(\\Omega) \\times \\bW_h(\\Omega)$ and suppose that $(\\vr^n)_{1\\le n \\le N}\\in [L_h^{+}(\\Omega)]^N$, $(\\bu^n)_{1\\le n \\le N} \\in [\\bW_h(\\Omega)]^N$ is a solution of the discrete problem \\eqref{scheme}.\nThen there exists\n\\begin{align*}\n& c=c\\Big(M_0, E_0, \\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r])},\n\\|(\\nabla r, \\partial_t r, \\bU, \\nabla\\bU, { \\nabla^2\\bU}, \\partial_t\\bU, \\partial_t^2\\bU,\\partial_t\\nabla\\bU,)\\|_{L^\\infty (Q_T;\\Rm^{58})})\\Big)> 0,\n\\end{align*}\nsuch that for any $m=1,\\ldots,N$, the following identity holds:\n\\begin{equation}\\label{strong1}\n\\begin{aligned}\n\t&\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_K\\Big(\\mu\\nabla\\bU_h^n\\cdot\\nabla(\\bu^n-\\bU^n_{ h})+ (\\mu +\\lambda)\\dv \\bU^n_{ h}\\dv (\\bu^n-\\bU^n_{ h})\\Big)\\dx\n\\\\\n\t&\\qquad +\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_Kr_K^{n-1}\\frac{\\bU^n_{ h,K}-\\bU^{n-1}_{ h,K}}{\\deltat}\\cdot (\\bu_K^{ n}-\\bU_{h,K}^{ n})\\dx\n\\\\\n\t &\\qquad+ { \\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| \\hat r_\\sigma^{n,{\\rm up}}\n[\\hat \\bU_{h,\\sigma}^{n,{\\rm up}}\\cdot{\\vc n}_{\\sigma,K}](\\bU_\\sigma^n-\\bU^n_{h,K})\\cdot(\\hat\\bu_\\sigma^{n,{\\rm up}}-\\hat\\bU_{h,\\sigma}^{n,{\\rm up}})}\n\\\\\n\t&\\qquad +\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K p(r_K^n)\\dv \\bU^n\\dx\n+\\deltat \\sum_{n=1}^m \\sum_{K\\in{\\cal T}}\\int_K p'(r^n_K)\\bu^n\\cdot\\nabla r^n\\dx\n{ + {\\cal R}^m_{h,\\deltat} =0},\n\\end{aligned}\n\\end{equation}\n{ where\n\\[\n|{\\cal R}_{h, \\deltat}^m|\\le c\\Big( h + \\deltat\\Big)\n\\]\n { and where we have used notation (\\ref{notation2-}) for $r^n$, $\\bU^n$ and (\\ref{vhat}--\\ref{vtilde}) for $\\vc U^n_h$, $\\vc U^n_{h,K}$, $r^n_K$, $\\vc u^n_{\\sigma}$.}\n}\n\\end{lm}\n{ Before starting the proof we recall an auxiliary algebraic inequality whose straightforward proof is left to the reader, and introduce some notations.}\n\\begin{lm}\\label{LL1}\n\tLet $p$ satisfies assumptions (\\ref{hypp}) and (\\ref{pressure1}).\n\tLet $00$ such that for all $\\vr\\in [0,\\infty)$ and $r\\in [a,b]$ there holds\n\t\\[\n\t\tE(\\vr|r)\\ge c(a,b)\\Big(1_{R_+\\setminus [a\/2,2 b]}{ (\\vr)}+\\vr^\\gamma 1_{R_+\\setminus [a\/2,2 b]}{ (\\vr)}+ (\\vr-r)^2 1_{[a\/2,2 b]}{ (\\vr)}\\Big),\n\t\\]\n\twhere $E(\\vr|r)$ is defined in (\\ref{E}).\n\\end{lm}\n{ If we take in Lemma \\ref{LL1} $\\vr=\\vr^n(x)$, $\\vr^n\\in L^+_h(\\Omega)$, $r=\\hat r^n(x)$, $a=\\underline r$, $b=\\overline r$ (where r is a function belonging to class (\\ref{dr,U}) and $\\underline r$, $\\overline r$\nare its lower and upper bounds, respectively), we obtain\n\\begin{equation}\\label{added}\nE(\\vr^n(x)|\\hat r^n(x))\\ge c(\\underline r,\\overline r)\\Big(1_{R_+\\setminus [\\underline r\/2,2 \\overline r]}{ (\\vr^n(x))}+(\\vr^n)^\\gamma(x) 1_{R_+\\setminus [\\underline r\/2,2 \\overline r]}{ (\\vr^n(x))}+ (\\vr^n(x)-\\hat r^n(x))^2 1_{[\\underline r\/2,2 \\overline r]}{ (\\vr^n(x))}\\Big)\n\\end{equation}\n}\n Now, for fixed numbers $\\underline r$ and $\\overline r$ { and fixed functions $\\vr^n\\in L^+_h(\\Omega)$, $n=0,\\ldots, N$, we introduce the residual and essential subsets of $\\Omega$ (relative to $\\vr^n$) as follows:}\n\\begin{equation}\\label{essres}\nN_{\\rm ess}^n=\\{x\\in\\Omega\\,\\Big|\\,\\frac 12\\underline r\\le \\vr^n(x)\\le 2\\overline r\\},\\;\nN_{\\rm res}^n= \\Omega\\setminus N_{\\rm ess}^n.\n\\end{equation}\nand we set\n\\[\n[g]_{\\rm ess}{ (x)}= g{ (x)} 1_{N^n_{\\rm ess}}{ (x)},\\; [g]_{\\rm res}{ (x)}= g { (x)}1_{N^n_{\\rm res}}{ (x)},\\;\\; { x\\in \\Omega},\\;\\;g\\in L^1(\\Omega).\n\\]\n\n Integrating inequality (\\ref{added}) we deduce\n\\begin{equation}\\label{rentropy}\nc(\\underline r,\\overline r)\\sum_{K\\in{\\cal T}} \\int_K{\\Big(\\Big[1\\Big]_{\\rm res}+\\Big[(\\vr^n)^\\gamma\\Big]_{\\rm res}+\\Big[\\vr^n-\\hat r^n\\Big]^2_{\\rm ess}\\Big)}\\dx\\le{\\cal E}(\\vr^n,\\bu^n\\Big| r^n,\\bU^n).\n\\end{equation}\nfor any pair $(r,\\bU)$ belonging to the class (\\ref{dr,U}) and any $\\vr^n\\in L_h(\\Omega)$.\n\n{ We are now ready to proceed to the proof of Lemma \\ref{strongentropy}.}\n\n\\begin{proof}\n\nWe start by projecting the momentum equation to the discrete spaces.\\label{7.2}\nSince $(r,\\bU)$ satisfies \\eqref{pbcont} and belongs to the class (\\ref{dr,U}), Equation (\\ref{mov2}) can be rewritten in the form\n\\begin{equation}\\label{str1}\nr\\partial_t\\bU+r\\bU\\cdot\\nabla\\bU +\\nabla p(r)=\\mu\\Delta\\bU +(\\mu+\\lambda)\\nabla\\dv\\bU.\n\\end{equation}\nWe write equation (\\ref{str1}) at $t=t_n$, multiply scalarly by $\\bu^n-\\bU^n_{h}$, and integrate over $\\Omega$.\nWe get, after summation from $n=1$ to $m$,\n\\begin{equation}\\label{strong0}\n\\begin{aligned}\n& \\sum_{i=1}^5{\\cal T}_i=0, \\qquad\\qquad\\qquad \\mbox{ with } && {\\cal T}_1 = -\\deltat\\sum_{n=1}^m\\int_\\Omega\\Big(\\mu\\Delta \\bU^n+ (\\mu+\\lambda)\n\\nabla\\dv\\bU^n\\Big)\\cdot(\\bu^n-\\bU^n_h)\\dx, &&\\\\\n&{\\cal T}_2 =\\deltat\\sum_{n=1}^m\\int_\\Omega r^{n}[\\partial_t\\bU]^n\\cdot (\\bu^n-\\bU^n_h)\\dx, && {\\cal T}_3 = \\deltat \\sum_{n=1}^m\\int_\\Omega r^n\\bU^n\\cdot\\nabla\\bU^n\\cdot (\\bu^n-\\bU^n_h)\\dx \\\\\n&{\\cal T}_4 = \\deltat\\sum_{n=1}^m\\int_\\Omega\\nabla p(r^n)\\cdot\\bu^n \\dx, && {\\cal T}_5 =-\\deltat\\sum_{n=1}^m\\int_\\Omega\\nabla p(r^n)\\cdot\\bU^n_h\\dx.\n \\end{aligned}\n\\end{equation}\n\n\n\n In the steps below, we deal with each of the terms ${\\cal T}_i$.\n\n\\vspace{2mm}\n\\textbf{Step 1: }\\textit{Term ${\\cal T}_1$.}\\label{7.3}\n Integrating by parts, we get:\n {\n\\begin{equation}\\label{cT1}\n \\begin{aligned}\n&{\\cal T}_1={\\cal T}_{1,1} + {\\cal R}_{1,1},\n\\\\\n& \\mbox{with }{\\cal T}_{1,1} = \\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_K\\Big(\\mu\\nabla\\bU_{h}^n:\\nabla(\\bu^n-\\bU_h^n)+\n(\\mu +\\lambda)\\dv \\bU_{ h}^n\\dv (\\bu^n-\\bU_h^n)\\Big)\\dx\n \\\\\n &\\mbox{and }\\;\n {\\cal R}_{1,1}=I_1+I_2,\\;\\mbox{with}\\\\\n &I_1=\n \\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_K\\Big(\\mu\\nabla(\\bU^n-\\bU_{ h}^n):\\nabla(\\bu^n-\\bU_h^n)+ (\\mu+\\lambda)\n \\dv((\\bU^n-\\bU_{ h}^n)\\dv(\\bu^n-\\bU_h^n)\\Big)\n \\dx\n \\\\\n &\n I_2=-\\deltat\\sum_{n=1}^m \\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)} \\int_{\\sigma} \\Big(\\mu\\bn_{\\sigma,K} \\cdot\\nabla\\bU^n \\cdot(\\bu^n- \\bU_h^n) + (\\lambda+\\mu)\\dv\\bU^n(\\bu^n-\\bU^n_h)\\cdot\\bn_{\\sigma,K}\\Big)\\dS\n \\\\\n & =\n -\\deltat\\sum_{n=1}^m\\ \\sum_{\\sigma\\in {\\cal E}} \\int_{\\sigma} \\Big(\\mu\\bn_{\\sigma} \\cdot\\nabla\\bU^n \\cdot\\Big[\\bu^n- \\bU_h^n\\Big]_{\\sigma,\\vc n_\\sigma} + (\\lambda+\\mu)\\dv\\bU^n\\Big[\\bu^n-\\bU^n_h\\Big]_{\\sigma,\\vc n_\\sigma}\\cdot\\bn_{\\sigma}\\Big)\\dS,\n\\end{aligned}\n\\end{equation}\n}\nwhere in the last line $\\vc n_\\sigma$ is a unit normal to $\\sigma$ and $[\\cdot]_{\\sigma,\\vc n_\\sigma}$ is the jump over sigma (with respect to $\\vc n_\\sigma$) defined in Lemma \\ref{Lemma6}. Employing estimate (\\ref{L1-3})\nwe easily verify that\n{\n$$\n|I_1|\\le h\\;c(M_0,E_0, \\|\\nabla\\vc U\\|_{L^\\infty(0,T; W^{1,\\infty}(\\Omega))}).\n$$\nSince the integral over any face $\\sigma$ of the jump of a function from $V_h(\\Omega)$ is zero, we may write\n$$\nI_2=-\\deltat\\sum_{n=1}^m\\ \\sum_{\\sigma\\in {\\cal E}} \\int_{\\sigma} \\Big(\\mu\\bn_{\\sigma} \\cdot\\Big(\\nabla\\bU^n -[\\nabla\\bU^n]_\\sigma\\Big)\\cdot\\Big[\\bu^n- \\bU_h^n\\Big]_{\\sigma,\\vc n_\\sigma}\n$$\n$$\n+ (\\lambda+\\mu)\\Big(\\dv\\bU^n-[\\dv\\bU^n]_\\sigma\\Big)\\Big[\\bu^n-\\bU^n_h\\Big]_{\\sigma,\\vc n_\\sigma}\\cdot\\bn_{\\sigma}\\Big)\\dS;\n$$\nwhence by using the { first order Taylor formula applied to functions $x\\mapsto \\nabla \\vc U^n(x)$ to evaluate the differences $\\nabla\\bU^n -[\\nabla\\bU^n]_\\sigma$, $\\dv\\bU^n-[\\dv\\bU^n]_\\sigma$, }\n and H\\\"older's inequality,\n$$\n\\begin{aligned}\n&|I_2| \\le \\deltat\\, h\\; c\\, \\|\\nabla^2\\bU\\|_{L^\\infty(Q_T;\\Rm^{27})} \\sum_{n=1}^m\\sum_{\\sigma\\in {\\cal E}} \\sqrt{|\\sigma|}\\sqrt h\\;\\Big(\\frac 1{\\sqrt h}\\,\\Big\\|\\Big[\\bu^n-\\bU_h^n\\Big]_{\\sigma,\\vc n_\\sigma}\\Big\\|_{L^2(\\sigma;\\Rm^3)}\\Big)\\\\\n&\\le \\deltat\\, h\\; c\\, \\|\\nabla^2\\bU\\|_{L^\\infty(Q_T;\\Rm^{27})} \\sum_{n=1}^m\\sum_{\\sigma\\in {\\cal E}}\\Big(|\\sigma|h+\n\\frac 1h\\,\\Big\\|\\Big[\\bu^n-\\bU_h^n\\Big]_{\\sigma,\\vc n_\\sigma}\\Big\\|_{L^2(\\sigma;\\Rm^3)}^2\\Big).\n\\end{aligned}\n$$\nTherefore,\n\\begin{equation}\\label{cR1.1}\n|{\\cal R}_{1,1}|\\le h\\, c(M_0, E_0,\\|\\vc U, \\nabla\\bU, \\nabla^2\\bU\\|_{L^\\infty(Q_T,\\Rm^{39})}),\n\\end{equation}\nwhere we have employed Lemma \\ref{Lemma6} and\n (\\ref{est0}) in Corollary \\ref{Corollary1}.\n\n\n\n}\n\n\\vspace{2mm}\n\\textbf{Step 2:}\\textit{ Term ${\\cal T}_2$.}\\label{7.4}\nLet us now decompose the term ${\\cal T}_2$ as\n\\begin{align*}\n\t & {\\cal T}_2={\\cal T}_{2,1}+ {\\cal R}_{2,1},\\\\\n\t& \\mbox{with } {\\cal T}_{2,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_Kr^{n-1}\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\cdot (\\bu^n-\\bU^n_h)\\dx,\\quad {\\cal R}_{2,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{\\cal R}_{2,1}^{n,K}, \\\\\n\t& \\mbox{and }{ {\\cal R}_{2,1}^{n,K}=\\int_K(r^n-r^{n-1})[\\partial_t\\vc U]^n\\cdot(\\bu^n-\\bU^n_h)\\dx }+ \\int_Kr^{n-1}\\Big([\\partial_t \\bU]^n-\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\Big) \\cdot(\\bu^n-\\bU^n_h)\\dx.\n\\end{align*}\nThe remainder ${\\cal R}_{2,1}^{n,K}$ can be rewritten as follows\n$$\n{\\cal R}_{2,1}^{n,K}=\\int_K\\Big[\\int_{t_{n-1}}^{t_n}r(t,\\cdot){\\rm d}t\\Big][\\partial_t\\vc U]^n\\cdot(\\bu^n-\\bU^n_{h})\\dx\n $$\n $$\n + \\frac 1k\\int_Kr^{n-1}\\Big[\\int_{t_{n-1}}^{t_n}\\int_s^{t_n}\\partial^2_t \\bU(z,\\cdot){\\rm d}z{\\rm d}s\\Big] \\cdot(\\bu^n-\\bU_{h}^n)\\dx;\n$$\n{ whence,\n\\[\n|{\\cal R}_{2,1}^{n,K}|\\le \\deltat\\Big[ (\\|r\\|_{L^\\infty(Q_T)}+\\|\\partial_t r\\|_{L^\\infty(Q_T)}) (\\|\\partial_t\\bU\\|_{L^\\infty(Q_T;\\Rm^3)}|K|^{5\/6}(\\|\\bu^n\\|_{L^6(K)}+ \\|\\bU^n_h\\|_{L^6(K)})\n \\]\n \\[\n +\\|\\partial^2_t \\bU^n\\|_{L^{6\/5}(\\Omega;\\Rm^3))} (\\|\\bu^n\\|_{L^6(K)}+ \\|\\bU^n_h\\|_{L^6(K)}).\n\\]\nConsequently, by the same token as in (\\ref{R5.2}) or (\\ref{R6.4}),\n\\begin{equation}\\label{cR2.1}\n|{\\cal R}_{2,1}|\\le \\deltat\\, c\\Big(M_0, E_0,\\overline r,\\|(\\partial_t r, \\bU, \\partial_t\\bU, \\nabla\\bU , )\\|_{L^\\infty(Q_T;\\Rm^{16})}, \\|\\partial^2_t \\bU\\|_{L^2(0,T; L^{6\/5}(\\Omega;\\Rm^3))}\\Big),\n\\end{equation}\n where we have used the H\\\"older and Young inequalities, the estimates (\\ref{L1-2}), (\\ref{L1-4}), (\\ref{L1-6}), (\\ref{sob1}), and to the energy bound (\\ref{est0}) from Corollary \\ref{Corollary1}.\n }\n\n\n\n\\vspace{2mm}\n{\\bf Step 2a:} {\\it Term ${\\cal T}_{2,1}$.} We decompose the term ${\\cal T}_{2,1}$ as\n\\begin{align*}\n&{\\cal T}_{2,1}={\\cal T}_{2,2}+ {\\cal R}_{2,2}, \\\\\n&\\mbox{with } {\\cal T}_{2,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_Kr_K^{n-1}\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\cdot (\\bu^{n}-\\bU^{n}_h)\\dx, \\;\n {\\cal R}_{2,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{\\cal R}_{2,2}^{n,K}, \\\\\n&\\mbox{and }{\\cal R}_{2,2}^{n,K}=\\int_K(r^{n-1}-r_K^{n-1})\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\cdot(\\bu^{n}-\\bU_h^n)\\dx;\n\\end{align*}\ntherefore,\n\\[\n|{\\cal R}_{2,2}^{n}|= |\\sum_{K\\in {\\cal T}}{\\cal R}_{2,2}^{n,K}| \\le h \\, c\\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)}\\|\\partial_t\\bU\\|_{L^\\infty(Q_T;\\Rm^{3})}\\|\\bu^n-\\bU^n_h\\|_{L^6(\\Omega;\\Rm^3)}.\n\\]\nConsequently, by virtue of formula (\\ref{est1}) in Corollary \\ref{Corollary1} and estimates (\\ref{sob1}), (\\ref{L1-5}),\n\\begin{equation}\\label{cR2.2}\n|{\\cal R}_{2,2}|\\le h \\, c(M_0, E_0, \\|(\\nabla r, \\bU,\\partial_t\\bU,\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{18})}).\n\\end{equation}\n\n{\n\\vspace{2mm}\n{\\bf Step 2b:} {\\it Term ${\\cal T}_{2,2}$.} We decompose the term ${\\cal T}_{2,2}$ as\n\\begin{align*}\n&{\\cal T}_{2,2}={\\cal T}_{2,3}+ {\\cal R}_{2,3}, \\\\\n&\\mbox{with } {\\cal T}_{2,3}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_Kr_K^{n-1}\\frac{\\bU^n_{h,K}-\\bU^{n-1}_{h,K}}{\\deltat}\\cdot (\\bu^{n}-\\bU^{n}_h)\\dx, \\;\n {\\cal R}_{2,3}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{\\cal R}_{2,3}^{n,K}, \\\\\n&\\mbox{and }{\\cal R}_{2,3}^{n,K}=\\int_K r_K^{n-1}\\Big(\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}-\\Big[\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\Big]_h\\Big)\n\\cdot(\\bu^{n}-\\bU_h^n)\\dx\\\\ &\n+\n\\int_K r_K^{n-1}\\Big(\\Big[\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\Big]_h-\\Big[\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\Big]_{h,K}\\Big)\n\\cdot(\\bu^{n}-\\bU_h^n)\\dx { =I_1^K+I_2^K}.\n\\end{align*}\n{ We have\n$$\n|I_2^K|=\\frac 1 \\deltat r_K^{n-1}\\Big|\\int_K \\Big(\\Big[\\int_{t_{n-1}}^{t_n}\\partial_t\\bU(z){\\rm d}z\\Big]_h- \\Big[\\int_{t_{n-1}}^{t_n}\\partial_t\\bU(z){\\rm d}z\\Big]_{h,K}\\Big)\n\\cdot(\\bu^{n}-\\bU_h^n)\\dx\\Big|\n$$\n$$\n\\le \\frac h \\deltat r_K^{n-1}\\int_{t_{n-1}}^{t_n}\\Big\\|\\Grad\\Big[\\partial_t\\bU(z)\\Big]_h\\Big\\|_{L^{6\/5}(K;\\Rm^3)}\n\\|\\bu^n-\\bU^n_h\\|_{L^6(K;\\Rm^3)}\n$$\nwhere we have used the Fubini theorem, H\\\"older's inequality and (\\ref{L2-1}), (\\ref{L1-3})$_{s=1}$.\nFurther, employing the Sobolev inequality on Crouzeix-Raviart space (\\ref{sob1}), H\\\"older's and Young's inequalities and estimate (\\ref{L1-3})$_{s=1}$, we get\n\n$$\n\\sum_{K\\in {\\cal T}}|I_2^K|\\le \\frac h \\deltat r_K^{n-1} \\|\\bu^n-\\bU^n_h\\|_{L^6(\\Omega;\\Rm^3)}\\int_{t_{n-1}}^{t_n}\\Big\\|\\Grad\\partial_t\\bU(z)\\Big\\|_{L^{6\/5}(\\Omega;\\Rm^3)}\n{\\rm d}z\n$$\n$$\n\\le\nc r_K^{n-1} \\Big(h|\\bu^n-\\bU^n_h|^2_{V^2_h(\\Omega;\\Rm^3)}+\\frac h \\deltat \\int_{t_{n-1}}^{t_n}\n\\Big\\|\\Grad\\partial_t\\bU(z)\\Big\\|^2_{L^{6\/5}(\\Omega;\\Rm^3)}\\Big)\n$$\n\nWe reserve the similar treatment to the term $I_1^K$. Resuming these calculations we get by using Corollary \\ref{Corollary1}\n\\begin{equation}\\label{cR2.3}\n|{\\cal R}_{2,3}|\\le h \\, c(M_0, E_0, \\|(r, \\bU,\\nabla\\bU, \\partial_t\\bU)\\|_{L^\\infty(Q_T;\\Rm^{16})},\\|\\partial_t\\nabla\\bU\\|_{L^2(0,T;L^{6\/5}(\\Omega;\\Rm^{9}))}).\n\\end{equation}\n}\n}\n{ \\bf Step 2c:} {\\it Term ${\\cal T}_{2,3}$.}\n{ We rewrite this term in the form\n\\begin{equation}\\label{cT2}\\begin{aligned}\n& {\\cal T}_{2,3}={\\cal T}_{2,4}+ {\\cal R}_{2,4},\\; {\\cal R}_{2,4}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{\\cal R}_{2,4}^{n,K}, \\\\\n&\\mbox{with } {\\cal T}_{2,4}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}} \\int_Kr_K^{n-1} \\frac {\\bU^n_{h,K}-\\bU^{n-1}_{h,K}} {\\deltat}\\cdot (\\bu^{n}_K-\\bU^{n}_{h,K})\\dx,\n\\\\ &\\mbox{and }{\\cal R}_{2,4}^{n,K}=\\int_Kr_K^{n-1}\\frac{\\bU_{h,K}^n-\\bU_{h,K}^{n-1}}{\\deltat}\\cdot\\Big((\\bu^{n}-\\bu^n_K) -(\\bU_h^n-\\bU_{h,K}^n)\\Big)\\dx.\n\\end{aligned}\n\\end{equation}\n{\nFirst we write, as in (\\ref{R2.1}),\n$$\n\\Big|\\frac{[{\\bU}^{n}-{\\bU}^{n-1}]_{h,K}}{\\deltat}\\Big|=\n\\Big|\\frac 1{|K|}\\int_K\\Big[\\frac 1\\deltat \\Big[\\int_{t_{n-1}}^{t_n} \\partial_t\\vc U(z,x) {\\rm d} z\\Big]_h\\Big]{\\rm d}x\\Big|\n$$\n$$\n=\\Big|\\frac 1{|K|}\\int_K\\Big[\\frac 1\\deltat \\int_{t_{n-1}}^{t_n} [\\partial_t\\vc U(z) \\Big]_h(x){\\rm d} z\\Big]{\\rm d}x\\Big|\\le \\|[\\partial_t\\vc U ]_h\\|_{L^\\infty(0,T;L^\\infty(\\Omega;\\Rm^3))}\\le \\|\\partial_t\\vc U \\Big\\|_{L^\\infty(0,T;L^\\infty(\\Omega;\\Rm^3))},\n$$\nNext we evaluate $\\bu^n-\\bu^n_K$ employing (\\ref{L2-1})$_{p=2}$, and $\\bU_h^n-\\bU_{h,K}^n$ by using\n(\\ref{L2-1})$_{p=\\infty}$, (\\ref{L1-3})$_{s=1}$. Finally we employ the H\\\"older inequality to get\n\\begin{equation}\\label{cR2.4}\n|{\\cal R}_{2,4}|\\le h \\; c(M_0,E_0,\\overline r, \\|(\\partial_t\\bU, \\bU, \\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{15})}, \\|\\partial_t\\nabla\\bU )\\|_{L^2(0,T;L^{6\/5}(\\Omega;\\Rm^{9}))}).\n\\end{equation}\n}\n}\n\n\\vspace{2mm}\n\\textbf{Step 3:} \\textit{ Term ${\\cal T}_3$.}\nLet us first decompose ${\\cal T}_3$ as\n\\begin{align*}\n& {\\cal T}_3={\\cal T}_{3,1} + {\\cal R}_{3,1}, \\\\\n&\\mbox{with } {\\cal T}_{3,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\int_K r_K^n\\bU_{h,K}^n\\cdot\\nabla\\bU^n\\cdot (\\bu_K^n-\\bU_{h,K}^n)\\dx,\n \\quad {\\cal R}_{3,1}=\\deltat\\sum_{n=1}^m \\sum_{K\\in{\\cal T}}{\\cal R}_{3,1}^{n,K},\n\\\\ &\\mbox{and }{\\cal R}_{3,1}^{n, K}=\\int_K(r^n-r_K^n)\\bU^n\\cdot\\nabla\\bU^n\\cdot(\\bu^n-\\bU^n_h)\\dx\n+ \\int_K r_K^n(\\bU^n-\\bU^n_h)\\cdot\\nabla\\bU^n\\cdot(\\bu^n-\\bU^n_h)\\dx \\\\\n& \\phantom{\\mbox{and }{\\cal R}_{3,1}^{n, K}=} +\\int_K r_K^n(\\bU_h^n-\\bU^n_{h,K})\\cdot\\nabla\\bU^n\\cdot(\\bu^n-\\bU^n_h)\\dx +\\int_K r_K^n\\bU^n_{h,K}\\cdot\\nabla\\bU^n\\cdot\\Big(\\bu^n-\\bu^n_K-(\\bU^n_h-\\bU^n_{h,K})\\Big)\\dx.\n\\end{align*}\nWe find that\n\\begin{align*}\n&|{\\cal R}_{3,1}^{n,K}| \\le h\\,\\Big[ |K|^{1\/2} (\\|\\bu^n\\|_{L^2(K;\\Rm^3)}+ \\|\\bU_h^n\\|_{L^2(K;\\Rm^3)})+|K|^{ 1\/2} (\\|\\nabla\\bu^n\\|_{L^2(K;\\Rm^3)}+ \\|\\nabla\\bU_h^n\\|_{L^2(K;\\Rm^3)})\\Big]\n\\\\ & \\qquad\\qquad\\qquad\\qquad\\qquad \\times\n\\Big(\\|r\\|_{L^\\infty(Q_T)}+\\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)}\\Big)\\,\\Big(\\|\\bU\\|_{L^\\infty(Q_T;\\Rm^3)} +\\|\\nabla\\bU\\|_{L^\\infty(Q_T;\\Rm^{9})}\\Big)^2,\n\\end{align*}\nwhere we have used several times H\\\"older's inequality and the standard first order Taylor formula ({ to evaluate\n$r^n-r^n_K$)}, along with the estimates\n(\\ref{L1-2}) (to evaluate $\\bU^n-\\bU^n_h$), (\\ref{L2-1}), (\\ref{L1-3})$_{s=1}$ (to evaluate\n$\\bU^n_h-\\bU^n_{h,K}$), (\\ref{L2-1}) (to evaluate $\\bu^n-\\bu^n_K$).\n\nConsequently, using again (\\ref{L1-3})$_{s=1}$ (to estimate $\\|\\nabla\\bU_h^n\\|_{L^2(K;\\Rm^3)}$), the definition\n of $ |\\cdot|_{V^2_h(\\Omega)}$ norm, the Sobolev inequality (\\ref{sob1}) and the energy\nbound (\\ref{est0}) from Corollary \\ref{Corollary1}, we conclude that\n\\begin{equation}\\label{cR3.1}\n|{\\cal R}_{3,1}|\\le h\\; c(M_0, E_0,\\overline r, \\|(\\nabla r, \\bU, \\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{15})}).\n\\end{equation}\n\n\nNow we shall deal wit term ${\\cal T}_{3,1}$. Integrating by parts, we get:\n\n\n\n\n\n\n\n\n\n\\begin{align*}\n\t\\int_K r_K^n\\bU_{h,K}^n\\cdot\\nabla\\bU^n\\cdot (\\bu_K^n-\\bU_{h,K}^n)\\dx &=\\sum_{\\sigma\\in {\\cal E}(K)}{ |\\sigma|} r_K^n [\\bU_{h,K}^n\\cdot{\\vc n}_{\\sigma,K}]\\bU_\\sigma^n\\cdot(\\bu_K^n-\\bU_{h,K}^n)\n\t\\\\\n\t&=\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| r_K^n [\\bU_{h,K}^n\\cdot{\\vc n}_{\\sigma,K}](\\bU_\\sigma^n-\\bU^n_{h,K})\\cdot(\\bu_K^n-\\bU_{h,K}^n),\n\\end{align*}\nthanks to the the fact that $\\sum_{\\sigma\\in {\\cal E}(K)}\\int_\\sigma \\bU_{h,K}^n\\cdot{\\vc n}_{\\sigma,K}{\\rm d} S=0$.\n\n{ Next we write\n\\begin{align*}\n& {\\cal T}_{3,1}= {\\cal T}_{3,2}+ {\\cal R}_{3,2}, \\quad {\\cal R}_{3,2}=\\deltat\\sum_{n=1}^m {\\cal R}_{3,2}^{n},\n\\end{align*}\n\\begin{align}\n&\n{\\cal T}_{3,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| \\hat r_\\sigma^{n,{\\rm up}}\n[\\hat \\bU_{h,\\sigma}^{n,{\\rm up}}\\cdot{\\vc n}_{\\sigma,K}](\\bU_\\sigma^n-\\bU^n_{h,K})\\cdot(\\hat\\bu_\\sigma^{n,{\\rm up}}-\\hat\\bU_{h,\\sigma}^{n,{\\rm up}}), \\label{cT3}\n\\end{align}\n\\begin{align*}\n& \\mbox{and }{\\cal R}_{3,2}^{n}=\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|(r_K^n-\\hat r^{n,{\\rm up}}_\\sigma) [\\bU_{h,K}^n\\cdot{\\vc n}_{\\sigma,K}](\\bU_\\sigma^n-\\bU^n_{h,K})\\cdot(\\bu_K^n-\\bU^n_{h,K})\n\\\\\n&+\n\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\hat r^{n,{\\rm up}}_\\sigma \\Big[\\Big(\\bU_{h,K}^n-\\hat \\bU_{h,\\sigma}^{n,{\\rm up}}\\Big)\\cdot{\\vc n}_{\\sigma,K}\\Big](\\bU_\\sigma^n-\\bU^n_{h,K})\\cdot(\\bu_K^n-\\bU^n_{h,K})\\\\\n&\n+\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\hat r^{n,{\\rm up}}_\\sigma [\\hat \\bU_{h,\\sigma}^{n,{\\rm up}}\\cdot{\\vc n}_{\\sigma,K}](\\bU_\\sigma^n-\\bU^n_{h,K})\n\\cdot\\Big((\\bu_K^n -\\hat\\bu_{h,\\sigma}^{n,{\\rm up}}) -(\\bU^n_{h,K}-\\hat\\bU_{h,\\sigma}^{n,{\\rm up}})\\Big).\n\\end{align*}\nWe may use several times the { Taylor formula (in order to estimate $r_K^n-\\hat r_\\sigma^{n,{\\rm up}}$,\n $\\vc U_\\sigma^n-{\\vc U}_{h,K}^n$, $\\vc U_{h,K}^n-\\hat{\\vc U}_{h,\\sigma}^{n,{\\rm up}}$)} to get the bound\n$$\n|{\\cal R}_{3,2}^n|\\le h\\, c \\|r\\|_{W^{1,\\infty}(\\Omega)}\\Big(1+ \\|\\bU\\|_{W^{1,\\infty}(\\Omega;\\Rm^3)}\\Big)^3\n\\sum_{K\\in {\\cal T}} h|\\sigma||\\bu^n_K|\n$$\n$$\n+ c \\|r\\|_{W^{1,\\infty}(\\Omega)}\\Big(1+ \\|\\bU\\|_{W^{1,\\infty}(\\Omega;\\Rm^3)}\\Big)^2\\sum_{K\\in{\\cal T}}\n\\sum_{\\sigma\\in {\\cal E}(K)} h|\\sigma| |\\bu_K^n-\\bu_\\sigma^n|,\n$$\nwhere by virtue of H\\\"older's inequality, (\\ref{sob4}), (\\ref{sob3}), (\\ref{L2+-1}) (\\ref{L2+-2}),\n$$\n\\sum_{K\\in {\\cal T}} h|\\sigma||\\bu^n_K|\\le c\\Big(\\sum_{\\sigma\\in {\\cal T}} h|\\sigma||\\bu^n_K|^6\\Big)^{1\/6}\n\\le c\\Big[\\Big(\\sum_{K\\in {\\cal T}}\\|\\bu^n-\\bu_K^n\\|^6_{L^6(K;\\Rm^3)}\\Big)^{1\/6}\n$$\n$$\n+\\Big(\\sum_{K\\in {\\cal T}}\\|\\bu^n\\|^6_{L^6(K;\\Rm^3)}\\Big)^{1\/6}\\Big]\n\\le c\\Big(\\sum_ {K\\in {\\cal T}}\\|\\nabla\\bu_n\\|^2_{L^2(K;\\Rm^9)}\\Big)^{1\/2},\n$$\n$$\n\\sum_{K\\in{\\cal T}}\n\\sum_{\\sigma\\in {\\cal E}(K)} h|\\sigma| |\\bu_K^n-\\bu_\\sigma^n|\\le c\\Big[\\Big(\\sum_{K\\in {\\cal T}}\\|\\bu^n-\\bu_K^n\\|^2_{L^2(K;\\Rm^3)}\\Big)^{1\/2}\n$$\n$$\n+ \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|\\bu^n-\\bu_\\sigma^n\\|^2_{L^2(K;\\Rm^3)}\\Big)^{1\/2}\\Big]\n\\le h\\,c\\Big(\\sum_{K\\in {\\cal T}} \\|\\nabla\\bu_n\\|^2_{L^2(K;\\Rm^9)}\\Big)^{1\/2},\n$$\nConsequently, we may use (\\ref{est0}) to conclude\n\\begin{equation}\\label{cR3.2}\n|{\\cal R}_{3,2}|\\le h\\, c\\Big(M_0, E_0, \\overline r, \\|\\nabla r, \\bU, \\nabla\\bU\\|_{L^{\\infty}(Q_T;\\Rm^{15})}\\Big).\n\\end{equation}\n}\n\n\n\\vspace{2mm}\n\\textbf{Step 4:}\\textit{ Terms ${\\cal T}_4$ and ${\\cal T}_5$.} \\label{7.6}\nWe decompose ${\\cal T}_4$ as\n\n\\begin{equation}\\label{cT4}\n\\begin{aligned}\n\t& {\\cal T}_4= {\\cal T}_{4,1}+ {\\cal R}_{4,1},\n\t\\\\\n\t& \\mbox{with }{\\cal T}_{4,1}=\\deltat \\sum_{n=1}^m \\sum_{K\\in{\\cal T}}\\int_K p'(r^n_K)\\bu^n\\cdot\\nabla r^n\\dx,\n\\quad\n{\\cal R}_{4,1}= \\deltat \\sum_{n=1}^m \\sum_{K\\in{\\cal T}}\\int_K \\Big( p'(r^n)-p'(r^n_K)\\Big)\\bu^n\\cdot\\nabla r^n\\dx;\n\\end{aligned}\n\\end{equation}\nwhence\n\\begin{equation}\\label{cR4.1}\n\t|{\\cal R}_{4,1}|\\le h\\, c(M_0, E_0,\\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r])}, \\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)}).\n\\end{equation}\n\nEmploying integration by parts, we infer\n\n\\begin{equation}\\label{cT5}\n\\begin{aligned}\n\t& {\\cal T}_5= {\\cal T}_{5,1} + {\\cal R}_{5,1}, \\mbox{ with }{\\cal T}_{5,1}=\n-\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K \\nabla p(r^n)\\cdot \\bU^n\\dx,\n\t\\\\ &\n{\\cal R}_{5,1}=\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K\\nabla p(r^n)\\cdot\\Big(\\bU^n-\\bU^n_h\\Big)\\dx\n\\end{aligned}\n\\end{equation}\nand\n\\begin{equation}\\label{cR5.1}\n|{\\cal R}_{5,1}|\\le h\\, c(\\overline r, | p'|_{C([\\underline r,\\overline r])},\\|{ \\nabla r},\\nabla \\bU\\|_{L^\\infty(Q_T;\\Rm^{12})}).\n\\end{equation}\n\\label{7.7}\nIntegrating by parts, we obtain\n$$\n{\\cal T}_{5,1}= \\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K p(r^n) \\dv \\bU^n\\dx;\n$$\nwhence\n$$\n\\begin{aligned}\n\t& {\\cal T}_{5,1}= {\\cal T}_{5,2} + {\\cal R}_{5,2}, \\mbox{ with }{\\cal T}_{5,2}=\n\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K p(r_K^n)\\dv \\bU^n\\dx,\n\t\\\\ &\n{\\cal R}_{5,2}=-\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K(p(r_K^n)-p(r^n))\\dv\\bU^n\\dx\n\\end{aligned}\n$$\nand\n\\begin{equation}\\label{cR5.2}\n|{\\cal R}_{5,2}|\\le h\\, c(\\overline r, |p'|_{[\\underline r,\\overline r]},\\|\\nabla r,\\nabla\\bU\\|_{L^\\infty(Q_T;\\Rm^{12})}).\n\\end{equation}\n\nGathering the formulae (\\ref{cT1}), (\\ref{cT2}), (\\ref{cT3}), (\\ref{cT4}), (\\ref{cT5}) and estimates for the residual terms (\\ref{cR1.1}), (\\ref{cR2.1}--\\ref{cR2.4}), (\\ref{cR3.1}--\\ref{cR3.2}), (\\ref{cR4.1}), (\\ref{cR5.1}), (\\ref{cR5.2}) concludes the proof of Lemma \\ref{strongentropy}.\n\\end{proof}\n\n\\section{{ End} of the proof of the error estimate (Theorem \\ref{Main})}\n\n{ In this Section we put together the relative energy inequality (\\ref{relativeenergy-}) and the identity (\\ref{strong1}) derived in the previous section. The final inequality resulting from this manipulation is formulated in the following lemma.}\n\\begin{lm}\\label{Gronwall}\nUnder assumptions of Theorem \\ref{Main} there exists a positive number\n\\[\n c=c\\Big(M_0, E_0, \\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r])},\n \\|(\\nabla r, \\partial_t r, \\partial_t\\nabla r, \\partial^2_t r, \\bU, \\nabla\\bU, { \\nabla^2\\bU}, \\partial_t\\bU, \\partial_t\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{65})}\\Big)\n\\]\n(depending tacitly also on $T$, $\\theta_0$, $\\gamma$, ${\\rm diam} (\\Omega)$, $|\\Omega|$),\nsuch that for all $m=1,\\ldots,N,$ there holds:\n\\[{\\cal E}(\\vr^m,\\bu^m|r^m,\\bu^m)\n{ +\\deltat\\frac \\mu 2\\sum_{n=1}^m\\sum_{{K}\\in {\\cal T}}\\int_K|\\Grad(\\vc u^n-\\vc U^n_h)|^2{\\rm d x}}\n\\]\n\\[\n\\le c\\Big[h^A+\\sqrt{\\deltat} + {\\cal E}(\\vr^0,\\bu^0|r^0,\\bU^0)\\Big] + c\\,\\deltat\\sum_{n=1}^m {\\cal E}(\\vr^n,\\bu^n|r^n,\\bu^n),\n\\]\nwhere $A$ is defined in (\\ref{A1}).\n\\end{lm}\n\n\n\\begin{proof}\nGathering the formulae (\\ref{relativeenergy-}) and (\\ref{strong1}), one gets\n\\begin{equation}\\label{relativeenergy-1}\n{\\cal E}(\\vr^m,\\bu^m\\Big| r^m, U^m)- {\\cal E}(\\vr^0,\\bu^0\\Big|r(0),\\bU(0)) + \\mu\\deltat\\sum_{n=1}^m\\Big|\\bu^n-\\bU^n_h\\Big|^2_{V^2_h(\\Omega;\\Rm^3)}\n\\end{equation}\n$$\n\\le{\\cal P}_1+ {\\cal P}_2 +{\\cal P}_3 + {\\cal Q}\n$$\nwhere\n\\begin{align*}\n&{\\cal P}_1=\n\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}|K|(\\vr_K^{n-1}-r^{n-1}_K)\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K}^{n} - \\bu_K^{n}\\Big),\n\\\\\n&{ {\\cal P}_2=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|\\Big(\\vr_\\sigma^{n,{\\rm up}}-\\hat r_\\sigma^{n,{\\rm up}}\\Big)\\Big({\\hat\\bU}^{n,{\\rm up}}_{h,\\sigma}-\n{\\hat\\bu}^{n,{\\rm up}}_\\sigma\\Big)\\cdot\\Big(\\bU^n_\\sigma-\\bU^n_{h,K}\\Big) \\hat \\bU_{h,\\sigma}^{n, {\\rm up}}\\cdot\\bn_{\\sigma,K},}\n\\\\\n&{\\cal P}_3=\n\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}} \\int_K \\Big(p(r_K^n)-p(\\vr_K^n)\\Big)\\dv\\bU^n\\dx\n\\\\\n&\\qquad \\qquad+\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\Big[\\int_K\\frac{r_K^n-\\vr^n_K}{r^n_K} p'(r^n_K) \\bu^n\\cdot\\nabla r^n\\dx\n+\\int_K \\frac{ r^n_K-\\vr^n_K} {r_K^n}p'(r_K^n) [\\partial_t r]^n\\dx\\Big],\n\\\\\n&{ {\\cal Q}=\n{\\cal R}^m_{h,\\deltat} +R^m_{h,\\deltat} +{ G}^m. }\n\\end{align*}\n\n\nNow, we estimate conveniently the terms ${\\cal P}_1$, ${\\cal P}_2$, ${\\cal P}_3$ in three steps.\n\n\\vspace{2mm}\n\n{\\bf Step 1:} {\\it Term ${\\cal P}_1$.}\n{ { We have\n$$\n\\Big|{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}\\Big| \\le \\int_{t_{n-1}}^{t_n}\\|[\\partial_t\\bU(z)]_h\\|_{L^\\infty(K;\\Rm^9)}{\\rm d z}\n\\le c \\int_{t_{n-1}}^{t_n}\\|\\partial_t\\bU(z)\\|_{L^\\infty(K;\\Rm^9)}{\\rm d z},\n$$\nwhere we have used (\\ref{ddd}).\n\nAccording to Lemma \\ref{LL1},\n$$\n|\\vr-r|^\\gamma 1_{R_+\\setminus[\\underline r\/2,2\\overline r]}(\\vr)\\le c(p) E^p(\\vr|r)\n$$\nwith any $p\\ge 1$. In particular, \n\\begin{equation}\\label{aaa}\n|\\vr-r|^{6\/5} 1_{R_+\\setminus[\\underline r\/2,2\\overline r]}(\\vr)\\le c E(\\vr|r)\n\\end{equation}\nprovided $\\gamma\\ge 6\/5$.\n\nWe get by using the H\\\"older inequality,\n$$\n\\Big|\\sum_{K\\in{\\cal T}}|K|(\\vr_K^{n-1}-r^{n-1}_K)\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K}^{n} - \\bu_K^{n}\\Big)\\Big|\\le c\\Big(\\|\\partial_t\\bU \\|_{L^\\infty(Q_T;\\Rm^3)}+\\|\\partial_t\\nabla\\bU \\|_{L^\\infty(Q_T;\\Rm^9)}\\Big)\\times\n$$\n$$\n\\Big[\\Big(\\sum_{K\\in{\\cal T}}|K||\\vr^{n-1}_K-r^{n-1}_K|^2 1_{[\\underline r\/2,2\\overline r]}(\\vr_K)\\Big)^{1\/2}\n+\n\\Big(\\sum_{K\\in{\\cal T}}|K||\\vr^{n-1}_K-r^{n-1}_K|^{6\/5} 1_{R_+\\setminus [\\underline r\/2,2\\overline r]}(\\vr_K)\\Big)^{5\/6}\\Big]\n\\times\n$$\n$$\n\\Big(\\sum_{K\\in{\\cal T}}|K| \\Big|{\\bU}_{h,K}^{n} - \\bu_K^{n}\\Big|^6\\Big)^{1\/6}\n\\le c(\\|(\\partial_t\\bU,\\partial_t\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{12})})\\Big({\\cal E}^{1\/2}(\\vr^{n-1},\\bu^{n-1}|r^{n-1},\\bU^{n-1})\n$$\n$$\n+\n{\\cal E}^{5\/6}(\\vr^{n-1},\\bu^{n-1}|r^{n-1},\\bU^{n-1})\\Big)\\,\\Big(\\sum_{K\\in {\\cal T}}\\| {\\bU}_{h,K}^{n} - \\bu_K^{n}\\|_{L^6(K;\\Rm^3)}^6\\Big)^{1\/6},\n$$\nwhere we have used (\\ref{aaa}) and estimate (\\ref{est4}) to obtain the last line.\n\n Now, by the Minkowski inequality,\n }\n$$\n\\Big(\\sum_{K\\in {\\cal T}}\\| {\\bU}_{h,K}^{n} - \\bu_K^{n}\\|_{L^6(K;\\Rm^3)}^6\\Big)^{1\/6}\\le\n\\Big(\\sum_{K\\in {\\cal T}}\\| ({\\bU}_{h,K}^{n} - \\bu_K^{n})-(\\bU_h^n-\\bu^{n})\\|_{L^6(K;\\Rm^3)}^6\\Big)^{1\/6}\n$$\n$$\n+\n\\|\\bU_h^n-\\bu^{n}\\|_{L^6(\\Omega;\\Rm^3)}\\le c \\Big|\\bu^n-\\bU^n_h\\Big|_{V^2_h(\\Omega;\\Rm^3)},\n$$\nwhere we have used estimate (\\ref{sob4-}) and the Sobolev inequality (\\ref{sob1}). Finally, employing\nYoung's inequality, and estimate (\\ref{est4}), we arrive at\n\\begin{multline}\\label{cP1}\n|{\\cal P}_1| \\le \\; c({ \\delta},M_0,E_0,\\underline r,\\overline r,\\|(\\bU,\\nabla\\bU,\\partial_t\\bU,\\partial_t\\nabla\\bU)\\|_{L^\\infty(Q_T,\\Rm^{15})} )\n\\\\\n\\times \\Big(\\deltat {\\cal E}(\\vr^0,\\bu^0|r^0,\\bU^0)+ \\deltat\\sum_{n=1}^m{\\cal E}(\\vr^n,\\bu^n|r^n,\\bU^n)\\Big)\n+ \\delta \\deltat\\sum_{n=1}^m\\Big|\\bu^n-\\bU^n_h\\Big|^2_{V^2_h(\\Omega;\\Rm^3)}\n\\end{multline}\nwith any $\\delta>0$.\n}\n\n\n\n{\\bf Step 2:} {\\it Term ${\\cal P}_2$.}\n{ We write ${\\cal P}_2=\\deltat \\sum_{n=1}^m{\\cal P}_2^n$ where Lemma \\ref{LL1} and the H\\\"older inequality yield, similarly as in the previous step,\n\\begin{equation*}\n\\begin{aligned}\n|{\\cal P}^n_2| & \\le c(\\underline r,\\overline r, \\|\\nabla\\bU\\|_{L^\\infty(Q_T;\\Rm^9)}) \\times\n\\\\ &\n\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h\n\\Big(E^{1\/2}(\\vr_\\sigma^{n,{\\rm up}},\\hat r_\\sigma^{n,{\\rm up}})+ E^{2\/3}(\\vr_\\sigma^{n,{\\rm up}},\\hat r_\\sigma^{n,{\\rm up}}\\Big)\\,|\\hat\\bU^{n,{\\rm up}}_{h,\\sigma}|\\,|{\\hat\\bU}_{h,\\sigma}^{n,{\\rm up}} -\\hat\\bu_\\sigma^{n,{\\rm up}}|\n\\\\\n& \\le c(\\underline r,\\overline r, \\|(\\bU,\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{12})})\\Big[\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h \\Big(E(\\vr_\\sigma^{n,{\\rm up}}|\\hat r_\\sigma^{n,{\\rm up}})\\Big)^{1\/2}\n\\\\\n&\n+\n\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h\nE(\\vr_\\sigma^{n,{\\rm up}}|\\hat r_\\sigma^{n,{\\rm up}})\\Big)^{2\/3}\\Big]\n \\times\n\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h\\Big|{\\hat\\bU}_{h,\\sigma}^{n,{\\rm up}} -\\hat\\bu_\\sigma^{n,{\\rm up}}\\Big|^6\\Big)^{1\/6}\n\\end{aligned}\n\\end{equation*}\nprovided $\\gamma\\ge 3\/2$.\nNext, we observe that the contribution of the face $\\sigma=K|L$ to the sums\n$\n\\sum_{K\\in {\\cal T}}$ $\\sum_{\\sigma\\in {\\cal E}(K)}$ $|\\sigma| h\nE(\\vr_\\sigma^{n,{\\rm up}}|\\hat r_\\sigma^{n,{\\rm up}})$\nand $ \\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h|{\\hat\\bU}_{h,\\sigma}^{n,{\\rm up}} -\\hat\\bu_\\sigma^{n,{\\rm up}}|^6$\nis less or equal than\n$2|\\sigma| h ( E(\\vr_K^{n}|\\hat r_K^{n}) + E(\\vr_L^{n}|\\hat r_L^{n})),\n$ and\nthan $2|\\sigma| h (|{\\bU}_{h,K}^{n} -\\bu_K^{n}|^6+ |{\\bU}_{h,L}^{n} -\\bu_L^{n}|^6)$, respectively. Consequently,we get\nby the same reasoning\nas in the previous step, under assumption $\\gamma\\ge 3\/2$,\n\\begin{equation}\\label{cP2}\n|{\\cal P}_2|\\le c(\\delta, M_0, E_0, \\underline r,\\overline r, \\|(\\bU,\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{12})})\\,\\deltat \\sum_{n=1}^m {\\cal E}(\\vr^n,\\bu^n|\nr^n,\\bU^n) + \\delta\\; \\deltat \\sum_{n=1}^m |\\bu^n-\\bU_{h}^n)|_{V^2_h(\\Omega;\\Rm^3)}^2.\n\\end{equation}\n}\n{\\bf Step 3:} {\\it Term ${\\cal P}_3$.}\nSince the pair $(r,\\bU)$ satisfies continuity equation (\\ref{cont2}) in the classical sense, we have for all $n=1,\\ldots,N$,\n\\[\n[\\partial_t r]^n +\\bU^n\\cdot\\nabla r^n=-r^n\\dv\\bU^n,\n\\]\nwhere we recall that $[\\partial_t r]^n(x)=\\partial_t r(t_n,x)$ in accordance with (\\ref{notation2-}).\nUsing this identity we write\n\\begin{align*}\n&{\\cal P}^n_3= {\\cal P}_{3,1} +{\\cal P}_{3,2},\\quad {\\cal P}_{3,i}=\\deltat\\sum_{n=1}^m {\\cal P}_{3,i}^n,\\\\\n& \\mbox{with } {\\cal P}_{3,1}^n=\n-\\sum_{K\\in {\\cal T}} \\int_K \\Big(p(\\vr_K^n)-p'(r_K^n)(\\vr_K^n-r_K^n)-p(r_K^n)\\Big)\\dv\\bU^n\\dx \\\\\n& \\mbox{and } {\\cal P}_{3,2}^n= \\sum_{K \\in {\\cal T}}\\Big[\\int_K\\frac{r_K^n-\\vr^n_K}{r^n_K} p'(r^n_K)( \\bu^n-\\bU^n)\\cdot\\nabla r^n\\dx.\n\\end{align*}\nNow, we apply Lemma \\ref{LL1} in combination with assumption (\\ref{pressure1}) to deduce\n\\begin{equation}\\label{cP3}\n|{\\cal P}_{3,1}|\\le c\\|\\dv \\bU\\|_{L^\\infty(Q_T)}\\deltat \\sum_{n=1}^m{\\cal E}(\\vr^n,\\bu^n|r^n,\\bU^n).\n\\end{equation}\nFinally, the same reasoning as in Step 2 leads to the estimate\n\\begin{equation}\\label{cP4}\n\\begin{aligned}\n|{\\cal P}_{3,2}|& \\le h\\;c(M_0, E_0,\\underline r, \\overline r, |p'|_{C([\\underline r,\\overline r])} \\|(\\nabla r,\\nabla\\bU)\\|_{L^\\infty(\\Omega;\\Rm^9)})\n\\\\ & + c(\\delta, \\|\\underline r, \\overline r, |p'|_{C([\\underline r,\\overline r])} \\|\\nabla r\\|_{L^\\infty(\\Omega;\\Rm^3)})\\;\n\\deltat \\sum_{n=1}^m{\\cal E}(\\vr^n,\\bu^n|r^n,\\bU^n) +\\delta \\,\\deltat \\sum_{n=1}^m|\\bu^n-\\bU^n_h|_{V^2_h(\\Omega;\\Rm^3)}^2.\n \\end{aligned}\n\\end{equation}\nGathering the formulae (\\ref{relativeenergy-1}) and (\\ref{cP1})-(\\ref{cP4}) with $\\delta$ sufficiently small (with respect to $\\mu$), we conclude the proof of Lemma \\ref{Gronwall}.\n\\end{proof}\n\n{ Finally, Lemma \\ref{Gronwall} in combination with the bound (\\ref{est4}) yields\n\\[{\\cal E}(\\vr^m,\\bu^m|r^m,\\bU^m)\\le c\\Big[h^A+\\sqrt{\\deltat}+\\deltat + {\\cal E}(\\vr^0,\\bu^0|r^0,\\bU^0)\\Big] + c\\,\\deltat\\sum_{n=1}^{m-1} {\\cal E}(\\vr^n,\\bu^n|r^n,\\bU^n);\n\\]\nwhence Theorem \\ref{Main} is a direct consequence of the standard discrete version of Gronwall's lemma. Theorem \\ref{Main} is thus proved.}\n\n\\section{Appendix: Fundamental auxiliary lemmas and estimates}\\label{3}\n\nIn this section we report several results related to the properties of the Sobolev spaces on tetrahedra\nand of the Crouzeix-Raviart (C-R) space. { We refer to the book Brezzi, Fortin \\cite{BRFO} for the general introduction to the subject.}\n\n\n\n\n\n\n\n\n\n{ We start with the inequalities that can be obtained by rescaling from the standard inequalities on a reference tetrahedron of size equivalent to one.}\n\n\\begin{lm}[{ Poincar\\'e, Sobolev and interpolation inequalities on tetrahedra}]\\label{Lemma2}\nLet $1\\le p\\le \\infty$. Let $\\theta_0 > 0$ and ${\\cal{T}} $ be a triangulation of $ \\Omega $ such that $ \\theta \\ge \\theta_0 $ where $ \\theta $ is defined in (\\ref{reg}). Then we have:\n\\begin{description}\n\\item{\\it (1) Poincar\\'e type inequalities on tetrahedra}\n\nLet $1\\le p\\le\\infty$. There exists $c=c(\\theta_0,p)>0$ such that for all\n$K\\in {\\cal T}$ and for all $v\\in W^{1,p}(K)$ we have\n\\begin{equation}\\label{L2-1}\n\\|v-v_K\\|_{L^p(K)}\\le c h\\|\\nabla v\\|_{L^p(K)},\n\\end{equation}\n\\begin{equation}\\label{L2-2}\n\\forall \\sigma\\in {\\cal E}(K), \\;\\|v-v_\\sigma\\|_{L^p(K)}\\le c h\\|\\nabla v\\|_{L^p(K)}.\n\\end{equation}\n\\item{\\it (2) Sobolev type inequalities on tetrahedra}\n\nLet $1\\le p0$ such that for all\n$K\\in {\\cal T}$ and for all $v\\in W^{1,p}(K)$ we have\n\\begin{equation}\\label{L2-3}\n\\|v-v_K\\|_{L^{p^*}(K)}\\le c \\|\\nabla v\\|_{L^p(K)},\n\\end{equation}\n\\begin{equation}\\label{L2-4}\n\\forall \\sigma\\in {\\cal E(}K), \\;\\|v-v_\\sigma\\|_{L^{p^*}(K)}\\le c \\|\\nabla v\\|_{L^p(K)},\n\\end{equation}\nwhere $p^*=\\frac{dp}{d-p}$.\n\\item{\\it (3) Interpolation inequalities on the tetrahedra}\n\nLet $1\\le p0$ such that for all\n$K\\in{\\cal T}$ and $v\\in W^{1,p}(K)$ we have\n\\begin{equation}\\label{interpol1}\n\\|v-v_K\\|_{L^{q}(K)}\n\\le c h^\\beta\\|\\nabla v\\|_{L^p(K;\\Rm^d)},\n\\end{equation}\n\\begin{equation}\\label{interpol2}\n\\|v-v_\\sigma\\|_{L^{q}(K)}\n\\le ch^\\beta \\|\\nabla v\\|_{L^p(K;\\Rm^d)},\n\\end{equation}\nwhere $\\frac 1q=\\frac \\beta p+\\frac {1-\\beta}{p^*}$.\n\\end{description}\n\\end{lm}\n{ Combining estimates (\\ref{L2-1}--\\ref{interpol2}) with the algebraic inequality}\n\\begin{equation}\\label{dod1*}\n\t\\Big(\\sum_{i=1}^L|a_i|^p\\Big)^{1\/p}\\le \\Big(\\sum_{i=1}^L|a_i|^q\\Big)^{1\/q}\n\\end{equation}\nfor all $(a=(a_1,\\ldots,a_L)\\in \\Rm^L$, $1\\le q\\le p<\\infty$, we obtain the following corollaries.\n\\begin{cor}[Poincar\\'e and Sobolev type inequalities on the Sobolev spaces]\\label{cor1}\nUnder the assumptions of Lemma \\ref{Lemma2}, we have:\n\\begin{description}\n\\item{\\it (1) Poincar\\'e type inequalities on the domain $\\Omega$}\n\n Let $1\\le p\\le \\infty$. There exists $c=c(\\theta_0,p)>0$ such that for all $v\\in W^{1,p}(\\Omega)$ we have\n\\begin{align}\\label{L2-5}\n\t\\|v-\\hat v\\|_{L^p(\\Omega)}\\equiv\\Big(\\sum_{K\\in {\\cal T}}\\|v-v_K\\|^p_{L^p(K)}\\Big)^{1\/p}\\le c h\\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)},\\\\\n\t\\label{L2-6}\n\t\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|v-v_\\sigma\\|^p_{L^p(K)}\\Big)^{1\/p}\\le c h\\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)}\n\\end{align}\nwhere $\\hat v$ and $v_\\sigma$ are defined by \\eqref{vhat} and \\eqref{vtilde}.\n\\item{\\it (2) Sobolev type inequalities on the domain $\\Omega$}\n\nLet $1\\le p0$ such that for all\n$v \\in W^{1,p}(\\Omega)$ we have\n\\begin{equation}\\label{L2-7}\n\\|v-\\hat v\\|_{L^{p^*}(\\Omega)}\n\\le c \\|\\nabla v\\|_{L^p(\\Omega)},\n\\end{equation}\n\\begin{equation}\\label{L2-8}\n\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|v-v_\\sigma\\|^{p^*}_{L^{p^*}(K)}\\Big)^{1\/p^*}\n\\le c \\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)}.\n\\end{equation}\n\\item{\\it (3) Interpolation inequalities on the domain $\\Omega$}\n\nLet $1\\le p0$ such that for all\n$v\\in W^{1,p}(\\Omega)$ we have\n\\begin{equation}\\label{L2-9}\n\\|v-\\hat v\\|_{L^{q}(\\Omega)}\n\\le c h^\\beta\\|\\nabla v\\|_{L^p(\\Omega)},\n\\end{equation}\n\\begin{equation}\\label{L2-10}\n\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|v-v_\\sigma\\|^{q}_{L^{q}(K)}\\Big)^{\\frac 1 q}\n\\le ch^\\beta \\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)},\n\\end{equation}\nwhere $\\frac 1q=\\frac \\beta p+\\frac {1-\\beta}{p^*}$.\n\\end{description}\n\\end{cor}\n\n\\begin{cor}[Poincar\\'e and Sobolev type inequalities on $V_h$]\\label{cor2}\nUnder assumptions of Lemma \\ref{Lemma2}, there holds:\n\\begin{description}\n\\item {\\it (1) Poincar\\'e type inequality in $V_h(\\Omega)$:}\nLet $1\\le p<\\infty$. There exists $c=c(\\theta_0,p)$ such that for all $v \\in V_h,$\n\\begin{equation}\\label{sob4-}\n\\|v-\\hat v\\|_{L^{p}(\\Omega)}\n\\le c h |v|_{V_h^p(\\Omega)},\n\\end{equation}\n\\begin{equation}\\label{sob5-}\n \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|v-v_\\sigma\\|_{L^p(K)}^p\\Big)^{\\frac 1 p}\n\\le c h | v|_{V_h^p(\\Omega)}.\n\\end{equation}\n\n\\item {\\it (2) Sobolev type inequality in $V_h(\\Omega)$:}\nLet $1\\le p0$ such that for all\n$v\\in V_h(\\Omega)$ we have\n\\begin{equation}\\label{L2+-1}\n\\|v-\\hat v\\|_{L^{q}(\\Omega)}\n\\le c h^\\beta|v|_{V_h^p(\\Omega)},\n\\end{equation}\n\\begin{equation}\\label{L2+-2}\n\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|v-v_\\sigma\\|_{L^q(K)}^q\\Big)^{\\frac 1 q}\n\\le ch^\\beta |v|_{V_h^p(\\Omega)},\n\\end{equation}\nwhere $\\frac 1q=\\frac \\beta p+\\frac {1-\\beta}{p^*}$.\n\\end{description}\n\\end{cor}\n\n{ The next fundamental lemma deals with the properties of the projection $v_h$ defined by (\\ref{vh})}.\n\n\\begin{lm} [Projection on $V_h$]\\label{Lemma1}\nLet $\\theta_0 > 0$ and ${\\cal{T}} $ be a triangulation of $ \\Omega $ such that $ \\theta \\ge \\theta_0 $ where $ \\theta $ is defined in (\\ref{reg}).\n\\begin{description}\n\\item{\\it (1) Approximation estimates on the tetrahedra}\n\nLet $1\\le p\\le \\infty$. There exists $c=c(\\theta_0,p)>0$ such that\n{\n\\begin{equation}\\label{ddd}\n\\forall v\\in W^{1,p}_0(\\Omega)\\cap L^\\infty(\\Omega), \\forall K \\in {\\cal{T}},\\;\\|v_h\\|_{L^\\infty(K)}\\le c \\|v\\|_{L^\\infty(K)},\n\\end{equation}\n}\n\\begin{equation}\\label{L1-2}\n\\forall v \\in W^{1,p}_0\\cap W^{s,p}(\\Omega), \\forall K \\in {\\cal{T}},~ || v-v_h||_{L^p(K)} \\le c h^{s}\\|\\nabla^s v\\|_{L^p(K;\\Rm^{d^s})},\n\\end{equation}\n\\begin{equation}\\label{L1-3}\n|| \\nabla( v- v_h) ||_{L^p(K;\\Rm^d)} \\le c h^{s-1} \\|\\nabla^s v\\|_{L^p(K;\\Rm^{d^s})}, s=1,2.\n\\end{equation}\n\n\\item{\\it (2) Preservation of divergence}\n\n\\begin{equation}\\label{L1-1}\n\\forall \\bv \\in W^{1,2}_{0}(\\Omega,\\R^d), \\forall q \\in L_h(\\Omega), \\sum_{K\\in {\\cal T}}\\int_K q ~\\dv\\bv_h \\dx = \\int_{\\Omega} q ~\\dv \\bv \\dx\n\\end{equation}\n\n\\item{(3) Approximation estimates of the Poincar\\'e type on the whole domain}\n\nLet $1\\le p<\\infty$. There exists $c=c(\\theta_0,p)>0$ such that for all $v\\in W^{1,p}_0(\\Omega)$,\n\\begin{equation}\\label{L1-4}\n\\|v-v_h\\|_{L^p(\\Omega)}\\le c h \\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)}.\n\\end{equation}\n\n\n\\item{ (4) Approximation estimates of the Sobolev type on the whole domain}\n\nLet $1\\le p0$ such that for all $v\\in W^{1,p}_0(\\Omega)$,\n\\begin{equation}\\label{L1-5}\n\\|v-v_h\\|_{L^{p^*}(\\Omega)}\\le c \\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)}.\n\\end{equation}\n\\end{description}\n\\end{lm}\n\n{ Statement {\\it (2)} of Lemma \\ref{Lemma1} is proved in\n\\cite{cro-73-con}, where one can find also the proof of item {\\it (1)} for $p=2$. We present here the proof of statements {\\it (1), (3), (4)} for arbitrary $p$ for the reader's convenience, since a straightforward reference is not available.\n\\\\ \\\\\n\\begin{proof}\n\n{\\textbf{ Step 1:}} We start with some generalities. First we complete the Crouzeix-Raviart basis (\\ref{CRB}) by functions $\\phi_\\sigma$ indexed also with $\\sigma\\in {\\cal E}_{\\rm ext}$ saying\n$$\n\\frac 1{|\\sigma'|}\\int_{\\sigma'}\\phi_\\sigma {\\rm d} S=\\delta_{\\sigma,\\sigma'},\\; \\;(\\sigma,\\sigma')\\in {\\cal E}^2\n$$\nand observe that\n\\begin{equation}\\label{+1}\n\\sum_{\\sigma\\in {\\cal E}(K)}\\phi_\\sigma(x)= 1\\;\\mbox{ for any $x\\in K$.}\n\\end{equation}\nA scaling argument yields\n\\begin{equation}\\label{+2}\n\\|\\phi_\\sigma\\|_{L^\\infty(\\Omega)}\\le c(\\theta_0),\\; h\\|\\nabla\\phi_\\sigma\\|_{L^\\infty(\\Omega;\\Rm^d)}\\le c(\\theta_0).\n\\end{equation}\nSecond, we define the projection $v\\to v_h$ for any $v\\in W^{1,p}(\\Omega)$ by saying\n$$\nv_h=\\sum_{\\sigma\\in {\\cal E}} v_\\sigma\\phi_\\sigma.\n$$\nWe notice that if $v\\in W^{1,p}_0(\\Omega)$ then $v_h$ coincides with (\\ref{CRP}). Moreover,\n\\begin{equation}\\label{+3}\nv_h=v\\;\\mbox{for any affine function $v$.}\n\\end{equation}\nThird, due to the density argument, it is enough to show the remaining statements {\\it (1), (3), (4)} for\n$v\\in W^{1,p}_0(\\Omega)\\cap W^{s,\\infty}(\\Omega)$, $s=1,2$, according to the case.\n\\\\ \\\\\n{\\textbf{Step 2:}} { We realize that ${\\rm supp }\\phi_\\sigma=K\\cup L$ and derive (\\ref{ddd}) directly by employing representation (\\ref{vh}), definition of $v_\\sigma$ and estimate (\\ref{+2}).}\n\nWe denote by $x_K=\\frac 1{|K|}\\int_K x{\\rm d}x$ the center of gravity of the tetrahedron $K$. We calculate by using (\\ref{+3}) and the first order Taylor formula\n$$\nv(x)-v_h(x)= v(x)-v(x_K) - [v-v(x_K)]_h(x)\n$$\n$$\n=(x-x_K)\\cdot\\int_0^1\\nabla v(x_K+t(x-x_K)){\\rm d} t\n- \\sum_{\\sigma\\in {\\cal E}(K)}\\phi_\\sigma(x) \\frac 1{|\\sigma|}\\int_\\sigma\\Big[(x-x_K)\\cdot\\int_0^1\\nabla v(x_K+t(x-x_K)){\\rm d} t\\Big]{\\rm d} S,\n$$\nwhere $x\\in K$.\nThis formula yields immediately the upper bound stated in (\\ref{L1-2})$_{s=1}$ if $p=\\infty$. If $1\\le p<\\infty$\nwe calculate the upper bound of the $L^p$-norm of each term at the right-hand side separately by using (\\ref{+2}), Fubini's theorem, H\\\"older's inequality and the change of variables $y= x_K+t(x-x_K)$ together with the convexity of $K$.\n\nThe same reasoning can be applied to prove (\\ref{L1-2})$_{s=2}$. Indeed, we observe\nthat\n$$\nv(x)-v_h(x)= v(x)-(x-x_K)\\cdot\\nabla v(x_K)-v(x_K) - [v -(x-x_K)\\cdot\\nabla v(x_K)-v(x_K)]_h(x)\n$$\nby virtue of (\\ref{+3}). Now we apply to the right hand side of the last expression the second order Taylor formula in the integral form, and proceed exactly as described before.\n\nFinally, one applies the same straightforward argumentation to get (\\ref{L1-3}). This completes the proof of statement {\\it (1)}.\n\\\\ \\\\\n{\\textbf{Step 3:}} Statement {\\it (3)} follows easily from (\\ref{L1-2})$_{s=1}$ and the algebraic inequality (\\ref{dod1*}).\n\\\\ \\\\\n{\\textbf{ Step 4:}} We use (\\ref{+1}) and (\\ref{+3}) to write\n$$\nv(x)-v_h(x)= \\sum_{\\sigma\\in {\\cal E}(K)} (v(x)-v_\\sigma)\\phi_\\sigma(x),\\;\\; x\\in K;\n$$\nwhence\n$$\n\\|v-v_h\\|_{L^{p^*}(K)}\\le c\\|\\nabla v\\|_{L^p(K;\\Rm^d)}\n$$\nwhere we have used the Sobolev inequality (\\ref{L2-4}) on the tetrahedron $K\\in {\\cal T}$ and the $L^\\infty$-bound (\\ref{+2}). We conclude the proof of statement {\\it (4)} by using the relation (\\ref{dod1*}). The proof of Lemma \\ref{Lemma1} is complete.\n\\end{proof}\n}\n\n\nThe following corollary is a direct consequence of (\\ref{L1-3}).\n\n\\begin{cor}[Continuity of the projection onto $V_h$]\\label{cor3}\nUnder assumptions of Lemma \\ref{Lemma1}, there exists $c=c(\\theta_0,p)>0$ such that\n\\begin{equation}\\label{L1-6}\n\\forall v \\in W^{1,p}_0(\\Omega),\\;|v_h|_{V_h^p(\\Omega)}\\le c \\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)},\n\\end{equation}\nwhere $1\\le p<\\infty$.\n\\end{cor}\n\nAlthough the non conforming finite element space $V_h$ is not a subspace of any Sobolev space, its elements enjoy the Sobolev type inequalities.\nThis important fact is formulated in the next lemma.\n\n\\begin{lm}[Sobolev inequality on $V_h$]\\label{Lemma2+}\nLet $\\Omega$ be a bounded domain of $\\R^d$.\nLet ${\\cal{T}}$ be a triangulation of the domain $\\Omega$ in simplices such that $\\theta \\ge \\theta_0 >0$ where $\\theta$ is defined in $(\\ref{reg})$.\nThen we have:\n\\begin{description}\n\\item{ (1) Sobolev inequality in $V_h(\\Omega)$ (case $1\\le p0$ such that forall $v \\in V_h(\\Omega)$,\n\\begin{equation}\\label{sob3}\n\\|v\\|_{L^q(\\Omega)} \\le c|v|_{V_h^p(\\Omega)}\n\\end{equation}\n\n\\end{description}\n\\end{lm}\n\\begin{proof}\n\n\\textbf{Step 1}\nLet $1 \\le r \\le \\alpha < \\infty$. Let $ u \\in V_h$. We call $v$ the element of $V_h$ such that $ v_\\sigma = |u_\\sigma|^\\alpha $. Then there exists $C$ only depending on $d,r,\\alpha$ such that\n\\begin{equation}\\label{leman3}\n ||u||_{L^r(\\Omega)}^\\alpha \\le || u ||_{L^{\\frac{r}{\\alpha}}(\\Omega)}.\n\\end{equation}\n\nTo prove ($\\ref{leman3}$) we remark that, using a change of variable, it is enough to show to prove the existence of $C$ for only the unit symplex $ \\hat{K}$. Let $u \\in \\mathbb{P}_1(\\hat{K}) $ and we call $v$ the element of $\\mathbb{P}_1(\\hat{K}) $ such that $ v_\\sigma = |u_\\sigma|^\\alpha $. Let $T(u) = ||u||_{L^r(\\hat{K})} $ and $ S(u) = || u ||_{L^{\\frac{r}{\\alpha}}(\\hat{K})}\n^{\\frac{1}{\\alpha}}$. These two functions are continuous, homogeneous of degree $1$ and non zero if $ u \\ne 0$. Since $\\mathbb{P}_1(\\hat{K}) $ is a finite dimensional space, we can choose a norm on $\\mathbb{P}_1(\\hat{K}) $ and take $C = (\\frac{M}{m})^\\alpha $ where $ M = \\max \\{ T(u), ||u||_{\\mathbb{P}_1(\\hat{K})}=1 \\} $ and $ m = \\min \\{ T(u), ||u||_{\\mathbb{P}_1(\\hat{K})}=1 \\} $.\n\\vspace{2em}\n\n\n\n\\textbf{ Step 2: Proof for $p=1.$} \\\\\nWe set $u=0$ outside $\\Omega$. For $\\sigma \\in {\\cal{E}_{\\intt}}, \\sigma=K|L$, we set $ |[u(x)]|=|u_K(x)-u_L(x)|$ for $x \\in \\sigma$. For $ \\sigma \\in {\\cal{E}}_{\\extt}\\cap {\\cal{E}}(K) $, we set $|[u(x)]| =|u_K(x)|$ for $x \\in \\sigma$. We first remark that there exists $C_{1,1}$ and $C_{1,2}$ only depending on $d$ such that\n$$ ||u||_{L^{\\frac{d}{d-1}}(\\Omega)} \\le C_{1,1}||u||_{BV(\\R^d)} \\le C_{1,2} || \\nabla_h u||_{L^1(\\Omega)} + C_{1,2} \\sum_{\\sigma \\in {\\cal{E}}} \\int_\\sigma |[u]| {\\rm d}S. $$\nWe now prove that there exits $C_{1,3}$ only depending on $d$ and $\\theta_0$ such that\n$$ \\sum_{\\sigma \\in {\\cal{E}}} \\int_\\sigma |[u]| {\\rm d}S \\le C_{1,3} || \\nabla_h u||_{L^1(\\Omega)}. $$\nLet $K \\in \\T $ and $ \\sigma \\in {\\cal{E}}(K) $. Let $x_\\sigma $ be the center of mass of $\\sigma$. We have, with$ u_K =u $ in $K$,\n$$ u_K(x) -u(x_\\sigma) = \\int_0^1 \\nabla u_K \\cdot (x-x_\\sigma) \\dx.$$\nThen if $ \\sigma=K|L$ we have\n$$ |u_K(x)-u_L(x)| \\le h_\\sigma \\Big( |\\nabla u_K| + |\\nabla u_L| \\Big). $$\nIntegrating this inequality on $\\sigma$ gives\n$$ \\int_\\sigma |[u]| {\\rm d}S \\le |\\sigma|h_\\sigma \\Big( |\\nabla u_K| + |\\nabla u_L| \\Big) \\le \\frac{2}{\\theta_0^d} \\Big( || \\nabla u||_{L^1(K)} +|| \\nabla u||_{L^1(L)} \\Big).$$\nSimilarly for $ \\sigma \\in \\E_{\\extt}\\cap \\E(K) $ we have\n$$ \\int_\\sigma |[u]| {\\rm d}S \\le \\frac{2}{\\theta_0^d} || \\nabla u||_{L^1(K)} $$\nThen there exists $C_{1,3}=C(d,\\theta_0) $ such that\n$$ \\stie \\int_\\sigma |[u]| {\\rm d}S \\le C_{1,3} || \\nabla_h u ||_{L^1(\\Omega)}. $$\nand then,\n$$ ||u||_{L^{1^*}(\\Omega)} \\le c(d,\\theta_0) || \\nabla_h u||_{L^1(\\Omega)}. $$\n\\textbf{ Step 3: Proof for $1 1$ and $\\alpha 1^*=p^* $. We call $v$ the element of $V_h$ such that $ v_\\sigma = |u_\\sigma|^\\alpha $ for $ \\sigma \\in \\E $. One has $v \\ne |u|^\\alpha $ but there exits $C_{2,1} $ only depending on $d$ and $p$ (see lemma $\\ref{leman3}$) such that\n$$ || u||_{L^{p^*}(\\Omega)}^\\alpha \\le C_{2,1} ||v||_{L^{1^*}(\\Omega)} \\le c(d,p,\\theta_0) ||\\nabla_h v||_{L^{1}(\\Omega)}. $$\nMoreover using a scalling argument we obtain\n$$ || \\nabla_h v ||_{L^1(K)} \\le c(d,p,\\theta_0) \\stiek |u_\\sigma|^{\\alpha-1}| \\nabla u_K | |K|. $$\nThen, using H\\\"older Inequality, we have, with $q=\\frac{p}{p-1}$ (so that $q( \\alpha-1) =p^*$),\n$$ || \\nabla_h v ||_{L^1(K)} \\le c(d,p,\\theta_0) || \\nabla u ||_{L^p(K)} || u||_{L^{p^*}(K)}^{\\frac{p^*}{q}}.$$\nSumming on $ K \\in \\T$ we obtain\n$$||u||_{L^{p^*}(\\Omega)} \\le C_2 || \\nabla_h u||_{L^p(\\Omega)}.$$\n\\textbf{Step 4: Proof for $p\\ge d.$} \\\\\nLet $ 1 \\le q < \\infty $. There exists $r=r(d,q)$ such that $r 0$ and $ {\\cal{T}} $ be a triangulation of $ \\Omega $ such that $ \\theta \\ge \\theta_0 $ where $ \\theta $ is defined in $ (\\ref{reg}) $. Then\nthe norms\n\\begin{equation}\\label{norms1}\n\\Big(\\sti |\\sigma| h |v_\\sigma|^p\\Big)^{1\/p}\\quad\\mbox{and}\\quad ||v||^p_{L^p(\\Omega)}\n\\end{equation}\nare equivalent on $V_h(\\Omega)$ uniformly with respect to $h>0$.\n\\end{lm}\n\nThe last lemma in this overview deals with the estimates of jumps over faces.\nThe reader can consult \\cite[Lemma 3.32]{ern-04-the} or \\cite[Lemma 2.2]{GHL2009iso} for its proof.\n\\begin{lm}[Jumps over faces in the Crouzeix-Raviart space]\\label{Lemma6}\nLet $ \\theta_0 > 0$ and $ {\\cal{T}} $ be a triangulation of $ \\Omega $ such that $ \\theta \\ge \\theta_0 $ where $ \\theta $ is defined in $ (\\ref{reg}) $. Then there exists $ c=c(\\theta_0)>0 $ such that for all $ v \\in V_h(\\Omega)$,\n\\begin{equation}\\label{tbound}\n\\sum_{\\sigma \\in {\\cal{E}}} \\frac{1}{h} \\int_\\sigma [v]_{\\sigma,\\bn_\\sigma}^2 \\dS \\le c |v|_{V_h^2(\\Omega)}^2,\n\\end{equation}\nwhere $[v]_{\\sigma,\\bn_\\sigma}$ is a jump of $v$ with respect to a normal $\\bn_\\sigma$ to the face $\\sigma$,\n\\[\n\\forall x\\in\\sigma=K|L\\in{\\cal E}_{\\rm int},\\quad [v]_{\\sigma,\\bn_\\sigma}(x)=\\left\\{\n\\begin{array}{c}\n v|_K(x)-v|_L(x)\\;\\mbox{if $\\bn_\\sigma=\\bn_{\\sigma,K}$}\\\\\nv|_L(x)-v|_K(x)\\;\\mbox{if $\\bn_\\sigma=\\bn_{\\sigma,L}$}\n\\end{array}\\right.\n\\]\nand\n\\[\n\\forall x\\in\\sigma\\in{\\cal E}_{\\rm ext},\\quad [v]_{\\sigma,\\bn_\\sigma}(x)=v(x),\\;\\mbox{with $\\bn_\\sigma$ an exterior normal to $\\partial\\Omega$}.\n\\]\n\\end{lm}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{} \n\n\\section{Introduction}\nFor many years the active galaxy phenomenon was regarded as an interesting sideline\nin astrophysics that related solely to our understanding of black holes. However,\nwith the recognition that every massive galaxy in the local universe has a \nsupermassive black hole at its heart \n\\citep{magorrian98, richstone98, ferrarese00, gebhardt00}\nwe now recognise that black hole formation is an integral part of galaxy formation,\nwith relationships between black hole and host galaxy that are consistent for\ngalaxies with or without AGN \\citep{onken04, barth05}.\nMoreover, it now seems possible that feedback from black hole growth may have a\nsignificant effect in shutting off star formation and defining the colour-luminosity \ndistribution of massive galaxies \\citep{benson03, croton06, bower06, delucia06}.\nHowever, our understanding of both the physics and the prevalence of AGN feedback is\nextremely limited, as reviewed in the next section. We also have only a partial\nunderstanding of the cosmological dependence of black hole growth within galaxies,\nyet if we wish to understand AGN feedback in the cosmological context this is\nsurely a crucial aspect of the problem.\nIn later sections of this article I discuss one particular scenario for black hole\ngrowth, namely that black holes and galaxies and their dark halos grow coevally,\nand compare with observation.\n\n\\section{AGN feedback - fact or fiction?}\nThe physics of supermassive black hole growth in galaxies and its cosmological\nevolution has recently been receiving wide attention because of the possible\neffects of AGN feedback on the formation of the host galaxy itself \\citep[e.g.][]{dimatteo05}.\nSince we now recognise that every massive galaxy (at least) has a black hole, such\nfeedback may be an integral part of the general galaxy-formation process.\n\\citet{benson03} have argued that something like AGN feedback is required to shut\ndown star formation in the most massive galaxies in order to recreate the observed\ngalaxy mass function, and it has been argued that such feedback\nis required to reproduce the observed colour-luminosity distribution of galaxies\n\\citep{bower06, croton06, delucia06}.\nHowever, the feedback process has to operate in every galaxy and in\nthe relatively recent universe,\nso rather than feedback from the luminous QSO or AGN phase of black hole growth,\nthese authors envisage a ``radio mode'' in which a radiatively-quiet, \nbut kinetically-powerful, jet or outflow provides the feedback in the recent\nuniverse. \n\nIt has long been suggested that radio sources may provide significant heat input to\ncluster gas \\citep[e.g.][]{miller88,pedlar88,pedlar90} but this idea wasn't really taken\nseriously until the discovery both of a lack of cooling gas in ``cooling flow'' clusters\nand of a possible link with radio structures in massive clusters such as Perseus\n(\\citealt{peterson_fabian06} and references therein). But most galaxies at the present\nday don't have powerful radio sources associated with them, so we need to consider\nthe evidence for ``quiet'' outflows in nearby galaxies.\n\nNearby Seyfert active galaxies do have low density outflowing gas in their nuclear regions,\nbut so far these do not appear energetically significant: in NGC\\,5548 for example the\nmass outflow rate is $\\dot{M} \\sim 0.3$\\,M$_{\\odot}$\\,year$^{-1}$ \nwith a maximum outflow velocity $\\sim 1000$\\,km\\,s$^{-1}$ \\citep{steenbrugge05}, \nso it seems unlikely that this\namount of feedback would significantly affect the host galaxy. Evidence for \nhigher velocities and therefore\nsignificantly higher mass and momentum outflow rates has been seen in a few QSOs\n\\citep[e.g.][]{pounds03}. High velocity QSO outflows are also seen in UV absorption lines,\noccurring in $\\sim 26$\\,percent of QSOs \\citep{trump06}, and it has been supposed that\nthese might be associated with an energetic disc wind \\citep[e.g.][]{proga_kallman04},\nbut if such outflows are restricted to the rare (at $z=0$) luminous QSO phase of\nblack hole activity they won't do the job. But kinetic jet output may be more significant\nthan outflowing winds \\citep[e.g.][]{omma04}, and radiatively-inefficient flows\nmay produce such outflows \\citep{narayan94, ho02}, or indeed rotating black holes may \nproduce output electrodynamically \\citep{blandford_znajek77,reynolds06} \nperhaps independently of\nvisible radiation. But a requirement of all these pictures is that, regardless of\nwhether or not we can see it, the source of the feedback energy is gravitational - accretion\nof matter onto the black hole. Despite many attempts over the past four decades,\nwe still have limited understanding\nof how black holes form in galaxies and why their luminous accretion phase shows such\nstrong cosmological evolution.\nIn the remainder of this article we discuss what we know about\nblack hole accretion and its cosmological history, and in particular discuss the\nhypothesis that black holes and galaxies grew coevally.\n\n\\section{QSO and AGN evolution}\nTwenty years ago a striking picture emerged of the cosmological evolution of luminous QSOs:\nthey had an optical luminosity function that had a broken power-law form and that\nevolved steadily to lower luminosity at lower redshifts without changing in normalisation\n\\citep{marshall83, boyle88}. A natural interpretation of this was that supermassive black\nholes formed relatively early in the universe, and that accretion onto them steadily\ndeclined with time to produce the apparent ``pure luminosity evolution''. More recently\nhowever this picture has fallen into disfavour, for two reasons. \n\nFirst, it is widely\nbelieved that dark matter haloes and their galaxies have grown hierarchically, with massive\nstructures continuing to build up in the relatively recent universe. It is often hypothesised\nthat QSOs are triggered by mergers between galaxies in such a hierarchical universe, and\nmodels seeking a cosmological explanation for QSO evolution have been based on a merger-driven\nbuild-up of black holes \\citep[e.g.][]{kauffmann_haehnelt00}. This link to galaxy build-up\nwould explain the qualitative similarity in the cosmological evolution of star formation \nand nuclear black hole activity \\citep[e.g.][]{dunlop97, boyle98, percival_miller99}.\n\nSecond, a more detailed look at the luminosity function has revealed departures from \npure luminosity evolution, with evolution in the slope of the luminosity function at both\nbright and faint magnitudes \\citep{hewett93, goldschmidt98, hopkins06a, fan06}.\nMore significantly, it seems that the peak in the comoving space density of AGN\/QSOs of\na given luminosity shifts systematically to lower redshifts with decreasing luminosity\n(\\citealt{steffen03}, \n\\citealt{cowie03}, \n\\citealt{ueda03}, \n\\citealt{zheng04}, \n\\citealt{barger05}, \n\\citealt{lafranca05}, \n\\citealt{hasinger05}, and \n\\citealt{hopkins06b}),\na phenomenon that has become known as cosmic downsizing. The downsizing is often\ninterpreted as reflecting an increasing prevalence of lower-mass black hole growth\nwith decreasing redshift (but note that, unlike the case of galaxy downsizing, \nthe black hole masses are not well determined at the redshifts covered by the above\nsurveys, so this interpretation can only be regarded as preliminary).\n\nDespite these concerns, the basic picture of twenty years ago must nonetheless be correct:\nthe comoving irreducible mass in black holes cannot decrease with cosmic time, so the\ndecrease observed in the integrated AGN luminosity density (and in the differential luminosity\nfunction) must indeed arise from a mean accretion rate that decreases with cosmic time.\nThe departures from pure luminosity evolution are then most likely indicating that we are\nnot observing a single fixed population of long-lived black holes but rather the \nstatistical changes in a relatively (compared to the Hubble time) short-lived\npopulation.\n\nConfirmation that luminous QSO lifetimes are shorter than the Hubble time comes from\nmeasurement of their clustering properties: the lack of clustering growth to\nlower redshift shows that QSOs cannot be a long-lived population of objects\n(\\citealt{croom05}, these proceedings). The upper limit on their mean lifetime\nis redshift dependent and does depend on the bias model assumed for QSOs, but\na reasonable model yields $2\\sigma$ limits on their existence\nwithin the 2QZ survey of $< 2$\\,Gyr and $< 1$\\,Gyr at $z=1$ and $z=2$ respectively.\n\nHence the picture that now emerges is that AGN evolution must be caused by a decline\nin overall accretion rate onto black holes, but with a characteristic timescale that\nindicates a cosmological influence on a population of objects whose active lives are\nshort. We can measure the characteristic timescale for the QSO population change \nfrom the optical luminosity function. If the characteristic break luminosity,\n$L^{\\star}$ varies as $(1+z)^{\\gamma}$, then the characteristic timescale \nmay be expressed as \n$$\n\\tau \\equiv \\frac{L^{\\star}}{|dL^{\\star}\/dt|} = \\frac{1+z}{\\gamma dz\/dt} = \\frac{1}{\\gamma H(z)}\n$$\nwhere for the optical LF, $\\gamma \\simeq 3$. In the next section we see whether\na cosmological origin for this timescale can be identified.\n\n\\section{The dark halo accretion rate}\nThe growth of dark halos can be calculated within the framework of hierarchical\ngrowth using the extended Press-Schechter \\citep{lc93,lc94}\napproach \\citep{miller06}. The timescale for growth of halos of mass $M_{\\rm H}$ is \n$$\n\\tau_{\\rm H} \\equiv \\frac{M_{\\rm H}}{\\left\\langle dM_{\\rm H}\/dt \\right\\rangle} = \\frac{1}{f(M_{\\rm H}) \\left | d\\delta_c\/dt \\right |},\n$$\nwhere $f(M_{\\rm H})$ is a slowly-varying function of mass, of order unity, \nand where $\\delta_c$ is the usual\nPress-Schechter redshift-dependent critical overdensity for collapse (see \\citealt{miller06}\nfor full details). For an Einstein-de-Sitter universe this has a simple form,\n$$\n\\tau_{\\rm H, EdS} = \\frac{1}{1.68 f(M_{\\rm H}) H(z) (1+z)},\n$$\nwhich has a similar value at $z \\sim 1$ to that observed in the QSO luminosity\nfunction. The timescale is insensitive to the choice of cosmology. It is tempting \nthen to suppose that the timescales for black hole growth and dark halo (and hence galaxy)\ngrowth are comparable and perhaps related.\n\n\\section{Coeval evolution of black holes and their hosts}\nOne of the striking features of the black hole\/bulge $M-\\sigma$ relation in\nthe local universe is its remarkably small scatter, with an intrinsic dispersion\nno larger than $\\sim 0.3$\\,dex \\citep{tremaine02}. It seems likely that feedback\nbetween black hole and galaxy growth is required to produce such a tight relation\n\\citep[e.g.][]{king05}, but whatever the mechanism it seems most natural to suppose\nthat the black hole has acquired its mass at the same time as the galaxy has acquired\nits mass: i.e., that black holes and galaxies have grown coevally. \\citet{miller06}\nhave investigated the hypothesis that the timescales for black hole and galaxy\ngrowth are the same and show the same cosmological dependence. The hypothesis\nis that, {\\em averaged across all galaxies at any given cosmological epoch}, \n\\begin{equation}\n\\tau_{\\rm BH}(z) \\equiv \\frac{M_{\\rm BH}}{\\left\\langle dM_{\\rm BH}\/dt \\right\\rangle}\n= \\frac{M_{\\rm H}}{\\left\\langle dM_{\\rm H}\/dt \\right\\rangle}\n\\equiv \\tau_{\\rm H}(z),\n\\label{eqn:pce}\n\\end{equation}\nwhere $M_{\\rm BH}$ and $M_{\\rm H}$ are the mass of a black hole and its dark\nhalo respectively. We call this\n``Pure Coeval Evolution'' (PCE). The hypothesis does not require\nthere to be a direct causal link between the two, but it may be that feedback processes\ndrive the system towards this behaviour.\n\nThere are two key predictions of this hypothesis. First, the mean Eddington ratio \nof black hole accretion, averaged over all galaxies, should\nrise dramatically to high redshifts, as shown in Babi\\'{c} et al. (these proceedings).\nNote that at any given epoch there is expected to be a wide range of individual\nEddington ratios: galaxies with the highest values, close to unity, would be those\nrecognised as AGN or QSOs. If Eddington ratios do have an upper bound around unity,\nwe do not expect the Eddington ratio of the most luminous\nAGN to show much cosmological evolution \\citep[see][]{kollmeier06}. Less luminous\nAGN and normal galaxies may show evidence for such evolution, however\n\\citep{netzer06}. Averaged over all galaxies, not just AGN, \nthe mean Eddington ratio should increase at higher redshift, implying a greater\nprevalence of visibly accreting black holes at higher $z$.\n\n\\begin{figure}\n \\begin{center}\n \\resizebox{10cm}{!}{ \n \\rotatebox{270}{\n \\includegraphics{LMiller_fig1.ps}\n }}\n \\end{center}\n \\caption{\nThe AGN bolometric luminosity density deduced from the best-fit model of\n\\citet{ueda03}, integrating over the range \n$10^{40}10^{11.5}$\\,M$_{\\odot}$\n(dashed curve). \nBoth curves assume average radiative efficiency $\\langle\\epsilon\\rangle=0.04$. \n}\n\\label{fig1}\n\\end{figure}\n\nSecond, we can predict the expected bolometric output from accreting black holes,\nas follows. The integrated bolometric luminosity density $\\rho_{\\rm L}$\nproduced by AGN depends on the\nnumber density of black holes, on the mean accretion rate onto those, and on the\nmean radiative efficiency \n$\\left\\langle\\epsilon\\right\\rangle$. If we adopt the simplest assumption, that\nequation\\,\\ref{eqn:pce} applies on average to black holes of all masses,\nand if we approximate the function $f(M_{\\rm H})$ as being independent of mass \n\\citep[see][]{miller06} then we can write\n\\begin{eqnarray}\n\\nonumber\n\\left\\langle\\rho_{\\rm L}\\right\\rangle \n&\\simeq &\nc^2 \\rho_{\\rm BH} \n\\left\\langle \n\\frac{\\epsilon}{\\left(1-\\epsilon\\right)M_{\\rm BH}}\n\\frac{dM_{\\rm BH}}{dt} \n\\right\\rangle\\\\\n&\\simeq &\nc^2 \\rho_{\\rm BH} \n\\left\\langle \n\\frac{\\epsilon f\\left(M_{\\rm H}\\right)}{\\left(1-\\epsilon\\right)} \n\\right\\rangle\n\\left |\n\\frac{d\\delta_{c}}{dt} \n\\right |,\n\\end{eqnarray}\nwhere $\\rho_{\\rm BH}$ is the cosmic black hole mass density. Here we adopt\nthe value for $\\rho_{\\rm BH}$ at $z=0$ estimated by \\citet{marconi04}.\nThe result of\nthis calculation is shown in Fig.\\,\\ref{fig1} for two different assumptions.\nThe dot-dashed curve shows the expected evolution if $\\rho_{\\rm BH}$ does not change\nwith redshift: the dashed curve shows the expected evolution if $\\rho_{\\rm BH}$\ntracks the mass density in massive halos, $M_{\\rm H} > 10^{11.5}$\\,M$_{\\odot}$,\ncalculated from the EPS mass function. We see remarkably good agreement with\nthe evolution in bolometric luminosity density, derived from hard X-ray surveys\n(solid curve and points with error bars),\nif the mean radiative efficiency has a constant value \n$\\left\\langle\\epsilon\\right\\rangle \\simeq 0.04$.\nThis is a promising result: there is no other free normalisation of the predicted\nluminosity density, and the value of \n$\\left\\langle\\epsilon\\right\\rangle$ required matches very well\nboth theoretical expectation ($\\epsilon \\la 0.06$ for accretion onto a Schwarzschild\nblack hole) and determinations based on a comparison of the X-ray background\nand the local relic black hole mass density \\citep[e.g.][]{marconi04}.\n\nWe can however also see that the prediction is too high at $z<0.5$, overproducing\nthe bolometric luminosity density by a factor 2 at $z=0$. This implies that\nblack hole growth and halo\/galaxy growth have decoupled by the present day.\nThis result may well be related to the phenomenon of downsizing noted\nat $z=0$ by \\citet{heckman04}. In that work it appears that the mean Eddington\nratio becomes a function of mass, a result that also appears to be confirmed\nby \\citet{netzer06}. Possible physical causes of decoupling could\nbe that either it is an apparent effect caused by a decrease in radiative\nefficiency at low Eddington ratios \\citep[e.g.][]{narayan95,beckert02}, or that \nblack hole growth really does slow down as a result of the changing environments\nof galaxies towards $z=0$, with an increasing prevalence of galaxies forming\ninto groups and clusters.\n\n\\section{Conclusions and further thoughts}\nIt seems that the hypothesis of Pure Coeval Evolution of black holes\nand dark halos reproduces rather well the observed bolometric luminosity\ndensity produced by AGN at $z<3$, implying that, broadly-speaking, black\nholes and their hosts grow together. The dramatic decrease in AGN activity\ntowards $z=0$ is thus seen to be a result of the slow-down in growth of galaxies\nthemselves. This parallel evolution does not necessarily imply a direct causal\nlink between them, but it is attractive to invoke such a causality, especially\nas this would also help explain the very tight $M-\\sigma$ relation between black\nholes and their hosts in the local universe \\citep{tremaine02}.\n\nOne of the main successes of this hypothesis is that it reproduces the observed\nbolometric luminosity density assuming a very reasonable value for the mean radiative\nefficiency, $\\langle\\epsilon\\rangle \\simeq 0.04$ - no previous model of AGN\nevolution has been able to predict the absolute value of the luminosity density.\nIt is worth noting that Pure Coeval Evolution produces a better match to the\nluminosity density evolution than current generations of semi-analytic models\n\\citep[e.g.][]{croton06}.\n\nA factor-two decoupling between black holes and galaxy\/halo growth does seem to occur\nat low redshift, more measurements of Eddington ratio as a function of black hole\nmass, environment and redshift should enable us to distinguish alternative explanations\nfor this effect.\n\nMore observational and theoretical work is needed to further test the scenario\nand to understand the physics of the coevolution process - it is not obvious\nwhy the growth of dark-matter-dominated halos with total mass $\\sim 10^{12}$\\,M$_{\\odot}$ should have\nsuch a direct influence on the growth of central black holes with mass\n$10^{6-8}$\\,M$_{\\odot}$. \nMeanwhile, other work in progress (Babi\\'{c} et al., these proceedings\nand in preparation) shows how both the AGN luminosity function and the X-ray\nbackground can be successfully reproduced within this framework, and it would be very interesting\nto extend this to higher redshifts.\n\nA consequence of the predicted high Eddington ratios at high redshifts is that we expect\na large fraction of galaxies to have nuclear outflows from their growing black holes\n\\citep[e.g.][]{king03,kingpounds03} - searching for evidence of this would be important\nboth for testing the hypothesis and for helping to understand the importance of feedback \nwhen galaxies formed.\n\n\\acknowledgements\nI thank Y.\\,Ueda for providing the data points used for the calculation shown\nin Fig.\\,\\ref{fig1}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nCurrent fluctuations in non-equilibrium open systems has been a major area of research in statistical physics over the last\nfew years \\cite{bodineau-derrida-prl2004, derrida-gers, derrida-gers-sep, bertini-jstatphys2006,\nbertini-landim-prl2005, prahofer-spohn-2002, prolhac-mallick-08,derrida-lecomte-wijland-pre2008,KM12}. The probability distribution of\nthe current of particles across a given point in space typically admits a large deviation form and the corresponding large deviation\nfunction (rate function) is often interpreted as an analogue of a free energy in non-equilibrium systems. For example, it was shown in \\cite{bodineau-derrida-prl2004} that for a large one dimensional system in contact with reservoirs at unequal densities at the two ends, there exists an additivity principle obeyed by the large deviation function, much like the free energy in equilibrium systems. This additivity principle has been exploited further to compute\ncumulants of the current distribution \\cite{prahofer-spohn-2002,prolhac-mallick-08,derrida-gers,derrida-gers-sep,bodineau-derrida-prl2004,bertini-jstatphys2006,bertini-landim-prl2005,derrida-lecomte-wijland-pre2008}. Several theoretical tools have been developed to study current fluctuations, \nnotable among them is the Macroscopic Fluctuation Theory~\\cite{bertini-jstatphys2006, derrida-gers, bodineau-derrida-prl2004,KM12} and the Bethe Ansatz~\\cite{derrida-gers-sep}. Most of these studies have focused on driven diffusive systems, both interacting (as in the simple symmetric exclusion process) and non-interacting, typically in a one-dimensional setting. \n\n\nAnother inherently out-of-equilibrium system, much studied recently, is the so-called {\\it active} run-and-tumble particle (RTP)~\\cite{Berg_book,TC_2008,Martens2012,patterson-gopinath}, a new incarnation of the persistent random walk \\cite{Sta87,Weiss}.\nSuch motion has been observed in certain bacteria such as E. Coli where the bacterium moves in straight runs, undergoes tumbling at the end of a run and chooses randomly a new direction for the next run~\\cite{Berg_book,TC_2008,Martens2012,patterson-gopinath,Sta87,Weiss}. These motions are inherently out-of-equilibrium \nsince they consume energy directly from the environment and self-propel themselves without any external force. There has been an enormous\namount of work concerning the collective properties of an assembly of such RTPs \\cite{Fodor17,bechinger_active_2016,cates_motility-induced_2015,Cates_Nature,SEB_16}. Even at the single-particle level, RTP displays interesting behaviour and several single-particle observables have been studied recently. These include the position distribution for a free RTP \\cite{Sta87,Martens2012,gradenigo}, non-Boltzmann \nstationary states for an RTP in a confining potential \\cite{Cates_Nature, ad-sm-gh,Mallmin_18,Sevilla_19,HP95}, effects of disordered potentials \\cite{Kardar-disorder}, first-passage properties \\cite{km-ad-sm,maes,pierre-satya-greg,ADP_2014, A2015, MLW_86}, the distribution of the time at which an RTP reaches its maximum displacement \\cite{anupam_rtp} and RTP subjected to stochastic resetting \\cite{me-sm-reset, review_resetting}. \n\n\n\n\n\n\nHowever, as far as we are aware, current fluctuations, even in a system of noninteracting RTPs have not been systematically studied. The purpose of this paper is to study the current fluctuations in the simplest setup \nwhere RTPs are noninteracting and initially confined on one-half of the real line (i.e. step-function initial condition). Such a setup was used before\nby Derrida and Gerschenfeld for noninteracting diffusive particles \\cite{derrida-gers} and they were able to compute the large-deviation form of the current\nor flux $Q_t$ of particles through the origin {\\it up to} time $t$. In this paper, we use exactly the same setup, but for a more general class of noninteracting particles, which includes both diffusive as well as run-and-tumble particles, and compute analytically the flux distribution up to time $t$. \n\nThus the main observable of our interest is the flux $Q_t$ defined as the number of particles that crossed the origin (either from left or right) {\\it up to} time $t$, starting from the step initial condition where the particles are uniformly distributed over only the left side of the origin. Let us denote its probability distribution by $P(Q,t) = {\\rm Prob.}(Q_t=Q)$. Clearly $Q_t$ is a history dependent quantity, since it involves counting of all the crossings of the origin up to time $t$. Our exact results rely on a simple but crucial observation, valid for this special step initial condition: each particle, starting from the left side of the origin, that crosses the origin an even number of times up to time $t$ does not contribute to the flux $Q_t$. But if it crosses the origin an odd number of times, it contributes unity to the flux $Q_t$. Hence, the flux $Q_t$ is exactly equal to the number $N^+_t$ of particles present on the right side of \nthe origin {\\it at} time $t$, i.e. $Q_t = N^+_t$. Thus the history dependent observable $Q_t$ gets related, via this observation, to $N^+_t$ which is an instantaneous observable at time $t$. As we will see later, it is much easier to compute the probability distribution $P(N^+,t)={\\rm Prob.}(N^+_t=N^+)$, rather than the distribution of $Q_t$ directly. Hence, knowing the distribution of $N^+_t$, we can compute the flux distribution from $P(Q,t) ={\\rm Prob.}(N^+_t=Q)$. Note that this equivalence holds for arbitrary dynamics of the particles, for example it holds both for diffusive as well as RTP dynamics of the particles. \n\n\nBased on this connection $Q_t = N^+_t$, we can apply our results for the flux distribution to another interesting problem. Consider for instance an ideal gas of noninteracting particles in \na box. Imagine that the box is divided into two halves by a removable wall. Initially, all the particles are on the left half of the box and at $t=0$ we \nlift the wall and let the particles explore the full box freely. At time $t$, we take a snapshot of the system and observe the locations of \nthe particles. Of course, on an average, one expects that, at long times, the particles will be uniformly distributed throughout the box. We can \nask: what is the probability that, at time $t$, all the particles are again back to the left half of the box? Clearly this is an extremely rare event but\nwhat is the probability of this event? How does it decay with time? But note that this is exactly the probability ${\\rm Prob.}(N^+_t=0)=P(Q_t=0,t)$. Hence, our computation of the flux distribution with step initial condition gives access to the probability of this rare event (corresponding to all the particles coming back to the left half of the box at time $t$), in a one-dimensional setting. \n \n\nThe rest of the paper is organised as follows. In section~\\ref{sec:model} we discuss the model and present a summary of our main results. Then in Sec.~\\ref{general-setting-sec} we introduce the general setting and show how the single-particle Green's function plays the central role in the analysis. Next in Secs.~\\ref{annealed-sec} and \\ref{quen-sec}, we calculate the annealed and quenched averages, respectively, of the probability distribution of the flux for both diffusive and run-and-tumble particles.\nThen in Sec.~\\ref{numerics} we give the numerical verifications of our results and\nfinally in Sec.~\\ref{conclu} we summarize and conclude. \n \n\\section{The model and the main results}\\label{sec:model}\n\nWe consider a set of $N$ noninteracting particles initially distributed uniformly with a density $\\rho$ on the negative real axis, as in Fig.~\\ref{model}. Without loss of generality, we label the particles $i=1,2, \\dots, N$ with $x_i(t)$ denoting the position of the $i$-th particle at time $t$. Each $x_i(t)$ evolves independently by a stochastic (or deterministic) evolution rule (the same law of evolution for each particle). For example, each particle can undergo independent Brownian motion. Alternatively, each particle can undergo independent RTP dynamics in one-dimension. \nThis RTP dynamics for a single particle is defined as follows. \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm]{dynamics.eps}\n\\caption{Schematic representation of an initial realization with all particles on the left of an arbitrary origin ($x=0$) on an infinite line $L \\rightarrow \\infty$. After time $t$ each particle undergoes some displacement depending on\nthe dynamics. The quantity of interest in our case is the number of particles on the right of the origin at time $t$.\n} \\label{model}\n\\end{center}\n\\end{figure}\n\n\\vspace{0.4cm}\n\\noindent {\\bf RTP dynamics}. The position of a single RTP\n$x(t)$ evolves via the Langevin equation\n\\begin{equation}\n\\frac{dx}{dt}= v_0\\, \\sigma(t)\\, \n\\label{RTP_evol.1}\n\\end{equation}\nwhere $v_0$ is the intrinsic speed during a run and $\\sigma(t)=\\pm 1$ is a dichotomous \ntelegraphic noise that flips from\none state to another with a constant rate $\\gamma$. The effective noise $\\xi(t)=v_0\\, \\sigma(t)$\nis coloured which is simply seen by computing its autocorrelation function\n\\begin{equation}\n\\langle \\xi(t) \\xi(t')\\rangle= v_0^2\\, e^{-2\\, \\gamma\\, |t-t'|}\\, .\n\\label{autocorr.1}\n\\end{equation}\nThe time scale $\\gamma^{-1}$ is the `persistence' time of a run that encodes the memory\nof the noise. In the limit $\\gamma\\to \\infty$, \n$v_0\\to \\infty$ but keeping the ratio $D_{\\rm eff}= v_0^2\/{2\\gamma}$ fixed, the noise $\\xi(t)$ reduces to\na white noise since\n\\begin{equation}\n\\langle \\xi(t) \\xi(t')\\rangle= \\frac{v_0^2}{\\gamma}\\, \\left[\\gamma\\, e^{-2\\gamma|t-t'|}\\right]\n\\to 2D_{\\rm eff}\\, \\delta(t-t')\\, .\n\\label{autocorr.2}\n\\end{equation}\nThus in this so called `diffusive limit', the persistent random walker $x(t)$ reduces to an\nordinary Brownian motion. For finite $\\gamma$ (i.e., persistence time scale of memory),\nthe RTP will be referred to as an ``active'' particle. In the diffusive limit, the active particle dynamics reduces\nto an ordinary Brownian motion, which we refer to as a ``passive'' motion.\n\nGiven the stochastic dynamics of the individual particles, starting from the step initial condition, our main object of\ninterest is the flux $Q_t$ of particles through the origin up to time $t$. If a trajectory crosses the origin from left to right,\nthis will contribute a $+1$ to the net current while if it crosses from right to left, its contribution is $-1$. The flux $Q_t$ is thus\nthe net contribution to the current up to time $t$. Let us denote by $P(Q,t,\\{x_i\\})$ the probability distribution ${\\rm Prob.}(Q_t=Q)$ \nfor a given initial condition where $x_i$'s denote the initial positions of the particles at time $t=0$. \nFollowing Derrida and Gerschenfeld, \nthe effect of the initial condition on the distribution can be studied in\ntwo alternative ways, in analogy with the disordered systems where the realisation of a disorder plays an analogous\nrole as the initial condition in our problem. \\textcolor{black}{It was indeed argued in \\cite{derrida-gers} that one has to distinguish between two different ways of \naveraging over the initial conditions: (i) the annealed average, where the probability distribution of the flux is averaged over all the realizations of the initial condition \nand (ii) the quenched average where the probability distribution is computed for the {\\it typical} initial configurations.} Instead of considering the distribution $P(Q,t,\\{x_i\\})$ directly, it turns out to be convenient to consider its \ngenerating function $\\langle e^{-p Q}\\rangle_{\\{x_i\\}}$, where the angular brackets $\\langle \\cdots \\rangle_{\\{x_i\\}}$ denote an average over the history, but with fixed initial condition $x_i$. The annealed and quenched averages are now defined as follows:\n\\begin{eqnarray}\n&&\\sum_{Q=0}^\\infty e^{-p Q} \\, P_{\\rm an}(Q,t) \\, = \\overline{\\langle e^{-p Q}\\rangle_{\\{x_i\\}}} \\;, \\label{def_ann} \\\\\n&&\\sum_{Q=0}^\\infty e^{-p Q} \\, P_{\\rm qu}(Q,t) \\, = \\exp{\\left[ \\overline{\\ln \\langle e^{-p Q}\\rangle_{\\{x_i\\}}} \\right]} \\;, \\label{def_quen}\n\\end{eqnarray}\nwhere $\\overline{\\cdots}$ denotes an average over the initial conditions. Note that in this problem $Q_t$ is always an integer. As mentioned in the introduction, for the step initial condition, we can compute both $P_{\\rm an}(Q,t)$ and $P_{\\rm qu}(Q,t)$ \\textcolor{black}{(see Fig.~\\ref{Pqu-compare} for a plot of these probability distributions)} for arbitrary dynamics of the particles by using the identity $Q_t = N^+_t$, where $N^+_t$ is the number of particles on the right side of the origin at time $t$. Indeed, the only quantity that enters the computation for independent particles is the single particle Green's function $G(x,x_0,t)$ denoting the probability density of finding the particle at position $x$ at time $t$, starting from $x_0$ at $t=0$. Let us first define a central object, that will appear in all our formulas \n\\begin{eqnarray}\\label{UzT}\nU(z,t) = \\int_0^\\infty G(x,-z,t) \\, dx \\quad, \\quad z\\geq 0 \\;,\n\\end{eqnarray}\nobtained by integrating the Green's function over the final position, with the initial position fixed at $x_0=-z \\leq 0$. If we can compute $U(z,t)$ for a given dynamics, we can express $P_{\\rm an}(Q,t)$ and $P_{\\rm qu}(Q,t)$ in terms of this central function $U(z,t)$. Our main results can now be summarised as follows. \n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{minipage}{0.45\\hsize}\n\\includegraphics[width=\\hsize]{Pan-compare-gm-1-t40.eps}\n\\end{minipage}\n\\begin{minipage}{0.45\\hsize}\n\\includegraphics[width=\\hsize]{Pqu-diffusion-rtp-Deff0-5-t40-gmt40.eps}\n\\end{minipage}\n\\end{center}\n\\caption{{(a) Annealed case: Semi-log plot of $P_{\\rm an}(Q,t)$ vs $Q$ for RTPs (red solid line) using Eqs.~(\\ref{ac-mut}), (\\ref{Poisson_largedev}) and (\\ref{psi_an}) compared to the diffusive case (black dashed curve) given by Eq. (\\ref{Poisson_largedev}) with $\\mu(t)=\\rho\\sqrt{\\frac{Dt}{\\pi}}$ for $\\rho = 1$ and $t=40$. For the RTP, we set $v_0=\\gamma=1$, corresponding to an effective diffusion constant $D_{\\rm eff}=v_0^2\/(2\\gamma)= 1\/2$, while we use, accordingly, $D=1\/2$ for diffusion. For such a (large) time where the scaling form (\\ref{Poisson_largedev}) is expected to hold, the RTP and the diffusive cases are almost indistinguishable. Finally, the blue dashed curve corresponds to the typical Gaussian approximation with mean and variance $\\mu(t)$ given in (\\ref{ac-mut}). (b)~ Quenched case: Semi-log plot of $P_{\\rm qu}(Q,t)$ vs $Q$ for the same set of parameters for RTPs (red solid line), obtained from Eqs.~(\\ref{Uzt-exact}) and~(\\ref{def_tildeI}), compared to the diffusive case (black dashed line) obtained from Eqs.~(\\ref{phi-1eq}) and (\\ref{fq-bas}). The blue dotted line corresponds to the Gaussian approximation with mean $\\mu(t)$ given in (\\ref{ac-mut}) and variance given in (\\ref{var_qu}). While the active and passive cases remain\nindistinguishable at small $Q$, the quenched distribution, at variance with the annealed one, carries a clear signature of activity \nat large $Q$. For example, in the quenched case, the maximum possible flux for the RTP is $Q= \\rho v_0 t = 40$ (large blue dot). \n}}\\label{Pqu-compare}\n\\end{figure} \n\n\n\\vspace*{0.4cm}\n\\noindent\n{\\bf Annealed case}: In this case, we show that $P_{\\rm an}(Q,t)$ is always a Poisson distribution \n\\begin{equation}\\label{Pan-model-def}\nP_{\\rm an}(Q=n,t) = e^{-\\mu(t)} \\frac{\\mu(t)^n}{n!} \\;, \\; n=0,1,2,\\cdots \\;,\n\\end{equation}\nwhere \n\\begin{equation}\\label{muan-mod-def}\n\\mu(t) = \\rho \\, \\int_0^{\\infty} dz ~U(z,t) \\;.\n\\end{equation} \nThe mean and the variance of $Q_t$ are both given by $\\mu(t)$, which \ncan be explicitly evaluated for different types of particle motion. For example, for a Brownian motion with diffusion constant $D$, using $G(x,x_0,t) = e^{-(x-x_0)^2\/(4 Dt)}\/\\sqrt{4 \\pi D t}$ in Eq. (\\ref{UzT}) we get \n\\begin{eqnarray} \\label{UzT_BM}\nU(z,t) = \\frac{1}{2} {\\rm erfc}\\left(\\frac{z}{\\sqrt{4 D t}} \\right) \\quad, \\quad {\\rm and} \\quad \\mu(t) = \\rho \\sqrt{Dt\/\\pi} \\;,\n\\end{eqnarray}\nwhere ${\\rm erfc}(z) = (2\/\\sqrt{\\pi}) \\int_z^\\infty e^{-u^2} du$. Our result for $P_{\\rm an}(Q=n,t)$ for the diffusive case is consistent with the result of Derrida and Gershenfeld \\cite{derrida-gers} obtained by a different method. \n\n\n\nIn the case of the RTP dynamics, we find explicitly that at all $t$,\n\\begin{equation}\\label{ac-mut}\n\\mu(t) = \\frac{1}{2}\\rho \\, v_0 \\, t\\, e^{-\\gamma t} \\,[I_0(\\gamma t) + I_1(\\gamma t)],\n\\end{equation}\nwhere $I_0(z)$ and $I_1(z)$ are modified Bessel functions of the first kind. Its asymptotic behaviours are given by\n\\begin{eqnarray} \\label{asympt_mu}\n\\mu(t) \\approx\n\\begin{cases}\n&\\dfrac{\\rho \\,v_0}{2}\\, t \\;, \\;\\;\\; {\\rm as}\\;\\;\\; t \\to 0 \\;, \\\\\n&\\\\\n&\\rho \\sqrt{\\dfrac{D_{\\rm eff}\\,t}{\\pi}} \\;, \\;{\\rm as}\\; t \\to \\infty \\;,\n\\end{cases}\n\\end{eqnarray}\nwhere $D_{\\rm eff} = v_0^2\/(2 \\gamma)$. Thus at late times, the RTP behaves like a diffusive particle with an effective diffusion\nconstant $D_{\\rm eff}$. \n\nNote that the Poisson distribution in Eq. (\\ref{Pan-model-def}) in the limit $Q \\to \\infty$, $\\mu(t) \\to \\infty$, keeping\nthe ratio $Q\/\\mu(t)$ fixed, can be written in a large deviation form (using simply Stirling's formula)\n\\begin{eqnarray}\\label{Poisson_largedev}\nP_{\\rm an}(Q,t) \\sim \\exp{\\left[- \\mu(t) \\, \\Psi_{\\rm an} \\left( \\frac{Q}{\\mu(t)}\\right) \\right]} \\;,\n\\end{eqnarray} \nwhere the rate function $\\Psi_{\\rm an}(q)$ is universal, i.e., independent of the particle dynamics, and is given by\n\\begin{eqnarray} \\label{psi_an}\n\\Psi_{\\rm an}(q) = q \\,\\ln q - q + 1 \\;, \\; q \\geq 0 \\;.\n\\end{eqnarray}\nIt has the asymptotic behaviours\n\\begin{eqnarray} \\label{asympt_psi_an}\n\\Psi_{\\rm an}(q) \\approx\n\\begin{cases}\n& 1 \\;, \\;\\; {\\rm as} \\;\\; q \\to 0 \\\\\n& \\dfrac{1}{2} (q-1)^2 \\;, \\;\\; {\\rm as} \\;\\; q \\to 1 \\\\\n& q \\, \\ln q \\;, \\;\\; {\\rm as} \\;\\; q \\to \\infty \\;. \n\\end{cases}\n\\end{eqnarray}\nThe quadratic behavior near the minimum at $q=1$ indicates typical Gaussian fluctuations for $Q$, with mean and variance both equal to $\\mu(t)$. Note that the dependence on the particle dynamics in Eq. (\\ref{Poisson_largedev}) enters only through the parameter $\\mu(t)$ but the function $\\Psi_{\\rm an}(q)$ is universal. \n\nWe also note that, from our general result in Eq. (\\ref{Pan-model-def}), it follows that \n\\begin{eqnarray}\\label{Pq0}\n{\\rm Prob.}(N_t^+ =0) \\Big|_{\\rm an} = P_{\\rm an}(Q=0,t) = e^{-\\mu(t)} \\;.\n\\end{eqnarray} \nThis result is valid for all $t$ and gives the probability of the rare event that all the particles are back on the left side of the origin at time $t$, as discussed in the introduction. {In Fig.~\\ref{Pan-qu-compare} (a) we show a plot of $P_{\\rm an}(Q=0,t)$ as a function of time, both for RTP and for diffusive particles.}\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{P0-an-log.eps}\n\\end{minipage}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{P0-qu-log.eps}\n\\end{minipage}\n\\end{center}\n\\caption{{(a) Semi-log plot of $P_{\\rm an}(Q=0,t) = e^{-\\mu(t)}$ vs $t$ for an RTP (red solid line) with $\\mu(t)$ given in Eq.~(\\ref{Pq0})\n compared to the diffusive case (black dashed line) corresponding to $\\mu(t)=\\rho \\sqrt{\\frac{D t}{\\pi}}$, for a density $\\rho=1$ in both cases. For the RTP, we set $v_0=\\gamma=1$, corresponding to an effective diffusion constant $D_{\\rm eff}=v_0^2\/(2\\gamma)= 1\/2$, while we set, accordingly, $D=1\/2$ in the case of diffusion. (b) Semi-log plot of $P_{\\rm qu}(Q=0,t)$ vs $t$ as given in Eq.~(\\ref{P_neq0_qu}) both for RTPs (red solid line) using the result for $U(z,t)$ in (\\ref{Uzt-exact}) and for Brownian particles (black dashed line) for which $U(z,t)$ is given in~(\\ref{UzT_BM}). In both annealed and quenched cases, the zero-net flux probabilities for active and passive systems differ at short times but do coincide in the large time limit.}}\\label{Pan-qu-compare}\n\\end{figure}\n\n\n\\vspace*{0.4cm}\n\\noindent\n{\\bf Quenched case}: In this case, the generating function in Eq. (\\ref{def_quen}), for arbitrary single particle dynamics, can again be expressed in terms \nof the central function $U(z,t)$ (\\ref{UzT}) as follows\n\\begin{eqnarray}\\label{P_qu_gen}\n\\sum_{Q=0}^\\infty P_{\\rm qu}(Q,t) e^{-p Q} \\, = \\exp{\\left[\\rho \\int_0^\\infty dz \\, \\ln{\\left[1-(1-e^{-p})U(z,t)\\right]} \\right]} \\;.\n\\end{eqnarray} \nThe quenched cumulants of $Q$ can then be extracted and expressed in terms of $U(z,t)$. For instance, \nthe quenched mean and the variance of $Q$ are given by \n\\begin{eqnarray}\n&&\\langle Q \\rangle_{\\rm qu} = \\rho \\, \\int_0^\\infty U(z,t) \\, dz \\;, \\label{mean_qu} \\\\\n&&\\sigma_{\\rm qu}^2 = \\langle Q^2 \\rangle_{\\rm qu} - \\langle Q \\rangle^2_{\\rm qu} = \\rho \\int_0^\\infty U(z,t)(1-U(z,t))\\, dz \\;. \\label{var_qu}\n\\end{eqnarray}\nIn addition, expanding the right hand side (rhs) in powers of $e^{-p}$ and matching with the left hand side (lhs), one can in principle obtain $P_{\\rm qu}(Q,t)$ for any integer $Q$ as a functional of $U(z,t)$. For example, \n\\begin{eqnarray}\\label{P_neq0_qu}\n{\\rm Prob.}(N_t^+ = 0) \\Big |_{\\rm qu} = P_{\\rm qu}(Q=0,t) = \\exp{\\left[\\rho \\int_0^\\infty \\, \\ln{\\left[1-U(z,t)\\right]} \\, dz\\right]} \\;,\n\\end{eqnarray}\nwhich is valid for all times $t \\geq 0$. However, the formula gets more complicated for higher values of $Q$. {In Fig.~\\ref{Pan-qu-compare} (b) we show a plot of $P_{\\rm qu}(Q=0,t)$ as a function of time, both for RTP and for diffusive particles.}\n\n\nFor the full quenched distribution, we first consider the diffusive motion of the particles. In this case, we obtain, in the scaling limit $Q \\to \\infty$, $t \\to \\infty$ keeping the ratio $Q\/\\sqrt{t}$ fixed, the same large deviation form as Derrida and Gerschenfeld~\\cite{derrida-gers},\n\\begin{equation}\\label{P-F-diff-largeQ}\nP_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho \\sqrt{D t} ~ \\Psi_{\\rm diff}\\left(\\frac{Q}{\\rho \\sqrt{D t}}\\right)\\right] \\;,\n\\end{equation}\nwhere the rate function $\\Psi_{\\rm diff}(q)$ has the following precise asymptotics\n\\begin{eqnarray} \\label{psi_diff_asympt}\n\\Psi_{\\rm diff}(q) \\approx\n\\begin{cases}\n&\\overline{\\alpha} - q + q \\ln(q\/\\overline{\\beta}) \\;, \\; {\\rm as} \\; q \\to 0 \\\\\n& \\\\\n& \\sqrt{\\dfrac{\\pi}{2}} \\left(q-\\dfrac{1}{\\sqrt{\\pi}} \\right)^2 \\;, \\; {\\rm as} \\; q \\to 1\/\\sqrt{\\pi} \\\\\n& \\\\\n& \\frac{1}{12} q^3 \\;, \\; {\\rm as} \\; q \\to \\infty \\;,\n\\end{cases} \n\\end{eqnarray}\nwith the two constants $\\overline{\\alpha}$ and $\\overline{\\beta}$ given explicitly by\n\\begin{eqnarray} \n&&\\overline{\\alpha} = -2 \\int_0^\\infty dz \\, \\ln\\left(1 - \\frac{1}{2} \\, {\\rm erfc}(z) \\right) = 0.675336 \\ldots \\label{C} \\\\\n&&{\\overline{\\beta}} = \\int_0^\\infty dz \\, \\frac{{\\rm erfc}(z)}{1-\\frac{1}{2} {\\rm erfc}(z)} = 0.828581 \\ldots \\label{A} \\;.\n\\end{eqnarray}\nNote that the large $q$ behavior $\\Psi_{\\rm diff}(q) \\approx q^3\/12$ coincides with the result of Derrida and Gerschenfeld \\cite{derrida-gers} obtained\nby a different method. The small $q$ behavior was not investigated in Ref. \\cite{derrida-gers}. Taking the $q \\to 0$ limit in Eq. (\\ref{P-F-diff-largeQ}) and using the small $q$ behavior in the first line of Eq. (\\ref{psi_diff_asympt}) implies that for large $t$ \n$P_{\\rm qu}(Q=0,t) \\sim \\exp \\left[ - \\overline{\\alpha} \\, \\rho \\sqrt{D\\,t}\\right]$ where the constant $\\overline{\\alpha}$ is given in Eq. (\\ref{C}). In fact, this result is valid not just at large time but at all times. Indeed, substituting $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4Dt})$ in our general formula (\\ref{P_neq0_qu}), it follows that at all times $t \\geq 0$,\n\\begin{eqnarray} \\label{P_Q0}\nP_{\\rm qu}(Q=0,t)\\Big \\vert_{\\rm diff} = \\exp \\left[ - \\overline{\\alpha} \\, \\rho \\sqrt{D\\,t}\\right] \\;.\n\\end{eqnarray}\nInterestingly, exactly the same rate function $\\Psi_{\\rm diff}(q)$ also appeared in the completely different context, namely as a large deviation function characterising the distribution of the number of eigenvalues (in a disk of radius $R$) of a complex Ginibre ensemble of $N\\times N$ Gaussian random matrices \\cite{castillo,bertrand}. \n\n\n\nWe then consider the quenched flux distribution for the RTP dynamics. First, we show that, for small $Q \\ll \\sqrt{t}$, the flux distribution decays as a stretched exponential at late times, e. g. $P_{\\rm qu}(Q=0,t) $ is given, for large $t$, by\n\\begin{eqnarray}\\label{P_Q0-1_RTP}\nP_{\\rm qu}(Q=0,t) \\Big \\vert_{\\rm RTP} \\sim \\exp \\left[ - \\overline{\\alpha} \\, \\rho \\sqrt{D_{\\rm eff}\\,t}\\right] \\;,\n\\end{eqnarray}\nwhere $\\bar{\\alpha}$ is the same constant as in Eq. (\\ref{C}) and $D_{\\rm eff} = v_0^2\/(2 \\gamma)$. One of the main results of this analysis is to find a new scaling limit $Q \\to \\infty$, $t \\to \\infty$, keeping the ratio $Q\/(\\rho\\,v_0 t)$ fixed where the quenched distribution admits a large deviation form [quite different from the diffusive case in Eq. (\\ref{P-F-diff-largeQ})]\n\\begin{equation}\\label{ac-sig}\nP_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho \\, v_0 \\, \\gamma \\, t^2 \\, \\Psi_{\\rm RTP}\\left(\\frac{Q}{\\rho \\, v_0 \\,t}\\right)\\right],\n\\end{equation}\nwhere the rate function $\\Psi_{\\rm RTP}(q)$ is given explicitly by\n\\begin{equation}\\label{rtp-ldf-model}\n\\Psi_{\\rm RTP}(q)=q-\\frac{q}{2}\\sqrt{1-q^2}-{\\rm sin}^{-1}\\left[ \\sqrt{\\frac{1-\\sqrt{1-q^2}}{2}} \\right]\\;, \\quad 0 \\leq q \\leq 1 \\;.\n\\end{equation}\nThe rate function has the asymptotic behavior\n\\begin{eqnarray}\\label{asympt_RTP}\n\\Psi_{\\rm RTP}(q) \\approx\n\\begin{cases}\n& \\dfrac{q^3}{6} \\quad,\\quad \\quad \\quad \\; q\\rightarrow 0 \\,\\\\ \n& \\\\\n& 1-\\dfrac{\\pi}{4} \\quad,\\quad ~ ~ ~ q = 1 \\;.\n\\end{cases}\n\\end{eqnarray}\nOne consequence of our result is the prediction of the probability of the rare event that the flux $Q$ up to time\n$t$ achieves its maximum possible value, namely $Q = \\rho \\, v_0\\, t$ -- this corresponds to the case where all\nthe particles move ballistically to the right up to time $t$. We find that the probability of this rare event is given by\n\\begin{eqnarray} \\label{P_qu_max}\nP_{\\rm qu}(Q = \\rho \\, v_0 \\, t,t) \\approx \\exp\\left[-\\left( 1 - \\frac{\\pi}{4}\\right) \\rho \\, v_0 \\, \\gamma \\, t^2 \\right] \\;.\n\\end{eqnarray}\nSuch a faster than exponential decay for the probability of this rare event is a nontrivial prediction of our\ntheory. \n\n\n\n\n\n\n\n\n\n\n\n\\section{The general setting and the single-particle Green's function}\\label{general-setting-sec}\n\nWe start with a step initial condition where $N$ particles are initially located on the negative half line at positions $\\{x_1, x_2, \\cdots, x_N \\}$ where \nall $x_i <0$. As stated before, for this step initial condition, the flux $Q_t$ up to time $t$ is identical in law to the number of particles $N^+_t$ to the\nright of the origin at time $t$. Let us introduce an indicator function ${\\cal I}_i(t)$ \nsuch that ${\\cal I}_i(t)=1$ if the $i{\\rm th}$ particle\nis to the right of the origin at time $t$, else ${\\cal I}_i(t)=0$. Hence we have\n\\begin{equation}\\label{N+defn}\n N_t^+= \\sum_{i=1}^N {\\cal I}_i(t) \\;.\n\\end{equation}\nFor fixed $x_i$'s the flux distribution is then given by\n\\begin{eqnarray}\\label{basic-defn}\nP(Q,t,\\{ x_i\\}) = {\\rm Prob.}(N_t^+ = Q) = \\left \\langle \\delta \\left[Q-\\sum_{i=1}^N {\\cal I}_i(t)\\right]\\right \\rangle_{\\{x_i\\}} \\;, \n\\end{eqnarray}\nwhere the angular brackets $\\langle \\cdots \\rangle_{\\{x_i\\}}$ denote an average over the history, but with fixed initial condition $x_i$. Taking the Laplace transform on both sides of Eq. (\\ref{basic-defn}) gives\n\\begin{equation}\\label{lpbasic}\n \\sum_{Q=0}^{\\infty} e^{-pQ} P(Q,t,\\{x_i\\}) = \\langle e^{-pQ}\\rangle_{\\{x_i\\}} = \\left \\langle \\exp[{-p \\sum_{i=1}^N {\\cal I}_i(t)}]\\right \\rangle_{\\{x_i\\}} \\;.\n \\end{equation}\nSince the ${\\cal I}_i$ can only take the values $0$ or $1$, one has the identity $e^{-p {\\cal I}_i} = 1 -(1-e^{-p}){\\cal I}_i$. Inserting this identity in Eq. (\\ref{lpbasic}) and using the independence of the random variables ${\\cal I}_i$'s we get\n \\begin{equation}\\label{sq-Ii}\n \\langle e^{-pQ}\\rangle_{\\{x_i\\}} = \\prod_{i=1}^N\\left[1- (1-e^{-p})\\langle {\\cal I}_i(t) \\rangle_{\\{x_i\\}} \\right],\n \\end{equation}\nwhere the right hand side (r.h.s.) implicitly depends on the $x_i$'s. The average $\\langle {\\cal I}_i(t) \\rangle_{\\{x_i\\}}$ is just the probability that the $i{\\rm th}$ particle is to the right of the origin at time $t$, starting initially at $x_i$ and hence we have \n\\begin{equation}\\label{I-propagator}\n \\langle {\\cal I}_i(t) \\rangle_{\\{x_i\\}} = \\int_0^{\\infty} G(x,x_i,t) dx = U(-x_i,t) \\;, \\quad x_i < 0 \\;,\n \\end{equation}\nwhere $G(x,x_i,t)$ is the single-particle Green's function, i.e., the propagator for a particle to reach $x$ at time $t$, starting initially at $x_i<0$. Note that $U(z,t)$ is defined in Eq. (\\ref{UzT}) and corresponds to the probability that a particle is on the positive side of the origin at time $t$, starting initially at $-z<0$. Inserting Eq.~(\\ref{I-propagator}) into Eq.~(\\ref{sq-Ii}), one obtains\n\\begin{equation}\\label{propagator}\n \\langle e^{-pQ}\\rangle_{\\{x_i\\}}= \\prod_{i=1}^N\\left[1- (1-e^{-p})U(-x_i,t)\\right] \\;, \\quad x_i < 0 \\;, \\quad \\forall i = 1, \\cdots, N \\;.\n \\end{equation}\nThis Eq.~(\\ref{propagator}) is general, i.e., valid for\n{\\em any} set of non-interacting particles undergoing a common dynamics in one-dimension. The information about the dynamics\nis entirely encoded in the function $U(z,t)$. \n\nFor instance, for simple diffusion, the single-particle Green's function is given by \n\\begin{equation}\\label{diff-pro2}\nG(x,x_i,t) = \\frac{1}{\\sqrt{4\\pi D t}}{{\\rm exp}\\left[-\\frac{(x-x_i)^2}{4Dt}\\right]} \\;,\n\\end{equation}\nwhich gives $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4Dt})$. For the RTP, on the other hand, the Green's function is known explicitly~\\cite{Weiss,me-sm-reset} \n\\begin{equation}\\label{ac-fullpd3}\nG(x,x_i,t) = \\frac{e^{-\\gamma t}}{2}\\left\\{ \\delta(x-x_i-v_0t) + \\delta(x-x_i+v_0t) +\\frac{\\gamma}{v_0}\\left[I_0(\\omega) + \\frac{\\gamma t I_1(\\omega)}{\\rho} \\right] \\Theta(v_0t-|x-x_i|) \\right\\},\n\\end{equation}\nwhere $\\omega$ is given by\n\\begin{eqnarray}\\label{def_omega}\n\\omega = \\frac{\\gamma}{v_0}\\sqrt{v_0^2t^2 -(x-x_i)^2} \\;.\n\\end{eqnarray}\nIn Eq. (\\ref{ac-fullpd3}), $\\Theta(z)$ is the Heaviside Theta function, and $I_0(\\omega)$ and $I_1(\\omega)$ are modified Bessel functions. Computing $U(z,t) = \\int_0^\\infty G(x,-z,t)\\,dx$ explicitly using Eq. (\\ref{ac-fullpd3}) is complicated. It is however much more useful, as we will see later, to work with the Laplace transform of $G(x,x_i,t)$ with respect to $t$, which has a much simpler expression, namely \n\\begin{equation}\\label{ac-Lap3}\n\\tilde{G}(x,x_i,s) = \\int_0^{\\infty} dt~ e^{-st} G(x,x_i,t) = \\frac{\\lambda(s)}{2s}e^{\\lambda(s)\\,|x-x_i|} \\;, \\quad \\lambda(s) = \\frac{\\sqrt{s(s+2\\gamma)}}{v_0} \\;.\n\\end{equation}\n\n\nThe relation in Eq. (\\ref{propagator}) is the central result of this section and we will analyse the annealed and the quenched cases separately in the next two sections. \n\n\n\n\\section{Flux distribution in the annealed case}\\label{annealed-sec}\n\nThe annealed distribution $P_{\\rm an}(Q,t)$ is defined in Eq. (\\ref{def_ann}) where the $\\overline{\\cdots}$ denotes an average over\nthe initial conditions. Performing this average in Eq. (\\ref{propagator}) gives\n\\begin{equation}\\label{ann-G-avg}\n\\langle\\overline{e^{-pQ}\\rangle_{\\{x_i\\}}}= \\prod_{i=1}^N\\left[1- (1-e^{-p})\\, \\overline{U(-x_i,t)}\\right] \\;,\n\\end{equation}\nwhere $U(-x_i,t)$ is defined in Eq. (\\ref{I-propagator}). To perform the average over the initial conditions with a fixed uniform density $\\rho$, \nwe assume that each of the $N$ particles is distributed independently and uniformly over a box $[-L,0]$ and then eventually take the limit \n$N \\to \\infty$, $L \\to \\infty$ keeping the density $\\rho = N\/L$ fixed. For this uniform measure, each $x_i$ is uniformly distributed in the box\n$[-L,0]$. Using the independence of the $x_i$'s we then get \n\\begin{equation}\\label{x_i-avg}\n\\langle\\overline{e^{-pQ}\\rangle_{\\{x_i\\}}} = \\prod_{i=1}^N \\left[1- (1-e^{-p}) \\int_{-L}^{0} U(-x_i,t) \\frac{dx_i}{L} \\right] = \\left[1- \\frac{1}{L}(1-e^{-p}) \\int_0^L U(z,t) dz \\right]^N \\;,\n\\end{equation}\nwhere, in the last equality, we made the change of variable $z=-x_i$. Taking now the limit $N \\to \\infty$, $L \\to \\infty$ keeping $\\rho = N\/L$ fixed gives\n\\begin{eqnarray}\\label{lap-pan}\n\\sum_{Q=0}^{\\infty} e^{-pQ} P_{\\rm an}(Q,t) = \\langle\\overline{e^{-pQ}\\rangle_{\\{x_i\\}}} = \\exp\\left[-\\mu(t) ~ (1-e^{-p})\\right] \\;, \\quad {\\rm where} \\;\\quad \\mu(t)= \\rho \\int_0^{\\infty} dz ~ U(z,t) \\;.\n\\end{eqnarray}\nBy expanding $\\exp\\left[-\\mu(t) ~ (1-e^{-p})\\right]$ in powers of $e^{-p}$ and comparing to the left hand side, we see that $Q$ can take only integer values $Q=n=0,1,2,\\cdots$ and the probability distribution is simply a Poisson distribution with mean $\\mu(t)$ as given in Eqs. (\\ref{Pan-model-def}) and (\\ref{muan-mod-def}). \n\nThis Poisson distribution, in the annealed case, is thus universal, i.e., holds for any dynamics. The details of the dynamics is encoded in the single\nparameter $\\mu(t)$ which can be computed explicitly for different types of dynamics. For example, for diffusing particles, using the explicit expression for the Brownian propagator, we get $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4Dt})$ and, hence, $\\mu(t) = \\rho \\sqrt{Dt\/\\pi}$ as mentioned in Eq. (\\ref{UzT_BM}). In contrast, for the RTP dynamics, $\\mu(t)$ is nontrivial. As discussed earlier, computing $U(z,t)$ from the Green's function in Eq. (\\ref{ac-fullpd3}) is difficult. Consequently, calculating $\\mu(t) = \\rho\\int_0^\\infty U(z,t)\\,dz$ is also hard. However, it turns out that its Laplace transform is much easier to manipulate, due to the simple nature of the formula in Eq. (\\ref{ac-Lap3}). The Laplace transform of $\\mu(t)$ is given by\n\\begin{equation}\\label{mus-RTP}\n\\tilde{\\mu}(s) = \\int_{0}^{\\infty} dt~ e^{-st}~ \\mu(t) = \\rho \\int_0^\\infty dz ~\\tilde{U}(z,s) \\;, \\quad {\\rm with} \\quad \\tilde U(z,s) = \\int_{0}^{\\infty} dt~ e^{-st}~ U(z,t) \\;,\n\\end{equation}\nwhere we have used the relation $\\mu(t) = \\rho \\int_0^\\infty U(z,t)\\,dz$. The Laplace transform of $U(z,t)$ can be computed as follows\n\\begin{eqnarray}\\label{LaplaceU_1}\n\\tilde U(z,s) = \\int_{0}^{\\infty} dt~ e^{-st}~ U(z,t) = \\int_0^\\infty dt \\, e^{-st} \\, \\int_0^\\infty G(x,-z,t) \\, dx \\;.\n\\end{eqnarray}\nExchanging the integrals over $x$ and $t$, and using the relation in Eq. (\\ref{ac-Lap3}) and integrating over $x$ we get\n\\begin{eqnarray}\\label{LaplaceU_2}\n\\tilde U(z,s) = \\frac{e^{-\\lambda(s) z}}{2s} \\;, \\quad {\\rm where} \\quad \\lambda(s) = \\frac{\\sqrt{s(s+2\\gamma)}}{v_0} \\;.\n\\end{eqnarray}\nInserting this relation in Eq. (\\ref{mus-RTP}) and performing the integral over $z$, we get\n\\begin{equation}\\label{mu-ac-lap}\n\\tilde{\\mu}(s) = \\frac{1}{2 s \\, \\lambda(s)} = \\frac{v_0}{2s\\sqrt{s(s+2\\gamma)}} \\;.\n\\end{equation}\nThis Laplace transform can be explicitly inverted, yielding the result in Eq. (\\ref{ac-mut}). \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Flux distribution in the quenched case}\\label{quen-sec}\n\n\nAs stated in Section \\ref{sec:model}, the quenched flux distribution is defined as\n\\begin{equation}\\label{quenched-definition}\n\\sum_{Q=0}^{\\infty} P_{\\rm qu} (Q,t) e^{-pQ} = \\exp \\left[\\overline{{\\ln}\\left[\\langle e^{-pQ}\\rangle_{\\{x_i\\}} \\right]} \\right] ,\n\\end{equation}\nwhere $\\overline{\\cdots}$ once again represents an average over the initial positions $\\{x_i\\}$. Our starting point is again Eq. (\\ref{propagator}). Taking the logarithm on both sides of (\\ref{propagator}) gives\n\\begin{equation}\\label{Nlog-quenched}\n{\\ln}\\left[\\langle e^{-pQ}\\rangle_{\\{x_i\\}} \\right] = \\sum_{i=1}^N {\\ln}\\left[1-(1-e^{-p})U(-x_i,t) \\right].\n\\end{equation}\nWe now perform the average over the initial positions, as in the annealed case, i.e., choosing each $x_i$ independently and uniformly from the box $[-L,0]$ and finally taking the limit $N \\to \\infty$, $L \\to \\infty$ keeping $\\rho = N\/L$ fixed. This gives\n\\begin{equation}\\label{Nlog-qu-avg}\n\\overline{{\\rm log}\\left[\\langle e^{-pQ}\\rangle_{\\{x_i\\}} \\right]}= \\frac{N}{L}\\int_{-L}^0 dx_i ~{\\ln}\\left[1-(1-e^{-p})U(-x_i,t) \\right] \\longrightarrow \\rho \\int_0^\\infty dz\\, \\ln \\left[ 1 - (1-e^{-p}) U(z,t)\\right] \\;.\n\\end{equation} \nTherefore the Laplace transform of the quenched flux distribution is given by\n\\begin{equation}\\label{Pqu2}\n\\sum_{Q=0}^{\\infty} P_{\\rm qu} (Q,t) e^{-pQ} = \\exp\\left[I(p,t)\\right] \\;,\n\\end{equation}\nwhere \n\\begin{eqnarray}\\label{def_Ip}\nI(p,t) = \\rho \\int_0^\\infty dz\\, \\ln \\left[ 1 - (1-e^{-p}) U(z,t)\\right] \\;.\n\\end{eqnarray}\nBefore extracting the full distribution $P_{\\rm qu} (Q,t)$ from this Laplace transform, it is useful to study first the asymptotic behaviors of $I(p,t)$ \nin the two limits : (i) $p \\rightarrow 0$ and (ii) $p \\rightarrow \\infty$. \n\\begin{itemize}\n\\item[$\\bullet$] $p\\to 0$ limit: Expanding $e^{-p}$ in powers of $p$ in Eq. (\\ref{def_Ip}), we get\n\\begin{equation}\\label{Ip0}\nI(p,t) = - p ~\\rho \\int_0^{\\infty} dz~U(z,t) + \\frac{p^2}{2} ~\\rho \\int_0^{\\infty} dz~ U(z,t)\\left[1-U(z,t) \\right] + {\\cal O}(p^3) \\;.\n\\end{equation} \nSubstituting this in Eq. (\\ref{Pqu2}) and expanding both sides in powers of $p$ we immediately get the mean and the variance of the flux $Q_t$ for the quenched case\nas stated in Eqs. (\\ref{mean_qu}) and (\\ref{var_qu}) respectively. \n \n\\item[$\\bullet$] $p\\to \\infty$ limit: In this case we expand $I(p,t)$ in Eq. (\\ref{def_Ip}) in powers of $e^{-p}$. The two leading terms are given by\n\\begin{equation}\\label{Ip-pinf1}\nI(p,t) = A(t) + B(t) e^{-p} + {\\cal O}(e^{-2p}) \\;,\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{atbt-pinf1}\nA(t) &=& \\rho \\int_0^{\\infty} {\\ln} [1-U(z,t)] dz \\\\ \nB(t) &=& \\rho \\int_0^{\\infty} \\frac{U(z,t)}{1-U(z,t)} dz \\;.\n\\end{eqnarray}\nSubstituting this expansion (\\ref{Ip-pinf1}) on the rhs of Eq. (\\ref{Pqu2}) and matching the powers of $e^{-p}$ on both sides of Eq. (\\ref{Pqu2}) immediately gives\n\\begin{eqnarray}\nP_{\\rm qu}(Q=0,t) &=& e^{A(t)} = \\exp{\\left[\\rho\\int_0^\\infty \\ln(1-U(z,t))dz\\right]} \\label{Pqu_eq_0} \\\\\nP_{\\rm qu}(Q=1,t) &=& B(t) \\, e^{A(t)} \\;. \\label{Pqu_eq_1}\n\\end{eqnarray}\nThe first line yields the general result mentioned in Eq. (\\ref{P_neq0_qu}). \n\n \n\\end{itemize}\n\n\n\n\nThese results so far are quite general, i.e., they hold for any dynamics -- the dependence on the dynamics comes\nonly through the function $U(z,t)$. In the following, we focus on two interesting dynamics, namely the diffusive and \nthe RTP and extract the large time behavior of $P_{\\rm qu}(Q,t)$ using Eqs. (\\ref{Pqu2}) and (\\ref{def_Ip}). \n\n\n\n\n\n\\subsection{$P_{\\rm qu}(Q,t)$ for simple diffusion}\\label{quen-sec-diff}\n\nIn this case, using the explicit expression $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4Dt})$, we get from Eq. (\\ref{def_Ip}) \n\\begin{equation}\\label{I-phi-correspondence}\nI(p,t) = \\rho \\sqrt{ 4 D t} \\int_0^{\\infty} dz~ {\\ln}\\left[1-\\frac{1}{2}(1-e^{-p}){{\\rm erfc}(z)}\\right]\n= -\\rho \\sqrt{D t}~ \\phi (p),\n\\end{equation}\nwhere \n\\begin{equation}\\label{phi-1eq}\n\\phi (p) = -2 \\int_0^{\\infty} dz~ {\\ln}\\left[1-\\frac{1}{2}(1-e^{-p}){{\\rm erfc}(z)}\\right ] \\;.\n\\end{equation}\nTherefore Eq. (\\ref{Pqu2}) reads for all time $t$\n\\begin{eqnarray} \\label{def_phi}\n\\sum_{Q=0}^{\\infty} e^{-pQ} \\,P_{\\rm qu} (Q,t) = \\exp\\left[-\\rho \\sqrt{D t}~ \\phi (p)\\right] \\;.\n\\end{eqnarray}\nIn the long time limit $t \\rightarrow \\infty$, we anticipate, and verify a posteriori, that \n$P_{\\rm qu}(Q,t)$ takes a large deviation form in the limit where $Q \\to \\infty$, $t \\to \\infty$ but\nwith the dimensionless ratio $q = Q\/(\\rho \\sqrt{D\\,t})$ fixed \n\\begin{equation}\\label{Pq}\nP_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho \\sqrt{D t}~ \\Psi_{\\rm diff} \\left(\\frac{Q}{\\rho \\sqrt{D t}}\\right)\\right], \n\\end{equation}\nwhere $\\Psi_{\\rm diff}(q)$ is a rate function that we wish to compute. Substituting this large deviation form (\\ref{Pq}) on the left hand side (lhs) of Eq. (\\ref{def_phi}) and replacing the discrete sum over $Q$ by an integral (which is valid for large $Q \\sim \\sqrt{t}$), we get\n\\begin{equation}\\label{Pq1}\n\\int_0^{\\infty} e^{-pQ} P_{\\rm qu}(Q,t) dQ \\sim \\rho \\sqrt{D t} \\int_0^{\\infty} e^{-\\rho \\sqrt{D t}\\, \\left[p \\, q+\\Psi_{\\rm diff}(q)\\right]} dq \\;.\n\\end{equation}\nFor large $t$, we can now evaluate the integral over $q$ in Eq. (\\ref{Pq1}) by a saddle point method, which gives\n\\begin{eqnarray}\\label{sp_diff}\n\\int_0^{\\infty} e^{-pQ} P_{\\rm qu}(Q,t) dQ \\sim \\exp{\\left[-\\rho \\sqrt{Dt}\\, \\underset{q}{\\min}[p\\,q+\\Psi_{\\rm diff}(q)] \\right]} \\;.\n\\end{eqnarray}\nComparing this with the rhs of Eq. (\\ref{def_phi}) we get\n\\begin{equation}\\label{legendre}\n\\underset{q}{\\min}\\left[p\\,q+\\Psi_{\\rm diff}(q)\\right]= \\phi(p) \\;.\n\\end{equation}\nInverting this Legendre transform one gets\n\\begin{equation}\\label{fq-bas}\n\\Psi_{\\rm diff}(q)=\\underset{p}{\\max}\\left[\\phi(p)-p\\,q\\right] \\;,\n\\end{equation}\nwhere $\\phi(p)$ is given in Eq. (\\ref{phi-1eq}). \n\nKnowing $\\phi(p)$ explicitly, one can plot the large deviation function $\\Psi_{\\rm diff}(q)$ using Eq. (\\ref{fq-bas}) -- see Fig. \\ref{diff-math-asymp}. Clearly,\n$\\Psi_{\\rm diff}(q)$ has a concave shape with a minimum at $q=q_{\\min}$, the value of $q_{\\min}$ will be computed shortly. The asymptotic behaviors of the\nrate function $\\Psi_{\\rm diff}(q)$ can also be extracted in the limits $q \\to q_{\\min}$, $q\\to 0$ and $q \\to \\infty$ by analysing $\\phi(p)$ respectively in the limits $p\\to 0$, $p\\to +\\infty$ and $p \\to -\\infty $ (where $\\phi(p)$ in Eq. (\\ref{phi-1eq}) has to be continued analytically to negative $p$). The results are summarised in Eqs. (\\ref{psi_diff_asympt}), (\\ref{C}) and (\\ref{A}) in section \\ref{sec:model}. Below we provide the derivation of these results. \n\n\n\n\n\n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{mathm-analytics-qtypandzero-diff.eps}\n\\end{minipage}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{mathm-analytics-qlarge-diff.eps}\n\\end{minipage}\n\\end{center}\n\\caption{Large deviation function $\\Psi_{\\rm diff}(q)$ vs $q$ for the diffusive case with quenched initial conditions. On both panels, the dashed black lines correspond to \nevaluating via Mathematica $\\Psi_{\\rm diff}(q)$ from Eq. (\\ref{fq-bas}) with $\\phi(p)$ given in Eq. (\\ref{phi-1eq}). On the left panel (a), the solid yellow curve corresponds to the small $q$ asymptotic behavior of $\\Psi_{\\rm diff}(q)$ in Eq.~(\\ref{Fq-qzero-soln}) and the dashed-dotted green curve corresponds to the quadratic behavior in Eq.~(\\ref{fq_s0}). On the right panel (b), we zoom in on the large $q$ tail. The solid violet curve corresponds to the leading asymptotic behavior $\\Psi_{\\rm diff}(q) \\approx q^3\/12$, while the dashed-dotted green curve is the quadratic behavior as in Eq.~(\\ref{fq_s0}). The violet and the green curves clearly demonstrate the non-Gaussian tail of $\\Psi_{\\rm diff}(q)$.}\\label{diff-math-asymp}\n\\end{figure}\n\n\n\n\\subsubsection{Typical Fluctuations : $Q \\sim \\langle Q \\rangle_{\\rm qu}$}\\label{qmi}\n In order to derive the result for $\\Psi_{\\rm diff}(q \\rightarrow q_{\\min})$, we need to analyze $\\phi(p)$ for $p \\rightarrow 0$. We expand $\\phi(p)$ in Eq. (\\ref{phi-1eq}) up to order $p^2$ and get\n\\begin{eqnarray}\\label{phi_s0}\n\\phi(p) = \\alpha p - \\beta p^2 + {\\cal O}(p^3) \\;, \n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n&&\\alpha=\\int_0^{\\infty} {\\rm erfc}(z) dz ~ = \\sqrt{\\frac{1}{\\pi}} \\label{alpha-defn} \\\\\n&&\\beta=\\frac{1}{4} \\int_0^{\\infty} (2\\, {\\rm erfc}(z) - {\\rm erfc}^2(z))\\, dz ~ = \\frac{1}{\\sqrt{8\\pi}}\\;.\n\\end{eqnarray}\nSubstituting $\\phi(p) = \\alpha \\, p - \\beta\\,p^2$ in Eq. (\\ref{fq-bas}) and maximising with respect to $p$ gives a quadratic form for the rate function \n \\begin{equation} \\label{fq_s0}\n \\Psi_{\\rm diff }(q) \\sim \\frac{(q-\\alpha)^2}{4\\beta}= \\sqrt{\\frac{\\pi}{2}} \\left(q-\\sqrt{\\frac{1}{\\pi}}\\right)^2 \\;.\n \\end{equation}\nThis form holds for $q$ close to $q_{\\min} = \\alpha = 1\/\\sqrt{\\pi}$ and gives the result in the second line in Eq. (\\ref{psi_diff_asympt}). Substituting this quadratic behavior in the large deviation form in Eq. (\\ref{Pq}) predicts\na Gaussian form for the quenched flux distribution for $q$ close to $q_{\\min}$\n\\begin{eqnarray}\\label{Gaussian_qu}\nP_{\\rm qu}(Q,t) \\sim \\exp{\\left[- \\frac{\\left(Q-\\langle Q\\rangle_{\\rm qu}\\right)^2}{2 \\sigma_{\\rm qu}^2}\\right]}\n\\end{eqnarray}\nwhere the mean the variance are given by \n \\begin{eqnarray}\n\\langle Q \\rangle_{\\rm qu} &=&\\rho \\sqrt{\\frac{D t}{\\pi}} \\label{diffusion-mean-quenched} \\\\\n\\sigma^2_{\\rm qu} &=& \\rho \\sqrt{\\frac{D t}{2\\pi}} \\label{diffusion-variance-quenched} \\;.\n\\end{eqnarray} \nNotice that these expressions for the mean and the variance, though derived here for large $t$, actually hold for all $t$, as one can verify directly from the formulae in Eqs. (\\ref{mean_qu}) and (\\ref{var_qu}) with $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4 Dt})$. Comparing with the annealed case, while the means in both cases are identical, both given by $\\mu(t) = \\rho \\sqrt{Dt\/\\pi}$, their variances and higher moments differ. For example, the variance in the annealed case is $\\mu(t)=\\rho \\sqrt{Dt\/\\pi}$ which differs by a factor $1\/\\sqrt{2}$ from the quenched case in Eq. (\\ref{diffusion-variance-quenched}). These results agree with those obtained in~\\cite{derrida-gers}. This typical quadratic behavior is shown by the dashed-dotted green curve in Fig. \\ref{diff-math-asymp}. \n\n\n \n\\subsubsection{Atypical fluctuations on the left of the mean: $Q \\ll \\langle Q \\rangle_{\\rm qu}$} \\label{qze}\n\nIn order to infer about the fluctuations of $P_{\\rm qu}(Q,t)$ around $Q \\rightarrow 0$, we need to evaluate how $\\Psi_{\\rm diff}(q)$ behaves when $q \\rightarrow 0$. This corresponds to the limit $p \\rightarrow \\infty$ for $\\phi(p)$ from Eq. (\\ref{fq-bas}). We use the large $p$ expansion in Eq. \\ref{Ip-pinf1}, and evaluate $A(t)$ and $B(t)$ from \nEq. (\\ref{atbt-pinf1}) using $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4Dt})$. This gives\n\\begin{eqnarray}\\label{At-diff}\nA(t) &=& -\\overline{\\alpha} \\, \\rho\\, \\sqrt{4Dt} \\;, \\quad {\\rm where} \\quad \\overline{\\alpha}= -2 \\int_0^{\\infty} {\\ln} \\left[1-\\frac{1}{2}{{\\rm erfc}(z)}\\right] dz = 0.675336\\ldots \\label{a-defn} \\\\ \nB(t) &=& \\overline{\\beta} \\, \\rho\\, \\sqrt{4Dt} \\;, \\quad {\\rm where} \\quad \\overline{\\beta} = \\int_0^{\\infty} \\frac{{\\rm erfc}(z)}{1-\\frac{1}{2}{\\rm erfc}(z)} dz = 0.828582 \\ldots \\;. \\label{b-defn}\n\\end{eqnarray}\nFrom Eqs. (\\ref{I-phi-correspondence}) and (\\ref{phi-1eq}) we get the two leading terms of $\\phi(p)$ for large $p>0$\n\\begin{equation}\\label{phip-zero-final}\n\\phi(p)= -\\frac{I(p,t)}{\\rho \\sqrt{D t}} \\approx \\overline{\\alpha}-\\overline{\\beta}~e^{-p} \\;.\n\\end{equation}\nPlugging this result for $\\phi(p)$ in Eq. (\\ref{fq-bas}) and maximizing with respect to $p$, we get the leading small $q$ behavior of $\\Psi_{\\rm diff}(q)$\n\\begin{equation}\\label{Fq-qzero-soln}\n\\Psi_{\\rm diff}(q)\\approx \\overline{\\alpha} - q +q ~ {\\ln}\\left(\\frac{q}{\\overline{\\beta}}\\right) \\;.\n\\end{equation}\nThis reproduces the first line of Eq. (\\ref{psi_diff_asympt}). In particular, for $q=0$, using $\\Psi_{\\rm diff}(q=0) = \\overline{\\alpha}$ in Eq. (\\ref{Pq}), we obtain $P_{\\rm qu}(Q=0,t) \\sim \\exp{(-\\overline{\\alpha} \\, \\rho \\sqrt{Dt})}$ as announced in Eq. (\\ref{P_Q0}). The small $q$ behavior of $\\Psi_{\\rm diff}(q)$ is shown by the solid yellow curve in Fig. \\ref{diff-math-asymp}(a). \n\n\n\n\\subsubsection{Atypical fluctuations on the right of the mean: $Q \\gg \\langle Q\\rangle_{\\rm qu}$}\\label{qla}\n\nIn order to derive the large $q$ asymptotics of $\\Psi_{\\rm diff}(q)$ from Eq. (\\ref{fq-bas}), we first need to continue $\\phi(p)$ in Eq. (\\ref{phi-1eq}) analytically to negative $p$\nand use its asymptotics in the limit $p \\to -\\infty$. For this, it is convenient to write first $p = -u$ where $u = |p|$. We write\n\\begin{equation}\\label{tildephi-diff}\n\\phi(p=-u) = \\tilde{\\phi}(u) = -2 \\int_0^{\\infty} dz ~{\\ln} \\left[1+ \\frac{\\left(e^u-1 \\right)}{2}{{\\rm erfc}(z)} \\right] \\quad \\underset{u \\to \\infty}{\\approx} \\quad -2 \\int_0^{\\infty} dz ~{\\ln} \\left[1+ \\frac{e^u}{2} {\\rm erfc}(z) \\right] \\;.\n\\end{equation}\nTo extract the large $u$ behavior of $\\tilde \\phi(u)$ from the integral on the rhs, it is convenient to take the derivative with respect to $u$\n\\begin{equation}\\label{qu-larphi}\n \\tilde{\\phi}'(u) \\approx -2\\int_0^{\\infty} dz \\, \\frac{~ \\frac{e^{u}}{2}\\,{\\rm erfc}(z)}{1 + \\frac{e^u}{2}\\,{\\rm erfc}(z)} \\;.\n\\end{equation} \nFor large $u$ the dominant contribution to this integral comes from large $z$ where ${\\rm erfc}(z) \\approx e^{-z^2}\/(z \\sqrt{\\pi})$. Hence we see that, for $z > \\sqrt{u}$ the integrand is essentially $0$ as $u \\to \\infty$, while, for $z < \\sqrt{u}$, the integrand is $1$ as $u \\to \\infty$. Hence, the integrand can be approximated by a Fermi function \n\\begin{eqnarray} \\label{phiprime2}\n\\tilde{\\phi}'(u) \\approx -2 \\int_0^{\\sqrt{u}} dz = - 2 \\sqrt{u} \\;.\n\\end{eqnarray}\nIntegrating it back, we get the leading order behavior for $\\tilde \\phi(u)$ for large $u$\n\\begin{equation}\\label{phi-large-diff}\n \\tilde{\\phi}(u) \\approx -\\frac{4}{3} u^{\\frac{3}{2}} \\;.\n\\end{equation}\nTherefore $\\phi(p) \\approx -(4\/3) (-p)^{3\/2}$ as $p \\to -\\infty$. Substituting this behavior in Eq. (\\ref{fq-bas}) and maximizing with respect to $p$ one gets $\\Psi_{\\rm diff}(q) \\approx q^3\/12$ as $q \\to \\infty$. This then gives the last line of the result in Eq. (\\ref{psi_diff_asympt}). As mentioned earlier, this leading large $q$ asymptotic behavior of $\\Psi_{\\rm diff}(q)$ coincides with the result of Ref. \\cite{derrida-gers} obtained by a different method. The large $q$ behavior of $\\Psi_{\\rm diff}(q)$ is shown solid the solid violet curve in Fig. \\ref{diff-math-asymp}(b). \n\n\n\n\n\n\\subsection{$P_{\\rm qu}(Q,t)$ for run-and-tumble particles}\\label{quen-act-sec}\n\nIn this case, our starting point again are Eqs. (\\ref{Pqu2}) and (\\ref{def_Ip}), except that the function $U(z,t)$ for the RTP is more complicated.\nIts Laplace transform is given in Eq. (\\ref{LaplaceU_2}). As shown in Appendix C of \\cite{pierre-satya-greg} it can be formally inverted to obtain $U(z,t)$ in real time\n\\begin{equation}\\label{Uzt-exact}\nU(z,t) = \\frac{1}{2}\\left[ e^{\\frac{-\\gamma z}{v_0}} + \\frac{\\gamma z}{v_0} \\int_1^{\\frac{v_0 t}{z}} dT \\frac{ e^{\\frac{-\\gamma z T}{v_0}}I_1(\\frac{\\gamma z}{v_0}\\sqrt{T^2-1})}{\\sqrt{T^2-1}}\\right] \\Theta(v_0 t-z) \\;.\n\\end{equation}\nHowever, it turns out that this expression is not very useful to extract the large deviation function at late times. \n\nBefore proceeding to compute the large deviation function at late times, it is useful to discuss the large $t$ behavior of $P_{\\rm qu}(Q,t)$ in different\nregimes of $Q$. In the following, we will first discuss the $Q \\to 0$ limit of $P_{\\rm qu}(Q,t)$, followed by the discussion of the typical\nfluctuations where $Q = {\\cal O}(\\sqrt{t})$. In this regime, we will recover the Gaussian fluctuations. When $Q\/(\\rho\\, \\sqrt{D\\,t}) \\gg 1$, we\nexpect to recover the large deviation regime for the diffusive behaviour discussed in the previous section. This is because, as explained in the \nintroduction, at late times, the RTP motion essentially reduces to that of a diffusive particle with an effective diffusion constant $D_{\\rm eff} = v_0^2\/(2 \\gamma)$.\nHowever, there exists yet another ``larger deviations regime'' where $Q \\sim {\\cal O}(t)$ where we will show that $P_{\\rm qu}(Q,t)$ carries the signature\nof activity and has a novel large deviation form\n\\begin{equation}\\label{ac-sig2}\nP_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho \\, v_0 \\, \\gamma \\, t^2 \\, \\Psi_{\\rm RTP}\\left(\\frac{Q}{\\rho \\, v_0 \\,t}\\right)\\right] \\;.\n\\end{equation}\nIn the following, we will indeed compute this rate function $\\Psi_{\\rm RTP}(q)$ and show that it is given by Eq. (\\ref{rtp-ldf-model}). \n \n\n\n\\subsubsection{Typical fluctuations: $Q \\sim \\langle Q\\rangle_{\\rm qu}$}\n\nIn order to extract the typical fluctuations of $Q_t$ around its mean value for the RTP case, we need to use the small $p$ expansion\nof $I(p,t)$ in Eq. (\\ref{Pqu2}). Quite generally, the small $p$ expansion of $I(p,t)$ is given in Eq. (\\ref{Ip0}). We use this expansion\non the rhs of Eq. (\\ref{Pqu2}) and approximate the sum on lhs by an integral. The resulting Laplace transform can be easily inverted\nand yields a Gaussian form \n\\begin{eqnarray}\\label{P_qu_Gauss}\nP_{\\rm qu}(Q,t) \\approx \\exp{\\left[-\\frac{(Q-\\langle Q \\rangle_{\\rm qu})^2}{2 \\, \\sigma_{\\rm qu}^2}\\right]} \\;,\n\\end{eqnarray} \nwhere $\\langle Q \\rangle_{\\rm qu}$ and $\\sigma^2_{\\rm qu}$ are given in Eqs. (\\ref{mean_qu}) and (\\ref{var_qu}) respectively where $U(z,t)$ is given in Eq. (\\ref{Uzt-exact}) --\nalternatively its Laplace transform is given by the simpler form in Eq. (\\ref{LaplaceU_2}). The mean value $\\langle Q \\rangle_{\\rm qu}$ can be computed explicitly. Indeed\n\\begin{eqnarray}\\label{av_qu_RTP}\n\\langle Q \\rangle_{\\rm qu} = \\rho \\int_0^\\infty U(z,t)\\, dz = \\mu(t)= \\frac{\\rho\\,v_0}{2} t\\,e^{-\\gamma t}\\left[ I_0(\\gamma\\,t) + I_1(\\gamma \\, t)\\right] \\;,\n\\end{eqnarray}\nwhere the last equality follows from Eq. (\\ref{ac-mut}). The variance $\\sigma_{\\rm qu}^2 = \\rho \\int_0^{\\infty} U(z,t)\\left[1-U(z,t)\\right] \\, dz$ is however difficult to compute explicitly using $U(z,t)$ from Eq. (\\ref{Uzt-exact}). However, it can be easily evaluated numerically. At large times, $\\langle Q \\rangle_{\\rm qu}$ and $\\sigma^2_{\\rm qu}$ converge to the diffusive limits given in Eqs. (\\ref{diffusion-mean-quenched}) and (\\ref{diffusion-variance-quenched}) respectively. \n\n\n\n\n\n\n\n\n\\subsubsection{Atypical fluctuations on the left of the mean: $Q \\ll \\langle Q \\rangle_{\\rm qu}$}\n\nExactly at $Q=0$ or $Q=1$, we have an exact\nexpression at all times $t$ for $P_{\\rm qu}(Q,t)$ in terms of $U(z,t)$, as given in Eqs. (\\ref{Pqu_eq_0})\nand (\\ref{Pqu_eq_1}). The function $U(z,t)$ for RTP appearing in these expressions is given in Eq. (\\ref{Uzt-exact}).\nGiven this rather complicated expression of $U(z,t)$, it is hard to obtain explicit formulae valid at all times for\n$P_{\\rm qu}(Q,t)$ even for $Q=0$ or $Q=1$. However, at late times, since $U(z,t)$ converges at late times\nto that of the diffusive limit in Eq. (\\ref{UzT_BM}) with an effective diffusion constant $D_{\\rm eff} = v_0^2\/(2 \\gamma)$, \nwe recover the diffusive results for this extreme left tail of $P_{\\rm qu}(Q,t)$.\nFor instance $P_{\\rm qu}(Q=0,t)$, which represents the probability of having no particle on the right side of the origin\nat time $t$, decays at late times as in the diffusive case \n\\begin{eqnarray} \\label{P_Q0_RTP}\nP_{\\rm qu}(Q=0,t)\\Big \\vert_{\\rm RTP} \\approx \\exp \\left[ - \\overline{\\alpha} \\, \\rho \\sqrt{D_{\\rm eff}\\,t}\\right] \\;,\n\\end{eqnarray} \nwhere $\\overline{\\alpha} = 0.675336\\ldots$ is given in Eq. (\\ref{C}). \n\n\n\\subsubsection{Atypical fluctuations on the right of the mean: $Q \\sim {\\cal O}(t) \\gg \\langle Q \\rangle_{\\rm qu}$}\n\nIn this section, we derive the result in Eq. (\\ref{ac-sig2}). We recall that in the diffusive case, the atypical \nfluctuations of $Q$ are encoded in the large deviation form in Eq. (\\ref{Pq}) with $Q \\sim \\rho \\sqrt{Dt}$. \nThe extreme fluctuations to the right of $\\langle Q \\rangle_{\\rm qu}$ in this case are described by the large\nargument behavior of the large deviation function ${\\Psi}_{\\rm diff}(q = Q\/(\\rho\\sqrt{D\\,t}))$, i.e., when $Q \\gg \\rho \\sqrt{Dt}$. \nThus, in the diffusive case, there is a single scale for the fluctuations of $Q$ at late times, namely $Q \\sim \\sqrt{t}$. \nIn contrast, for the RTP, in addition to the scale $\\sqrt{t}$ that describes the moderate large deviations around\nthe mean, there is yet another scale where $Q \\sim t$. This comes from the fact that each particle in time $t$ can move a maximum distance \n$v_0 t$, where $v_0$ is the velocity. So for an initial density $\\rho$, the maximum possible flux through the origin is $Q_{\\rm max} = \\rho v_0 t$. \nHence $Q \\sim t$ describes the scale of fluctuations at the very right tail of the distribution $P_{\\rm qu}(Q,t)$. \n\n\nTo extract this extreme right tail, we again start from Eqs. (\\ref{Pqu2}) and (\\ref{def_Ip}) with $U(z,t)$, for RTP, given by its Laplace transform \nin Eq. (\\ref{LaplaceU_2}). Before extracting the large deviation form of $P_{\\rm qu}(Q,t)$, we first analyse $U(z,t)$ in the limit $z \\sim t$. Inverting the Laplace transform of $U(z,t)$ in Eq. (\\ref{LaplaceU_2}) we get\n\\begin{eqnarray}\\label{U_Bromwich1}\nU(z,t) = \\int_\\Gamma \\frac{ds}{2\\pi i} \\exp{\\left[t\\left( s - \\sqrt{s(s+2\\gamma)} \\frac{z}{v_0\\,t}\\right)\\right]} \\;,\n\\end{eqnarray} \nwhere $\\Gamma$ represents the Bromwich contour in the complex $s$-plane. In the limit $t \\to \\infty$, $z \\to \\infty$ with the ratio $z\/t$ fixed,\nthe integral can be evaluated by the saddle-point method, which yields (up to pre-exponential factors)\n\\begin{equation}\\label{Uzt_saddle}\nU(z,t) \\approx \\exp \\left[-\\gamma t \\left(1-\\sqrt{1-\\frac{z^2}{v_0^2 t^2}}\\right)\\right] \\, \\Theta(v_0 \\,t-z ) ~.\n\\end{equation}\nWe have checked numerically that this approximation (\\ref{Uzt_saddle}) works very well, at large times, as one would expect. \n\nTo extract the large $Q \\sim t \\gg \\langle Q\\rangle_{\\rm qu}$ behavior from Eq. (\\ref{Pqu2}) we need to analytically continue\n$I(p,t)$ to $p$ negative and study the limit $p \\to -\\infty$, as in the diffusive case. Setting $p=-u$ with $u>0$, and approximating the discrete sum\non the lhs of Eq. (\\ref{Pqu2}) by an integral, we get\n\\begin{eqnarray}\\label{laplace1}\n\\int_0^\\infty P_{\\rm qu}(Q,t)\\, e^{u Q} \\,dQ \\approx e^{\\tilde I(u,t)} \\;,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{def_tildeI}\n\\tilde I(u,t) = \\rho \\int_0^\\infty dz\\, \\ln \\left( 1+ (e^u-1)\\, U(z,t)\\right) \\underset{u \\to \\infty}{\\approx} \\rho \\int_0^\\infty dz\\, \\ln \\left( 1+ e^u\\, U(z,t)\\right) \\;.\n\\end{eqnarray}\nTo extract the large $u$ behavior of $\\tilde I(u,t)$ we follow the same procedure as in the diffusive case and take a derivative with respect to $u$. \nWe get\n\\begin{eqnarray}\\label{derivative_1}\n\\frac{d\\tilde I(u,t)}{du} \\approx \\rho \\int_0^\\infty \\frac{dz}{1+e^{-u} \\frac{1}{U(z,t)}} \\;.\n\\end{eqnarray}\nFor large $u$, this integral is dominated by the region where $z \\sim t$ where we can use the approximate form of $U(z,t)$ given in Eq. (\\ref{Uzt_saddle}). Substituting this form for $U(z,t)$ in Eq. (\\ref{derivative_1}), we get\n\\begin{eqnarray}\\label{derivative_2}\n\\frac{d\\tilde I(u,t)}{du} \\approx \\rho \\int_0^{v_0\\,t} \\frac{dz}{1+\\exp{\\left[-\\left(u-\\gamma\\,t + \\gamma\\,t \\sqrt{1-\\frac{z^2}{v_0^2\\,t^2}}\\right)\\right]}} \\;.\n\\end{eqnarray}\nWe now analyse this integral in two different cases, assuming $u \\sim t \\gg 1$:\n\n\\vspace*{0.3cm}\n\\noindent $\\bullet$ If $u > \\gamma t$: in this case as $t \\to \\infty$ it is clear that the integrand in Eq. (\\ref{derivative_2}) is always $1$ for any $z$. Hence \n\\begin{eqnarray}\\label{derivative_3}\n\\frac{d\\tilde I(u,t)}{du} \\approx \\rho v_0 t \\quad \\quad {\\rm if} \\quad u > \\gamma t \\;.\n\\end{eqnarray}\n\n \\begin{figure}\n\\begin{center}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{wx-plot.eps}\n\\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{wpr-plot.eps}\n\\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{active-large-deviation-full.eps}\n\\end{minipage}\n\\end{center}\n\\caption{ (a) Plot of $W(x)$ vs $x$ with $W(x)$ given by Eq.~(\\ref{phi-full}). The non-analytical point at $x=1$ is shown by a black dot\nwhere $W(x)$ and its first two derivatives are continuous, while the third derivative is discontinuous, as in Eq. (\\ref{third-der-phi}). \n(b) Plot of $W'(x)$ vs $x$ where $W'(x)$ can be read off from Eq. (\\ref{def_W}). Note that $W'(x) <1$ for $x<1$ and saturates at $1$ at $x=1$. (c) Large deviation function $\\psi_{\\rm {RTP}}(q)$ vs $q$ for the run-and-tumble case. Black solid line represents the analytical expression given in Eq.~(\\ref{rtp-ldf-model}) valid at very large time. Dashed lines ($t=100$ (red-dashed), $t=400$ (violet-dotted) and $t=600$ (blue dashed-dotted)) correspond to the finite time evaluation of $\\underset{x}{\\max}[\\frac{qu}{\\gamma t}-\\frac{\\tilde{I}(u,t)}{\\rho v_0 \\gamma t^2}]$ using Eq. (\\ref{def_tildeI}) -- \ntogether with Eq.~(\\ref{def_tildeI}) with $\\rho=v_0=1, \\gamma=0.5$ -- with Mathematica.\n}\\label{phi-full-plot}\n\\end{figure}\n \n \n\\vspace*{0.3cm}\n\\noindent $\\bullet$ If $u < \\gamma t$: this case is a bit more complicated to analyze. Since $u<\\gamma t$ the argument of the exponential in Eq. (\\ref{derivative_2}) can be either positive or negative. Accordingly, the integrand will either $0$ or $1$ for large $u \\sim t \\gg 1$. The value\nof $z$ for which the argument of the exponential changes sign is given by\n\\begin{eqnarray}\\label{zstar}\nz^*(u) = \\frac{v_0}{\\gamma} \\sqrt{u(2\\gamma\\,t - u)} \\;.\n\\end{eqnarray}\nThus, if $z>z^*(u)$ the integrand is $0$ while for $z1 \\;.\n\\end{cases}\n\\end{eqnarray}\nPerforming these integrals explicitly, we get\n\\begin{eqnarray}\\label{phi-full}\nW(x) =\n\\begin{cases}\n \\frac{(x-1)}{2}\\sqrt{x(2-x)}+{\\rm sin}^{-1}(\\sqrt{x\/2})\\quad,\\quad x<1\\\\ \n\\\\\n\\frac{\\pi}{4}+x-1\\quad,\\quad x>1 \\;.\n\\end{cases}\n\\end{eqnarray}\nThe function $W(x)$ is plotted vs $x$ in Fig.~\\ref{phi-full-plot} (a). Interestingly, while $W(x)$ and its first two derivatives are continuous \nat $x=1$, its third derivative is discontinuous. Indeed one has\n\\begin{eqnarray}\\label{third-der-phi}\nW^{'''}(x \\rightarrow 1^-) &=& -1 \\nonumber \\\\\nW^{'''}(x \\rightarrow 1^+) &=& 0 \\;.\n\\end{eqnarray}\nUsing this result for $W(x)$ in Eq. (\\ref{phi-full}) gives us an expression for $\\tilde I(u,t)$ in (\\ref{full-intI_1}). We then substitute this expression for $\\tilde I(u,t)$ in Eq. (\\ref{laplace1}) to get\n\\begin{eqnarray}\\label{PqW}\n\\int_0^\\infty P_{\\rm qu}(Q,t) e^{uQ} dQ \\sim \\exp{\\left[\\rho v_0 \\gamma t^2 W\\left(\\frac{u}{\\gamma t}\\right)\\right]} \\;.\n\\end{eqnarray}\nInverting formally this Laplace transform, we obtain\n\\begin{eqnarray} \\label{Inverse1}\nP_{\\rm qu}(Q,t) \\sim \\int_{\\Gamma} \\frac{du}{2 \\pi i} \\exp{\\left[- u Q + \\rho v_0 \\, \\gamma t^2 W\\left( \\frac{u}{\\gamma t}\\right) \\right]} \\;.\n\\end{eqnarray}\nRescaling $u\/(\\gamma t)= x$ we get, up to pre-exponential factors,\n\\begin{eqnarray} \\label{Inverse2}\nP_{\\rm qu}(Q,t) \\sim \\int_{\\Gamma} \\frac{dx}{2\\pi i} \\exp{\\left[ - \\rho v_0 \\gamma t^2 \\left(-W(x) + x q \\right) \\right]} \\;, \\quad {\\rm where} \\; q = \\frac{Q}{\\rho v_0 t} \\;.\n\\end{eqnarray}\nwhere $\\Gamma$ is the Bromwich contour in the complex $x$-plane. Performing this integral by using a saddle-point for large $t$, we get\n\\begin{eqnarray} \\label{Inverse3}\nP_{\\rm qu}(Q,t) \\sim \\exp{\\left[- \\rho v_0 \\gamma t^2 \\Psi_{\\rm RTP} \\left( q = \\frac{Q}{\\rho v_0 t}\\right) \\right]} \\;,\n\\end{eqnarray}\nwith the rate function given by\n\\begin{eqnarray}\\label{Psi_RTP}\n\\Psi_{\\rm RTP}(q) = \\underset{x}{\\max} \\left[q\\,x - W(x) \\right] \\;,\n\\end{eqnarray}\nwhere $W(x)$ is given explicitly in Eq. (\\ref{phi-full}). It is easy to verify that the maximum of the function $q\\,x-W(x)$ occurs at $x=x^* = 1-\\sqrt{1-q^2} < 1$. Since $x^*<1$ we use the branch of $W(x)$ in the first line of Eq. (\\ref{phi-full}). Substituting this value of $x^*$ in Eq. (\\ref{Psi_RTP}) we get the result in Eq. (\\ref{rtp-ldf-model}). The asymptotic behaviours of this function $\\Psi_{\\rm RTP}(q)$ are given in Eq. (\\ref{asympt_RTP}) and a plot of this function is shown in Fig. \\ref{phi-full-plot} (c). Note that for small $q$, i.e. $Q \\ll \\rho v_0 t$, $\\Psi_{\\rm RTP}(q)$ behaves as $\\Psi_{\\rm RTP}(q) \\sim q^3\/6$. Substituting this behavior in Eq. (\\ref{Inverse3}) gives\n\\begin{eqnarray}\\label{matching1}\nP_{\\rm qu}(Q,t) \\Big \\vert_{\\rm RTP} \\sim \\exp{\\left( - \\frac{\\gamma Q^3}{6 \\rho^2 v_0^2 t}\\right)} \\sim \\exp{\\left( - \\frac{Q^3}{12 \\rho^2 D_{\\rm eff}t}\\right)} \\;, \\quad {\\rm where} \\quad D_{\\rm eff} = \\frac{v_0^2}{2 \\gamma} \\;.\n\\end{eqnarray}\nOn the other hand, for the diffusive case, from Eq. (\\ref{P-F-diff-largeQ}), using $\\Psi_{\\rm diff}(q) \\approx q^3\/12$ for large $q$, i.e. $Q \\gg \\rho \\sqrt{D\\,t}$, one gets \n\\begin{eqnarray}\\label{matching2}\nP_{\\rm qu}(Q,t) \\Big \\vert_{\\rm diff} \\sim \\exp{\\left( - \\frac{Q^3}{12 \\rho^2 D_{}t}\\right)}\n\\end{eqnarray}\nComparing these two tails (\\ref{matching1}) and (\\ref{matching2}), one sees that for the RTP, on a scale where $\\rho \\, \\sqrt{D_{\\rm eff}t} \\ll Q \\ll \\rho \\, v_0 \\, t$ these two behaviors match perfectly, supporting the expectation that, at large $t$, even for moderately large fluctuations to the right of the mean, the flux distribution for the RTP and the diffusive case coincide, once one identifies the effective diffusion constant as $D_{\\rm eff} = v_0^2\/(2\\gamma)$. \n\n\n\n\n\n\\section{Numerical Results}\\label{numerics}\n\n\nThis section is dedicated to Monte Carlo simulations. We check numerically the analytical results previously obtained and characterize the properties of the physical realizations corresponding to large $Q$ values. \n\n\\subsubsection{The Importance Sampling strategy}\\label{IS-strategy}\n\n\n \\noindent In order to obtain the tails of $P_{an}(Q,t)$ and $P_{qu}(Q,t)$ we have employed Importance Sampling, a general method used to reduce the variance of observables whose expectation value is dominated by rare realizations, in this case rare trajectories. In the context of the evaluation of large deviation functions, very popular implementations of importance sampling ideas are cloning algorithms or transition path sampling \\cite{giardina2011simulating}.\n Here we use an implementation of Importance Sampling, similar to transition path sampling, the details of the technique can be found in \\cite{hartmann-epjb-2011, alberto-importance}.\nBasically we sample realizations with an exponential\nbias on their flux, $e^{-\\theta Q}$. The adjustable parameter $\\theta$ allows to explore atypical realizations:\n a negative $\\theta$ favours realizations with large $Q$, while a positive $\\theta$ favours small $Q$. The sampling is done using a standard Metropolis algorithm as discussed in \\cite{hartmann-epjb-2011, alberto-importance} and error bars are smaller than the symbol size.\n\n\nTo proceed we note that the flux depends only on the particle positions at time $t$ :\n\\begin{equation}\\label{dis-diff}\n x_i(t) = x_i(0) + \\Delta x_i(t), \\quad \\forall i \\;.\n\\end{equation}\nWe wrote this quantity as the sum of two contributions: (i) the initial (negative) position,\n $x_i(0) <0$ and (ii) the total displacement, $\\Delta x_i(t)$. The latter depends on the stochastic process we are considerning: for the diffusive particles it is a Gaussian number of zero mean and standard deviation $\\sqrt{2 D t}$.\nFor active particles, it can be expressed as \n\\begin{equation}\\label{rtp-pos-velo}\n \\Delta x_i(t) =\\pm v_0 (T_1-T_2),\n\\end{equation}\nwhere $T_1$, is the total time spent moving in the initial particle direction, $T_2=t-T_1$ the time spent in the opposite direction.\nThe signs $+$ or $-$ correspond to the initial direction and they are chosen with equal probability. The times $T_1$ and $T_2$ are determined as follows: the run times $\\tau_1, \\tau_2,\\ldots, \\tau_n$ are drawn from an exponential distribution of rate $\\gamma$, the last run being the first time interval for which $\\sum_{i=1}^{n} \\tau_i > t$. Then $\\tau_n$ is replaced by $t-\\sum_{i=1}^{n-1} \\tau_i$ and $T_1, T_2$ are computed.\n\nThe choice of the initial conditions is the delicate point that makes the annealed case different from the quenched one: in the annealed case, averages are performed over all initial conditions while in the quenched case the initial condition is fixed and typical. We first study the annealed case for active particles. At large times, their behavior is statistically equivalent to the one of diffusive particles with the effective constant $D_{\\text{eff}}=v_0^2\/(2 \\gamma)$. Then we discuss the quenched case and recover the exact results of previous sections. At variance with the annealed case one expects that when $Q \\simeq \\rho v_0 t $ active particles should have a clear non-diffusive nature even at long times. Unfortunately, the biased Monte Carlo used in this paper does not allow to sample configurations with $Q \\simeq \\rho v_0 t$.\n\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{snapshot-active-diffusion-annealed-initial-histo-new.eps}\n\\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{winners-displacement-annealed-new.eps}\n\\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{mcs-psiann-ac-new-com.eps}\n\\end{minipage}\n\\end{center}\n\\caption{Monte Carlo results for the annealed case. RTP versus Diffusion. The initial particle position is sampled with $\\rho=1$ and $L=2000$. Particles evolve up to time $t=400$ with $v_0=1, \\gamma=0.05$ for RTP (points), $D_{\\text{eff}}=10$ for diffusion (lines). We performed $10^7$ realizations.\n (a) Initial particle concentration. For typical value of $Q$ (violet circles\/black solid lines) the profile is flat, as expected. It has been obtained by setting $\\theta =0$ as importance sampling parameter which corresponds to $Q \\simeq \\langle Q \\rangle \\approx 35.45$. For large values of $Q$ (red squares\/yellow dashed lines) we observe an accumulation of particle close to origin. There we set $\\theta =-1$ as importance sampling parameter which corresponds to $ Q \\approx 93 \\gg \\langle Q \\rangle$.\n(b) Histogram (normalized to the number of realizations) of the displacement for particles with a positive final position with $\\theta=0$, and $\\theta=-1$. \n (c) $P_{an}(Q,t)$ vs $Q$; points represent Monte Carlo results (with $L=500$ (red squares) and $L=2000$ (green circles)) while the dashed line is the Poisson distribution with mean $\\mu=35.4573$ (Eq.~\\ref{ac-mut}).\nIn the tails the agreement improves with increasing $L$. \\label{annealed-ac-ld-causes}} \n\\end{figure}\n\n\n\n\\subsection{Annealed case} \n\n \n \\noindent The flux of RTPs depends on the evolution of the particles with an initial position $x(0)$ in $[-v_0 t, 0]$. But what is their number?\n Typically we expect $N \\sim \\rho v_0 t$, but, in the annealed case, rare realizations with large or small $N$ can occur. To capture them\nwe consider a large interval $[-L,0]$ with $L \\gg \\rho v_0 t$ and draw $L \\rho$ initial positions evenly distributed in the interval. Then only the partciles with $x(0) > -v_0 t$ are evolved.\n \n \\noindent Here we study RTP with $\\rho=1, v_0=1, \\gamma=0.05$. At time $t=400$, the average flux predicted by Eq.~\\ref{ac-mut} is $\\mu=35.4573$, very close to the one predicted in diffusive limit $\\sim \\sqrt{t\/(2 \\gamma \\pi)} =35.6825$. Then typical realizations ($\\theta=0$) are expected to be similar to the diffusive ones with $D_\\text{eff}=v_0^2\/(2 \\gamma)=10$. When the bias is applied ($\\theta =1$) the sampled realizations have a larger flux, $Q \\approx 93 $ and it is instructive to characterize their statistical properties for RTP and diffusion.\n \n \n\\noindent In Fig.~\\ref{annealed-ac-ld-causes}(a) we show the initial profile of particles. While for $\\theta=0$ it is flat as expected, for $\\theta=-1$ it displays an accumulation of particles around the origin and total number of particles in the interval $[-v_0 t, 0]$ is much larger than $\\rho v_0 t$. In Fig.~\\ref{annealed-ac-ld-causes}(b) we show the histogram (normalized to the number of realizations) of displacement for particles with a positive final position. The peak for $\\theta=0$ has the same location of the peak for $\\theta =-1$. The only difference is that more particles have a positive $x(t)$ for $\\theta =-1$ than for $\\theta=0$.\nThis suggests that, in the annealed case, larger values of the flux are essentially due to rare fluctuations of the initial conditions that are completly insensitive to the nature of the particle motion. \n\n \n \n\n\\noindent For this reason the non-gaussian tails of the flux are Poissonian both for diffusive and active particles. In Fig.~\\ref{annealed-ac-ld-causes}(c) \\footnote{Note that to obtain a reliable histogram one has to re-weight the sampled values of $Q$ by a factor $Z(\\theta) e^{\\theta Q}$. $Z(\\theta)$ is a normalization constant that is determined following the method explained in \\cite{hartmann-epjb-2011, alberto-importance}. } we observe that the agreement between simulations and Poissonian tails increases when $L$ is large, this confirms that the origin of anomalous fluctuations of the flux is in the rare realizations with large initial concentrations of particles close to the origin. This mechanism cannot work for the quenched case where the initial concentration is always flat.\n\n \n \n \n\n\\subsection{Quenched case} \nIn the quenched case the initial condition is fixed and the number of particles with an initial position in $[-v_0 t, 0]$ is always $N =\\rho v_0 t$.\nIn practice, we fix the position of the first particle\n$x_1$ using a uniform random number between $0$ and $-1\/(2 \\rho)$ and the positions of all the other $N-1$ particles are then slaved to $x_1$ according to\n\\begin{equation}\\label{eqp-inc}\n x_i (0) = x_1 (0) - i,\n\\end{equation}\n\nWe first check the agreement between our Monte Carlo simulations and the analytical predictions.\nIn Fig.~\\ref{diffusion-quenched-plots} we compare\nthe exact large deviation function $\\Psi_{\\rm diff}(\\frac{Q}{\\rho\\sqrt{D t}})$ and the exact probability distribution function $P_{qu}(Q,t)|_{\\rm diff}$ with the ones obtained from our Monte Carlo simulations. \nFor the quenched active case, the results are shown in Fig~\\ref{active-quenched-final-plots} where we also plot the large deviation function $\\Psi_{\\rm RTP}(\\frac{Q}{\\rho v_0 t})$, using for the Monte Carlo data the definition $\\Psi_{\\rm RTP}(q) = -\\frac{{\\rm log} P_{\\rm qu}(Q,t)}{\\rho v_0 \\gamma t^2}$ with $q=\\frac{Q}{\\rho v_0 t}$. We note that the Monte Carlo results perfectly match the predictions for both RTP and diffusion.\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{mcs-fq-com2.eps}\n\\end{minipage}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{prob-diff-quen.eps}\n\\end{minipage}\n\\end{center}\n\\caption{Quenched case for passive dynamics. Monte Carlo simulations (red points) and exact results obtained from Mathematica (dashed black line) using Eqs.~\\ref{phi-1eq} and \\ref{fq-bas}. Simulations were perfomed using $\\rho=1, D=0.2, t=10^4, N=2000$.\n(a) $\\Psi_{\\rm diff}(q)$ vs $q$.\n(b) $P_{qu}(Q,t)|_{\\rm diff}$ vs $Q$. The two functions are related via $P_{qu}(Q,t)|_{\\rm diff}=\\exp[-\\rho \\sqrt{D t} \\Psi_{\\rm diff}(\\frac{Q}{\\rho\\sqrt{D t}})] $.\n }\n\\label{diffusion-quenched-plots}\n\\end{figure}\n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{mathm-analytics-mcs-t400et100active.eps}\n\\end{minipage}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{unscaled-active-prob-t400-norm.eps}\n\\end{minipage}\n\\end{center}\n\\caption{Quenched case for active dynamics. Monte Carlo simulations (points) and exact results obtained from Mathematica (dashed black line) using Eqs.~\\ref{Uzt-exact} and~\\ref{def_tildeI}. Simulations were perfomed using $\\rho=v_0=1, \\gamma=0.5$ and $t=100$ (green squares) and $t=400$ (red circles). (a) $\\Psi_{\\rm RTP}(q)$ vs $q$. The violet solid curve represents $\\frac{q^3}{6}$, indicating that $t=400$ is also not large enough to see the ${q^3}$ behavior; also see Fig.~\\ref{phi-full-plot}(c) which shows that Eq.~\\ref{rtp-ldf-model} becomes a good fit to the exact results only at very large times.\n(b) $P_{qu}(Q,t)|_{\\rm RTP}$ vs $Q$ for $t=400$ in the semi-log scale. Note that $P_{qu}(Q,t)|_{\\rm RTP}$ decays slower than the Gaussian expected for the typical fluctuations (green solid line). The two functions are related via\n$P_{qu}(Q,t)|_{\\rm RTP} = \\exp [-\\rho v_0 \\gamma t^2 \\Psi_{\\rm RTP}(\\frac{Q}{\\rho v_0 t})]$.}\\label{active-quenched-final-plots}\n\\end{figure}\n\n\n\n\n\n\nThe difference between RTP and diffusive behaviour in the quenched case is shown is Fig.~\\ref{comp-rtp-diff-gmt40} (a). There we compare the exact distribution of RTP with $\\rho=v_0=1, \\gamma=0.5$ and $t=80$ (red) with the diffusive one with $D_\\text{eff}=v_0^2\/(2 \\gamma)=1$.\n Both distributions deviate from the Gaussian (solid green) but they become separable from each other only at extremely small probabilities, when $Q \\simeq 60 \\approx \\rho v_0 t$. The Importance Sampling strategy enables us to explore the non-Guassian tails of the distribution but not the extremely rare configurations where the finerprints of the activity are present. Indeed in Fig.~\\ref{comp-rtp-diff-gmt40} (b) we can hardly see a difference in histogram of the displacement of particles with positive final position between diffusion and RTP, both for $\\theta =0$ and $\\theta=-3$. This is because the displacements involved are still very small compare to $v_0t =80$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{quenched-diffusion-rtp-Deff1-t80-gmt40.eps}\n\\end{minipage}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{winners-displacement-quenched.eps}\n\\end{minipage}\n\\end{center}\n\\caption{ (a) $P_{\\rm qu}(Q,t)$ vs $Q$. RTPs with $\\rho=v_0=1, \\gamma=0.5$ and $t=80$ (long-dashed red lines, triangles) are compared with the diffusive ones with $D_\\text{eff}=v_0^2\/(2 \\gamma)=1$ (short-dashed black lines, circles). Exact results (lines) and Monte Carlo simulations (points). The solid green line represents the Gaussian valid at small $Q$.\n(b) Histogram (normalized to the number of realizations) of the displacement for particles with a positive final position with $\\theta=0$ (violet circles for RTP\/ black -solid lines for diffusion), and $\\theta=-3$ (red squares for RTP\/ yellow-dashed lines for diffusion).}\n\\label{comp-rtp-diff-gmt40}\n\\end{figure}\n\n\n\\section{Conclusion}\\label{conclu}\n\n\nIn this paper, we have presented a general framework to study current fluctuations for non-interacting particles executing a common random dynamics in one dimension and starting from a step initial condition. The probability distribution $P(Q,t, \\{x_i\\})$ depends on the initial positions of the particles $\\{x_i\\}<0$. The initial positions are distributed uniformly on the negative axis with a uniform density $\\rho$. There are two different ways to perform the average over the initial positions, namely (i) annealed and (ii) quenched averages, in analogy with disordered systems: here the initial condition plays the role of the disorder. In the annealed case, the distribution $P(Q,t,\\{x_i\\})$ is averaged directly over the initial positions. In contrast, in the quenched case, one considers the configurations of $x_i$'s that lead to the most likely current distribution (i.e., the {\\it typical} current distribution). In both cases, we have shown that, for noninteracting particles, the distribution can be fully characterized in terms of the single particle Green's function, which in general will depend on the dynamics of the particles. In this article, we have focused mostly on two different dynamics: a) when the single particle undergoes simple diffusion and b) when the single particle undergoes run-and-tumble dynamics (RTP). \n\nFor the annealed case, we have shown that $P_{\\rm an}(Q,t)$, at all times, is given by a Poisson distribution, with parameter $\\mu(t)$ given by the exact formula in Eq.~(\\ref{muan-mod-def}). We provide exact formula for $\\mu(t)$ in the RTP case (for the diffusive case this was known already\nfrom Ref. \\cite{derrida-gers}). For the quenched diffusive case we show that our formalism correctly recovers the large deviation result obtained in Ref. \\cite{derrida-gers}) using a different approach. For the RTP case, we showed that there is a new large deviation regime with $Q \\sim t$, where $P_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho \\, v_0 \\, \\gamma \\, t^2 \\, \\Psi_{\\rm RTP}\\left(\\frac{Q}{\\rho \\, v_0 \\,t}\\right)\\right]$. One of the main results of this paper is an explicit computation of the rate function $\\Psi_{\\rm RTP}(q)$ given by\n\\begin{equation}\\label{rtp-ldf-model_conclusion}\n\\Psi_{\\rm RTP}(q)=q-\\frac{q}{2}\\sqrt{1-q^2}-{\\rm sin}^{-1}\\left[ \\sqrt{\\frac{1-\\sqrt{1-q^2}}{2}} \\right]\\;, \\quad 0 \\leq q \\leq 1 \\;.\n\\end{equation}\n\n\nOur method gives access to another physical observable, namely the probability of an extremely rare event that \nthere is no particle on the right side of the origin at time $t$. We have shown that this is just the probability of having\nzero flux up to time $t$, i.e. $P(Q=0,t)$, both for the annealed and the quenched case. For the annealed case, this is just\n$P_{\\rm an}(Q=0,t) = e^{-\\mu(t)}$. For the quenched case, we have that, both for the diffusive and RTP cases, this probability \ndecays at late times as a stretched exponential $P_{\\rm qu}(Q=0,t) \\sim e^{-\\bar{\\alpha} \\sqrt{D_{\\rm eff}\\,t}}$, where we computed the constant $\\bar{\\alpha} = 0.675336\\ldots$ analytically [see Eq. (\\ref{C})]. For diffusive particles, $D_{\\rm eff} = D$ while for RTP's, $D_{\\rm eff} = v_0^2\/(2 \\gamma)$. \n\nWe have also verified our analytical predictions by numerical simulations. Computing numerically the large deviation function is far from trivial. Even for the diffusive case the large deviation function predicted for $P(Q,t)$ (both annealed and quenched) in Ref. \\cite{derrida-gers} was\nnever verified numerically. In this paper, we used a sophisticated importance sampling method to compute numerically this large deviation \nfunction in the diffusive case up to an impressive accuracy of order $10^{-200}$. We further used the same technique to compute the large deviation function in the RTP case. \n\nThe formalism developed in this paper can be easily generalized in different directions. For instance, one can compute the flux distribution exactly for the case where there are, initially, arbitrary densities $\\rho_{\\rm left}$ and $\\rho_{\\rm right}$ to the left and to the right of the origin respectively, both the diffusive and for the RTP cases. One could also generalise this result in higher dimensions, with step-like initial conditions, where for instance one region of the space is initially occupied by particles with uniform density. For the diffusive case, the flux distribution in the presence of hard-core repulsions between particles was studied in Ref. \\cite{derrida-gers-sep} (for the simple symmetric exclusion process). It would be interesting to see whether our formalism can be generalized to study the flux distribution for RTP's with hard core repulsions. \n\n\n\n\\begin{acknowledgments}\nWe acknowledge support from the project 5604-2 of the Indo-French Centre for the Promotion of Advanced Research (IFCPAR). \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzigz b/data_all_eng_slimpj/shuffled/split2/finalzigz new file mode 100644 index 0000000000000000000000000000000000000000..c6414a90812d0e9047920453a5143498af085566 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzigz @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction}\n\\acresetall\n\nCryptocurrencies have become an increasingly popular medium for decentralized transactions in recent years. The computation of a cryptographic \\ac{PoW} is the foundational concept of decentralized consensus. It is computed by so-called miners, which are rewarded with a fraction of cryptocurrency for their effort. Currently, \\ac{PoW} computations are built by iterating a cryptographic hash function until the output has a dedicated form. This energy-intensive process presents a prime candidate for hardware acceleration.\n\nSeveral cryptocurrencies adopt so-called ASIC-resistant \\ac{PoW} algorithms. These types of \\ac{PoW} aim to deter dedicated hardware miners in favor of CPU- and GPU-based miners, which are more generally available to the public, thereby sustaining the distributed ledger idea. The Haven Protocol \\cite{Haven} is one of the projects with such a goal. For its \\ac{PoW}, Haven leverages on a custom ASIC-resistant hash function named as \\textit{CryptoNight-Haven}. \n\n\n\n\n\n\nIn this project, we challenge the ASIC-resistance claims of CryptoNight-Haven, by implementing the \\ac{PoW} as an RTL kernel on FPGA. We target the Xilinx Varium C1100 Blockchain Accelerator Card \\cite{VariumC1100}. The card employs Xilinx' recent Ultrascale+ architecture, \\ac{XRT} integration over a high-speed PCIe Gen 4 bus connection, and 8~GB of \\ac{HBM}. Xilinx demonstrated that the Varium C1100 accelerates transaction validation in Hyperledger Fabric \\cite{androulaki2018hyperledger} over an Intel Xeon Silver 4114 CPU with a factor of 14$\\times$.\n\nOur CryptoNight-Haven accelerator aims at a full hardware-based computation of the hash rather than software-assisted. It employs a pipelined datapath with multi-hash computation in a single kernel, and targets multiple kernel instantiations on a single FPGA with nonce-based \\ac{HBM} partitioning. We verified the computation modules under simulation; however, its memory interface demands improvements for random access rather than the simulator's straightforward memory models. \n\nOur CryptoNight-Haven miner RTL, testbench, and host-code (a software patch to XMRig \\cite{xmrig} with \\ac{XRT} integration) are publicly available at: \\url{https:\/\/github.com\/KULeuven-COSIC\/CryptoNightHaven-FPGA-miner}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Algorithm}\n\nThe Cryptonight-Haven algorithm is a variant of Cryptonight \\cite{Cryptonote}. It chains together multiple well-known cryptographic primitives: AES, Keccak, Blake, etc. Collectively, these primitives are used to initialize a large scratchpad data buffer, on which semi-random operations are performed. \n\n\n\\begin{figure}[t]\n\\centering\n\\input{figures\/overview2.tex}\n\\caption{The dataflow Cryptonight-Haven. \n}\n\\label{fig:overview}\n\\end{figure}\n\n\\Cref{fig:overview} shows an overview of the computation flow. First, the input passes through a Keccak module, extracting the state. The explode module takes the first 32 bytes of the state and expands them to 10 AES round keys. These keys are used to perform AES rounds on the remaining bytes (divided into 8 blocks of 128-bits) of the Keccak state. After 10 rounds, the AES output is written to the memory buffer. Next, the blocks are XOR'ed with each other, and they again undergo 10 rounds of AES with the same keys as before. This process is repeated until 4 MB of data has been generated.\n\nThe Shuffle step performs a semi-random operation; either AES, a division, or a multiplication followed by addition. Corresponding input and output data yield intensive memory accesses to irregular addresses.\n\nThe Implode operation is similar to Explode. Bytes 32-63 of the state are used to generate AES round keys. Bytes are read from the start of the memory buffer and XOR'ed with state bytes 64-191. Subsequently, these bytes are put through 10 AES rounds --one round per key-- and the next bytes in the memory buffer are read. After reading the entire 4 MB scratchpad buffer twice, the 10 AES rounds are repeated 16 additional times without reading from the memory buffer.\n\nFinally, the new state passes through a Keccak permutation to generate the final state. Depending on the 2 LSBs of this state, it is subsequently hashed using either of the Blake, Skein, JH or Groestl schemes, resulting in the final 256-bit output.\n\n\n\\section{Implementation}\n\nOur single accelerator kernel employs a datapath with a module hierarchy similar to \\Cref{fig:overview}. The hash computations in the first and last steps of the computation stages contribute a negligible overhead. The Explode step consists of a simple implementation that generates 4 MB of data with simple binary operations and stores it into \\ac{HBM} memory. It heavily benefits from AXI burst transfers with multiple outstanding transactions. The Implode step has double the latency of Explode for accessing the memory twice. For the underlying computations, both the Explode and Implode modules employ 10 AES cores.\n\nIn contrast, Shuffle's computation is demanding due to the multitude of iterations and underlying memory accesses. Additionally, data dependencies between consecutive accesses prevent optimizations, e.g. burst transfers. The memory size prevents using on-chip memory --BRAM or URAM-- that allows single-cycle data access. Irregular addresses are another obstacle that prevents caching parts of the memory on-chip. Hence, minimizing data transfer overheads is the most advantageous strategy.\n\nThe memory accesses of Shuffle form one of the foundations for CryptoNight-Haven's ASIC-resistance claims. Implementing these computations on an ASIC requires significant chip resources to be reserved for implementing memory, leaving a limited amount of silicon for the computation. In contrast, the Varium C1100 natively has 8~GB of \\ac{HBM} available, and our accelerator heavily benefits from it. Moreover, the partitioning of \\ac{HBM} into a number of \\acp{PC} allows us to instantiate each Shuffle unit with a dedicated memory port, resulting in a scalable design.\n\n\nWe pipelined Shuffle for simultaneous computation of up to 128 hashes to boost the computation performance. That requires an identical increase in memory consumption --easily accommodated by the 8~GB HBM-- where individual nonces for each hash computation partition memory regions. In line with this pipelining, we split the computation into various stages that communicate over AXI-Stream interfaces connected with FIFOs. Our pipelining approach allows the time-critical Shuffle module to be clocked at 500~MHz while the other modules remain at 200~MHz.\n\n\n\\section{Future Work}\n\nIn its current state, our design computes hashes correctly within the Vivado simulation environment. That employs a simplified view of memory, which restricts the memory model to AXI accessed BRAM. However, when instantiated as an \\ac{XRT} kernel on the Varium C1100, the hash computations are inconsistent with simulation. We have enhanced the design with a set of AXI-lite accessible status registers that collect additional performance and debug information on the hardware execution. The further roadmap we envision is as follows:\n\n\\begin{enumerate}\n \\item Enhancing our Vivado simulation with random AXI access latencies and AXI protocol checkers.\n \\item Progressing with \\ac{XRT} kernel construction, by replacing the BRAM with HBM under Vitis \\texttt{hw\\_emu} based \\ac{XRT} executions.\n \\item Extending Shuffle's memory accesses with Xilinx' \\ac{RAMA} IP.\n\\end{enumerate}\n\nAfter these steps enable the correct computation of the CryptoNight-Haven \\ac{PoW}, we have already taken the first steps to integrate the accelerator into XMRig \\cite{xmrig} using \\ac{XRT} APIs. The accelerator should be compared thoroughly to existing CPU and GPU-based miners for Cryptonight-Haven, hopefully showing increased throughput and\/or energy efficiency. Finally, we also aim to compare to related work: FPGA-based miners were proposed for the ASIC-resistant \\ac{PoW} Lyra2REv2~\\cite{Lyra2-FPGA, Lyra2-standalone}, Scrypt \\cite{MRSA21}, and X16R \\cite{9786081}.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro.sec}\n\nAs an Italian-Chinese collaboration project, the experiment ARGO-YBJ \n(Astrophysical Radiation with Ground-based Observatory at YangBaJing) \nis under way over the next few years at Yangbajing High Altitude Cosmic Ray \nLaboratory (4300 m a.s.l., 606 $g\/cm^2$), 90 km North to Lhasa \n(Tibet, P.R. China). \nThe aim of the experiment is the study of cosmic rays, mainly \ncosmic $\\gamma$-radiation, at an energy threshold of $\\sim 100$ $GeV$, by \nmeans of the detection of small size air showers at high altitude. \n\nThe apparatus consists of a full coverage detector of dimension \n$\\sim 71\\times 74\\>m^2$ realized with a single layer of Resistive Plate \nCounters (RPCs). The area surrounding the central detector core, up to $\\sim\n100\\times 100\\>m^2$, consists in a guard ring partially ($\\sim 50\\> \\%$) \ninstrumented with RPCs. \nThese outer detector improves the apparatus performance, enlarging the \nfiducial area, for the detection of showers with the core outside the \nfull coverage carpet. \nA lead converter $0.5$ $cm$ thick will cover uniformly the RPC plane in order\nto increase the number of charged particles by conversion of shower photons \nand to reduce the time spread of the shower front.\nThe site coordinates (longitude $90^{\\circ}$ 31' 50'' E, latitude \n$30^{\\circ}$ 06' 38'' N) permits the monitoring of the Northern hemisphere in \nthe declination band $-10^{\\circ}<\\delta <70^{\\circ}$. \n\nARGO-YBJ will image with high sensitivity atmospheric showers induced by\nphotons with $E_{\\gamma}\\geq 100$ $GeV$, allowing to bridge the\nGeV and TeV energy regions and to face a wide range of fundamental issues in\nCosmic Ray and Astroparticle Physics (Abbrescia et al. (1996)):\n\\begin{verse}\n1) {\\it \\underline {Gamma--Ray Astronomy}}, at a $\\sim 100$ $GeV$ threshold energy. \nSeveral galactic and extragalactic point candidate sources can be \ncontinuously monitored, with a sensitivity to unidentified sources better than \n$10\\%$ of the Crab flux.\\\\\n2) {\\it \\underline {Diffuse Gamma--Rays}} from the Galactic plane, \nmolecular clouds and SuperNova Remnants at energies $\\geq 100\\>GeV$.\\\\\n3) {\\it \\underline {Gamma Ray Burst physics}}, by allowing the extension of the \nsatellite measurements over the full GeV\/TeV energy range.\\\\\n4) {\\it \\underline {$\\overline{p}\/p$ ratio}} at energies $300\\>GeV\\div TeV$ not \naccessible to satellites, with a sensitivity adequate to distinguish \nbetween models of galactic or extragalactic $\\overline{p}$ origin.\\\\\n5) {\\it \\underline {Sun and Heliosphere physics}}, including cosmic ray \nmodulations at $10\\>GeV$ threshold energy, the continuous \nmonitoring of the large scale structure of the interplanetary \nmagnetic field and high energy gamma and neutron flares from the Sun.\n\\end{verse}\n\nAdditional items come from using ARGO-YBJ as a traditional EAS array \ncovering the full energy range from $10^{11}$ to $10^{16}\\>eV$. \nSince the detector provides a high granularity space-time picture of the \nshower front, detailed study of shower properties as, for instance, \nmulticore events, time and lateral distributions of EAS particles, \nmultifractal structure of particle densities near the core, can be \nperformed with unprecedented resolution. \nDetector assembling will start late in 2000 and data taking with the first \n$\\sim$ 750 $m^2$ of RPCs in 2001. \n\nIn order to investigate both the RPCs performance at 4300 $m$ a.s.l. and \nthe capability of the detector to sample the shower front of atmospheric \ncascades, during 1998 for the first time a full coverage carpet of \n$\\sim 50\\>m^2$ has been put in operation in the Yangbajing Laboratory. \nThe present paper is a report of this test-experiment. \n\n\\section{The test experiment at Yangbajing}\n\\label{testexp.sec}\n\nThe basic elements of ARGO-YBJ detector are RPCs of dimensions $280\\times \n125\\>cm^2$. The detector is organized in modules of 12 chambers whose \ndimensions are $5.7\\times 7.9\\>m^2$. \nThis group of RPCs represent a logical subdivision (Cluster) of the \napparatus: the detector consists of 117 Clusters in the central part and \n28 Cluster in the guard ring for a total of 1740 RPCs. The proposed \nlay-out allows to achieve an active area of $\\sim 92\\%$ the total. \nThe trigger and the DAQ systems are built following a two level\narchitecture. The signals from the Cluster are managed by Local Stations.\nThe information from each Local Station is collected and elaborated in the\nCentral Station. According to this logic a module of 12 RPCs (i.e. the Cluster)\nrepresents the basic detection unit.\n\nA Cluster prototype, very similar to that proposed for ARGO-YBJ \nexperiment, has been installed in the Yangbajing Laboratory. It \nconsists of 15 chambers distributed in 5 columns with an active area of \n$\\sim 90\\%$. The total area is about $6.1\\times 8.7$ $m^2$. \n\nThe detector consists of single-gap RPCs made of bakelite (with volume \nresistivity \n$\\rho > 5\\cdot 10^{11}\\>\\Omega\\cdot cm$) with a $280\\times 112\\>cm^2$ area. \nThe RPCs read-out is performed by means of Al strips 3.3 $cm$ wide and 56 \n$cm$ long at the edge of which the front-end electronics is connected.\nThe FAST-OR of 16 strips defines a logical unit called pad: ten pads \n($56\\times 56\\>cm^2$) cover each chamber. The FAST-OR signal from each pad \nis sent, via coaxial cable, to dedicated modules that generate the trigger \nand the STOP signals to the TDCs. Each channel of the TDCs \nmeasures, with $1$ $ns$ clock, the arrival times (up to \n16 hits per channel) of the particles hitting a pad, for a total number \nof 150 timing channels. \nThe RPCs have been operated in streamer mode with a gas mixture of \nargon ($15\\%$), isobutane ($10\\%$) and tetrafluoroethane $C_2H_2F_4$ \n($75\\%$), at a voltage of 7400 $V$, about 500 $V$ above the plateau knee. \nThe efficiency of the detector, as measured by a small telescope selecting \na cosmic ray beam, is $>95\\%$, and the intrinsic time resolution \n$\\sigma_t\\sim 1$ $ns$. The description of the results concerning the RPCs\nperformance at YangBaJing are given in Bacci et al. (1999). \n\n\\section{Data Analysis}\n\\label{datana.sec}\n\nDifferent triggers based on pad multiplicity have been used to collect \n$\\sim 10^6$ shower events with $0.5$ $cm$ of lead on the whole carpet in \nApril-May 1998.\nThe integral rate as a function of the pad multiplicity is shown in Fig. 1 \nfor showers before and after the lead was installed. \nA comparison at fixed rate indicates an increasing of pad multiplicity due to \nthe effect of the lead of $\\sim 15\\div 20\\%$, as expected according to our \nsimulations. \n\n\\begin{figure}[htb]\n\\vfill \\begin{minipage}{.47\\linewidth}\n\\begin{center}\n\\mbox{\\epsfig{file=frequenze.eps,height=8.cm,width=8.cm}}\n\\end{center}\n\\caption{\\em The integral rate as a function of the pad multiplicity. }\n\\end{minipage}\\hfill\n\\hspace{-0.5cm}\n\\begin{minipage}{.47\\linewidth}\n\\begin{center}\n\\mbox{\\epsfig{file=linee.eps,height=8.cm,width=8.cm}}\n\\end{center}\n\\caption{\\em Time profile observed in a typical event. Straight lines are \nfit to experimental hits. }\n\\end{minipage}\\hfill\n\\end{figure}\n\n\n\\subsection{Detector Calibration}\n\\label{detcali.sec}\n\nThe relative time offset among different pads are measured as follows: \n\\begin{verse}\n1) We construct the time distributions of each TDC channel adding all \nthe delays in the individuals events and compute the relative time \nmean values $\\overline{t_i}$. \\\\\n2) We compute the mean value of the $\\overline{t_i}$ distribution: \n$={ { \\sum_{i=1}^{N} \\overline{t_i} } \\over N}$.\\\\\n3) We define the time offset as $\\Delta t_i=\\overline{t_i}-$. These values \nare used to correct the times provided by each TDC channel. \n\\end{verse}\nTo check the consistency of the procedure, we fit on an event-by-event \nbasis the corrected time values to a plane, construct for each TDC channel \nthe distribution of time residuals $\\delta t_i=t_{plane}-t_i$ and calculate \nthe mean values $<\\delta t_i>$. These are distributed with a spread of \n$\\sim$ $0.6$ $ns$. \n\n\\subsection{ Event Reconstruction}\n\nIn this test-experiment we are not able to determine the core position, \ntherefore we use a plane as the fitting function. Moreover, the estimated \narrival direction is relatively free from the curvature effect because we \nsample only a small portion of the shower front. \nIn this approximation the expected particle arrival time is a linear \nfunction of the position.\nThe time profile observed in a typical event is shown in Fig. 2. Here $x,y$ \nare orthogonal coordinates which identify the pad position. Straight \nlines are one-dimensional fits to experimental hits along two different $x$ \nvalues. \n\nWe performe an optimized reconstruction procedure of the shower direction \nas follows:\n\\begin{verse}\n1) Unweighted plane fit to hits for each event with pad multiplicity \n$\\geq$ 25. \\\\ \n2) Rejection of out-lying points by means of a 2.5 $\\sigma$ cut and iteration \nof the fit until this condition is not verified.\\\\\n\\end{verse}\nAfter these iterations a fraction $\\leq 10\\%$ of the time signals that\ndeviate most from the fitted plane are excluded from further analysis. \nThe distribution of time residuals $\\delta t$ = $t_{plane}-t_i$\n(Fig. 3) exhibits a long tail due to time fluctuations and to the\ncurved profile of the shower front, more pronounced for low multiplicity. \nThe width of these distributions is related to the time thickness of the \nshower front. Since the position of the shower core is not reconstructed, \nthe experimental result concerns a time thickness averaged on different radial \ndistances. Increasing pad multiplicity select showers with core near the \ndetector, as confirmed by MC simulations. Taking into account the total \ndetector jitter of $1.3$ $ns$ (RPC intrinsic jitter, strip length, \nelectronics time resolution) the time jitter of the earliest particles in \nhigh multiplicity events ($116\\div 120$ hits) is estimated $\\sim 1.1$ $ns$. \n\n\n\\begin{figure}[htb]\n\\vfill \\begin{minipage}{.47\\linewidth}\n\\begin{center}\n\\mbox{\\epsfig{file=residui.eps,height=8.cm,width=8.cm}}\n\\end{center}\n\\caption{\\em Distribution of time residuals for events with different pad \nmultiplicity (all channel added).}\n\\end{minipage}\\hfill\n\\hspace{-0.5cm}\n\\begin{minipage}{.47\\linewidth}\n\\begin{center}\n\\mbox{\\epsfig{file=teta.eps,height=8.cm,width=8.cm}}\n\\end{center}\n\\caption{\\em Even-odd angle difference distribution for events with \ndifferent pad multiplicity.}\n\\end{minipage}\\hfill\n\\end{figure}\n\n\n\\subsection{Angular Resolution}\n\nThe angular resolution of the carpet has been estimated by dividing \nthe detector into two independent sub-arrays and comparing the two\nreconstructed shower directions. \nEvents with N total hits have been selected according to the \nconstraint $N_{odd}\\simeq N_{even}\\simeq N\/2$. \nThe even-odd angle difference $\\Delta \\theta_{eo}$ is shown in Fig. 4 for \nevents in different multiplicity ranges. \nWe note that these distributions narrow, as expected, with the increase of \nthe shower size. \nTo see the dependence of the angular resolution on the lead sheet, we show \nin Fig. 5 the median $M_{\\Delta \\theta_{eo}}$ of the distribution of \n$\\Delta \\theta_{eo}$ as a function of pad multiplicity for showers \nreconstructed before and after the lead was added. \nThe improvement of the angular resolution is a factor $\\sim$ 1.4 for $N=50$\nand decreases with increasing multiplicity. \n\nAssuming that the angular resolution for the entire array is Gaussian \n(Alexandreas et al. (1992)), a standard deviation \n$\\sigma_{\\theta}\\sim 2.1^{\\circ}$ is found for events with a pad \nmultiplicity $\\geq 100$.\n\n\\section{Conclusions}\n\\label{conclu.sec}\n\n\\begin{figwindow}[1,r,%\n{\\mbox{\\epsfig{file=risol.eps,height=7.cm,width=7.cm}}},%\n{\\em Median of $\\Delta \\theta_{eo}$ distribution as a function of\npad multiplicity.}]\nA Resistive Plate Counters carpet of $\\sim$ 50 $m^2$ has been put in \noperation at the Yangbajing Laboratory to study the high altitude \nperformance of RPCs and the detector capability of imaging with high \ngranularity a small portion of the EAS disc, in view of an enlarged use \nin Tibet (ARGO-YBJ experiment). \n\nIn this paper we have presented the results of this test experiment \nconcerning the carpet capability of reconstructing the shower features. \nIn particular, we have focused on the angular resolution \nin determining the arrival direction of air showers, the most important \nparameter for $\\gamma$-ray astronomy studies. \n\nThe effect of a $0.5$ $cm$ lead sheet on the whole carpet has been \ninvestigated. An increase $15\\div 20\\%$ of the hit multiplicity is found. \nThe improvement of the angular resolution depends on the shower density. \n\nThe test confirms that RPCs can be operated efficiently to sample air showers \nat high altitude with excellent space and time resolutions. \nThe results are consistent with data assumed in the computation of the \nperformance of the ARGO-YBJ detector. \n\\end{figwindow}\n\n\\vspace{1ex}\n\\begin{center}\n{\\Large\\bf References}\n\\end{center}\nAbbrescia M. et al., {\\it Astroparticle Physics with ARGO}, Proposal (1996). \nThis document can be downloaded at the URL: \nhttp:\/\/www1.na.infn.it\/wsubnucl\/cosm\/argo\/argo.html\\\\\nAlexandreas D.E. et al., Nucl. Instr. Meth. A311 (1992) 350.\\\\\nBacci C. et al. (ARGO-YBJ coll.), (1999) submitted to Nucl. Instr. Meth.\\\\\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nWe will study the so-called source-type self-similar solution to the Navier-Stokes\nequations. Our motivation for this is as follows.\nFirst of all, it will give us a\nparticular self-similar solution to the Navier-Stokes equations, which\ncharacterises the decaying process in the late stage of evolution. Second,\nit is likely to give useful information as to how we may handle more general\nsolutions.\n\nIt has been shown under mild conditions that no nontrivial smooth backward self-similar\nsolution exists to the Navier-Stokes equations. On the other hand it is known that\nnontrivial forward self-similar solutions {\\it do} exist, but their explicit\nfunctional forms are not known, except for some asymptotic results. \nIt is of interest to see they actually behave, because\nsuch solutions contain the important information regarding more general\nsolutions.\nThis is particularly the case when the governing equations are exactly\nlinearisable, e.g. the Burgers equations. While it is not expected\nthat the Navier-Stokes equations are exactly soluble in general, we might still obtain\ninsights into the nature of their solutions.\n\nIn \\cite{CP1996} the existence of forward self-similar solutions\nfor small data was proved by using a fixed-point theorem in a Besov space (see below).\nThere initial data for self-similar solution (3D)\nis assumed to be homogeneous of degree $-1$ in velocity\n$$\\bm{u}_0(\\lambda \\bm{x})=\\lambda^{-1} \\bm{u}_0(\\bm{x})$$\nand the existence of a self-similar solution of the form\n$$\\bm{u}(\\bm{x},t)=\\frac{1}{\\sqrt{t}}\\bm{U}\\left(\\frac{\\bm{x}}{\\sqrt{t}} \\right)$$\nhas been established under the assumption that initial data is small in some Besov space.\nMoreover, it has been proved that the self-similar profile $U$ satisfies (in their notations)\n$$U=S(1) u_0 +W,$$\nwhere $S(1)$ denotes a heat operator at time 1 and $\\|W\\|_{L^3}$ is small.\nIn Jia-Sverak(2014), using a locally H{\\\"o}lder class in $\\mathbb{R}^3\\backslash\\{0\\}$\nthe smallness assumption has been removed and\nit is furthermore shown that\n$$|U(x)-e^{\\triangle}u_0(x)| \\leq \\frac{C(M)}{(1+|x|)^{1+\\alpha}},$$\nwhere $0 < \\alpha <1$ and $C(M)$ denotes a constant with some norm $M$ of $u_0$. \nThose studies indicates that the self-similar solution is close to the heat flow in the late\nstage. However, studies on the determination of a specific functional form of\nself-similar solutions are few and far between, except for an attempt in\\cite{Brandolese2009}.\nWe also note that the existence of generalised self-similar solutions (in the sense that scaling holds\nonly at a set of discrete values of $\\lambda$) was studied subsequently, e.g. \\cite{CW2017, BT2019}.\n\nBy definition the source-type solution for nonlinear parabolic PDEs is a solution in a scaled space,\nwhich starts from a Dirac mass in \\textit{some} dependent variable and ends up like a near-identity of\nthe Gaussian function in the long-time limit. It is a counterpart to the fundamental solution to nonlinear PDEs.\nIt is important to choose the right unknown to alleviate the difficulty of analysis.\n\nWe will construct an approximate solution valid in the long-time limit, using\nthe vorticity curl $\\nabla \\times \\bm{\\omega}$.\nWe will explain why this is the most convenient variable for our purpose.\n\nThe unknown whose $L^1$-norm is marginally divergent is suitable for describing the late-stage\nevolution. This is because it satisfies the same scaling as the Dirac mass and both of them\nbelong to a Besov space near $L^1$. \nIn one dimension, in the limit of $t \\to 0$, we have roughly\n$$u \\sim \\dfrac{1}{x},\\; \\mbox{marginally}\\,\\notin L^1(\\mathbb{R}^1),$$\nwhich shows that the velocity is convenient in this case.\n\nIn two dimensions it is the vorticity which is the most convenient, \nas can be seen from\n$$\\omega \\sim \\dfrac{1}{r^2},\\;\\mbox{marginally}\\,\\notin L^1(\\mathbb{R}^2),$$\nwhere $|\\bm{x}|=r$.\nRecall that those scaling properties of velocity in 1D or vorticity in 2D,\nare the same as that of the Dirac mass:\n$\\lambda^d \\delta(\\lambda \\bm{x})=\\delta(\\bm{x})$\nin $d$-dimensions.\nNow consider Besov spaces whose norms are given by\n$$\\|\\bm{u}\\|_{B^s_{pq}} \\equiv \\left\\{\\sum_{j=1}^\\infty\n\\left( 2^{sj} \\|\\Delta_j (\\bm{u}) \\|_{L^p}\\right)^q \\right\\}^{1\/q},$$\nwhere $1 \\leq p,q \\leq \\infty, s \\in \\mathbb{R}$ and $\\Delta_j (\\bm{u})$ represents band-filtered velocity\nat frequency $2^j$.\nIt is known that in $\\mathbb{R}^d$ the Dirac delta mass is embedded as\n\\begin{equation}\\label{delta}\n\\delta(\\bm{x}) \\in B_{p,\\infty}^{-d+d\/p},\\;\\;\\mbox{for}\\;\\; p \\geq 1,\n\\end{equation}\nsee e.g. \\cite{HL2017}. In particular we have\n$\\delta(\\bm{x}) \\in B_{1,\\infty}^{0}$ for any $d.$\nWhile the velocity $u \\sim \\dfrac{1}{r} \\notin L^3(\\mathbb{R}^3),$ we have \n$$u \\sim \\frac{1}{r} \\in B_{3,\\infty}^{0}(\\mathbb{R}^3),$$ and correspondingly for the vorticity curl $\\chi$ \n$$\\chi \\sim \\frac{1}{r^3} \\in B_{1,\\infty}^{0}(\\mathbb{R}^3).$$ \nHence in three dimensions this $\\chi$ and the Dirac mass belong to the same\nfunction class $B_{1,\\infty}^{0}(\\mathbb{R}^3)$, with $p=1$ in (\\ref{delta}).\n\\section{Illustrationg the ideas with Burgers equations}\n\\subsection{1D Burgers equation}\nWe consider the Burgers equation \\cite{Burgers1948} \n\\begin{equation}\\label{1DBurgers}\n\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x}\n=\\nu \\frac{\\partial^2 u}{\\partial x^2},\n\\end{equation}\nwhich satisfies static scale-invariance under\n$$x \\to \\lambda x, t \\to \\lambda^2 t, u \\to \\lambda^{-1} u.$$\nThis means that if $u(x,t)$ is a solution, so is \n$\\lambda u(\\lambda x, \\lambda^2 t) :=u_\\lambda(x,t),$\nfor any $\\lambda (>0)$.\nIt is readily checked that \n$$\\|u_\\lambda\\|_{L^p}=\\lambda^{\\frac{p-1}{p}}\\|u\\|_{L^p},$$\nwhich shows that the $L^1$-norm is scale-invariant.\n\nLet us clarify the two kinds of critical scale-invariance;\none is deterministic where the additional terms arising in the governing equations\nunder dynamic\nscaling is minimised in number and the other is statistical in nature where the\nadditional terms under dynamic scaling is maxmised in number so that\nconservative form (that is, a divergence form) is completed in the advection term.\nIn the former the dependent variable has the same physical dimension\nas kinematic viscosity, whereas in the latter the argument of the Hopf\ncharacteristic functional (the independent variable) has the same physical \ndimension as the reciprocal of kinematic viscosity \\cite{Ohkitani2020}.\nThis approach provides a viewpoint from which the problem\nappears in the simplest possible form.\n\nCritical scale-invariance is achieved with the velocity potential $\\phi,$ which is defined by\n$u=\\partial_x \\phi$. If $\\phi(x, t)$ is a solution, so is $\\phi(\\lambda x, \\lambda^2 t).$\nUnder dynamic scaling for the velocity potential $\\phi(x,t)=\\Phi(\\xi,\\tau)$ we have\n\\begin{equation}\\label{ScaledBurgers0}\n\\frac{\\partial \\Phi}{\\partial \\tau}\n + \\frac{1}{2}\\left(\\frac{\\partial \\Phi}{\\partial \\xi}\\right)^2\n =a \\xi\\frac{\\partial \\Phi}{\\partial \\xi}+\\nu \\frac{\\partial^2 \\Phi}{\\partial \\xi^2},\n \\end{equation}\nwhose linearisation has the Ornstein-Uhlenbeck operator.\nThis will be called type 1 (deterministic) scale-invariance where the number of additional terms is\nminimised, that is, only the drift term.\nUnder dynamic scaling for velocity\n$u(x,t)=\\frac{1}{\\sqrt{2a(t+t_*)}}U(\\xi,\\tau),\\;\n\\xi=\\frac{x}{\\sqrt{2 a(t+t_*)}},\\;\n\\tau=\\frac{1}{2 a}\\log(1+2at)$ with $2at_*=1$,\nwe find\n\\begin{equation}\\label{ScaledBurgers}\n\\frac{\\partial U}{\\partial \\tau}\n +U \\frac{\\partial U}{\\partial \\xi}\n=a \\frac{\\partial}{\\partial \\xi}\\left( \\xi U \\right)\n+\\nu \\frac{\\partial^2 U}{\\partial \\xi^2},\n\\end{equation}\nwhose linaerisation is the Fokker-Planck equation. This will be called type 2 (statistical)\nscale-invariance where the number of additional terms is maximised\nin the sense that a divergence form is completed with the addition of $aU$ term.\nAs it is of second-order it has two independent\nsolutions, of which we will focus on the Gaussian one.\nSee \\textbf{Appendix A(a)} for the other non-Gaussian possibility.\n\nEquation (\\ref{ScaledBurgers}) is exactly soluble and its steady solution is called the source-type solution\n\\cite{BKW1999, BKW2001}:\n\\begin{equation}\\label{Burgers_source1}\nU(\\xi)=\\frac{U(0) \\displaystyle{\\exp \\left( -\\frac{a \\xi^2}{2 \\nu} \\right)}}\n{1 -\\displaystyle{\\frac{U(0)}{2\\nu}\\int_0^{\\xi}} \n \\exp \\left( -\\frac{a \\eta^2}{2 \\nu}\\right) d\\eta}.\n\\end{equation}\nThe name has come from the time zero asymptotics (with $t_*=0$)\n$$\\lim_{t \\to 0}\\frac{1}{\\sqrt{2at}} U(\\xi)=M\\delta(x),$$\nwhere $M\\equiv\\int_{\\mathbb{R}^1} u_0(x) dx$ and\n$U(0)=\\sqrt{\\frac{8a\\nu}{\\pi}}\\tanh\\frac{M}{4\\nu}\n\\approx \\sqrt{\\frac{a}{2\\pi \\nu}}M\\;\\; (\\mbox{for}\\;M\/\\nu \\ll 1).$\\footnote{Here we have taken the virtual time\n origin $t_*=0$. There are two ways of handling it; one is to take the limit\n $\\tau \\to \\infty$ first and then let $a\/\\nu \\to \\infty$ keeping $2at_*=1$. The other one is\nto consider $\\lambda(t)=\\sqrt{2at}$ from scratch and consider the steady equations.}\nSee \\cite{EZ1991, BKW1999, BKW2001}.\n\nIt is also known that for\n$u_0 \\in L^1$ we have \n$$t^{\\frac{1}{2}\\left(1-\\frac{1}{p}\\right)}\n\\left\\| u(x,t)- \\frac{1}{\\sqrt{2at}} U(\\xi)\\right\\|_{L^p}\n\\to 0\\;\\;\\mbox{as}\\;\\; t \\to \\infty,$$\nwhere $1 \\leq p \\leq \\infty$ and $\\xi=\\frac{x}{\\sqrt{2at}}.$\n\nThe simplest method for solving (\\ref{ScaledBurgers}), without linearisation,\nis as follows. Rewrite the equation \n\\begin{eqnarray}\n \\frac{U^2}{2}&=&a\\xi U +\\nu \\frac{dU}{d\\xi} \\nonumber\\\\\n &=&\\nu \\exp \\left(-\\frac{a \\xi^2}{2\\nu}\\right)\n \\frac{d}{d\\xi} \\left(U \\exp \\left( \\frac{a \\xi^2}{2\\nu} \\right)\\right).\n \\nonumber\n\\end{eqnarray}\nBy changing variables to\n$\\tilde{U}=U \\exp \\left( \\frac{a \\xi^2}{2\\nu}\\right),\\eta = \\frac{1}{2\\nu}\n\\int_0^\\xi \\exp \\left(-\\frac{a \\zeta^2}{2\\nu} \\right) d\\zeta,$ we find\n$$\\frac{d \\tilde{U}}{d \\eta}=\\tilde{U}^2,$$\nwhich is readily integrable. Alternatively we may solve the equation (2.1) by regarding it as a\nBernoulli equation.\n\nIt is in order to comment on the significance of source-type solution.\nWhen we recast (\\ref{Burgers_source1}) as\n\\begin{equation}\\label{Burgers_source2}\nU(\\xi)=-2\\nu \\frac{\\partial}{\\partial \\xi}\\log\\left(\n{1 -\\displaystyle{\\frac{U(0)}{2\\nu}\\int_0^{\\xi}} \\exp \\left( -\\frac{a \\eta^2}{2 \\nu}\\right) d\\eta}\n\\right),\n\\end{equation}\nwhich is reminiscent of the celebrated Cole-Hopf transform. In other words, the source-type solution\nencodes the vital information of the nonlinear term in the case of the Burgers equation. Note that the error-function\n$\\int_0^{\\xi} \\exp \\left( -\\frac{a \\eta^2}{2 \\nu}\\right) d\\eta$ itself is a self-similar solution to\nthe heat equation. This suggests that studying source-type solution of the Navier-Stokes equations may give a hint on the\nnature of their long-time evolution.\n\n\n\\subsection{Successive approximations}\nThe operator $L = \\triangle^* \\equiv \\triangle +\\frac{a}{\\nu}\\partial_\\xi(\\xi \\cdot )\n$ is not self-adjoint. It is possible to find a function $G$ such that\n$L^{\\dagger} G(\\xi) =-\\delta(\\xi)$ holds, where\n$L^{\\dagger}\\equiv \\partial_\\xi^2 -\\frac{a}{\\nu}\\xi \\partial_\\xi$ is the adjoint of $L$.\nIn fact $G(\\xi) \\propto D\\left( \\sqrt{\\frac{a}{2\\nu}} \\xi\\right),$ where $D(\\cdot)$ denotes\nthe Dawson's integral, see e.g. {\\bf Appendix A(a)}. \nHowever, because $G$ decays slowly at large distances\n$G(\\xi) \\propto \\frac{1}{\\xi}$ as $|\\xi| \\to \\infty$, it cannot be used as a Green's function,\nat least, in the usual manner.\n\nThe inversion formula for $\\triangle^*$ can be obtained in an alternative method.\nRecall that on the basis of a formal analogy with $\\frac{1}{a}=\\int_0^\\infty e^{-at} dt \\; (a>0),$\nthe fundamental solution to the Poisson equation in 1D is given by\n$$ (\\nu \\triangle)^{-1} \\equiv -\\int_{0}^{\\infty} ds e^{\\nu s \\triangle} =\\frac{|\\xi|}{2\\nu}*,$$\nwhere * denotes convolution.\nLikewise for the fundamental solution to the Fokker-Planck equation in 1D we write\n$$(\\nu \\triangle^*)^{-1} \\equiv -\\int_{0}^{\\infty} ds e^{\\nu s \\triangle^*} =\\int_{-\\infty}^{\\infty}\nd\\eta g(\\xi,\\eta),$$\n where\n $$g(\\xi,\\eta) \\equiv \\frac{-1}{\\sqrt{2\\pi\\nu}} {\\rm f.p.}\\int_{\\sqrt{a}}^{\\infty}\n \\frac{d\\sigma}{\\sigma^2-a}e^{-\\frac{1}{2\\nu}\\left(\\sigma \\xi -\\eta\\sqrt{\\sigma^2-a}\\right)^2}$$\n and f.p. denotes the finite part of Hadamard, e.g. \\cite{Bureau1955, Yosida1956}.\n It can be verified by changing the variable $\\sigma=\\sqrt{\\frac{a}{1-e^{-2a\\tau}}},$ in the solution to\n the Fokker-Planck equation\n$$e^{\\nu \\tau \\triangle^*}f=\\left( \\frac{a}{2\\pi \\nu(1-e^{-2 a \\tau})}\\right)^{1\/2}\n \\int_{\\mathbb{R}^1}e^{a\\tau} f(e^{a\\tau} \\eta)\\exp \\left( -\\frac{a}{2\\nu} \\frac{(\\xi-\\eta)^2}{1-e^{-2 a \\tau}}\\right) d\\eta.$$\nAs we will consider the 3D Navier-Stokes equations, for which methods of exact solutions are not known,\nwe treat (\\ref{ScaledBurgers}) by approximate methods as a preparation. Because the inversion of\n$\\triangle^*$ is cumbersome, we will seek a workaround by which we can dispense with it.\n\nFirst we convert it to an integral equation by\nthe Duhamel principle for the Fokker-Planck operator\n$\\triangle^*$. \n\\begin{align*}\n U(\\tau) = & e^{\\nu \\tau \\triangle^*} U(0)\n -\\int_{0}^{\\tau} e^{\\nu(\\tau-s)\\triangle^*}\\partial\\, \\frac{U(s)^2}{2}ds &\\\\\n = & e^{\\nu \\tau \\triangle^*} U(0)\n -\\int_{0}^{\\tau} e^{\\nu s \\triangle^*}\\partial\\,\\frac{U(\\tau-s)^2}{2}ds. &\n\\end{align*}\nThe long-time limit $U_1=\\lim_{\\tau \\to \\infty} e^{\\nu s \\triangle^*} U(0)$ is given by\n$$U_1=\\left( \\frac{a}{2\\pi \\nu}\\right)^{1\/2} M e^{-\\frac{a}{2\\nu}\\xi^2}\\;\\;\\mbox{with}\n\\;\\;M=\\int_{-\\infty}^{\\infty} U(0) d\\xi.$$\nWe may consider a number of different iteration schemes. For example, the following option (1), also known as the Picard iteration, requires\nthe inversion $(\\triangle^*)^{-1}$:\n$$\\mbox{Successive approximation (1):}\\;\\;U_{n+1} =U_1-\\int_{0}^{\\infty} e^{\\nu s \\triangle^*}\\partial\\,\\frac{U_n^2}{2}ds,$$\n $$\\mbox{in particular, for}\\;\\;n=1:\\;\\; U_2=U_1-\\int_{0}^{\\infty} e^{\\nu s \\triangle^*}\\partial\\,\\frac{U_1^2}{2}ds.$$\nIt is noted that $U_n$ is a steady function at each step.\n\nAlternatively we first consider the steady equation \n$$\\triangle^* U\\equiv \\triangle U +\\frac{a}{\\nu}(\\xi U)_\\xi=\\frac{1}{\\nu}\\left( \\frac{U^2}{2}\\right)_\\xi$$\nand then introduce iteration schemes:\n$$\\mbox{Iteration scheme (2a):}\\;\\;\n\\triangle U_{n+1} +\\frac{a}{\\nu}(\\xi U_{n+1})_\\xi =\\frac{1}{\\nu}\\left( \\frac{U_{n}^2}{2}\\right)_\\xi,\\;\\;(n \\geq 0)$$\n$$\\mbox{For}\\;\\;n=1:\\;\\;\\triangle U_{2} +\\frac{a}{\\nu}(\\xi U_{2})_\\xi =\\frac{1}{\\nu}\\left( \\frac{U_{1}^2}{2}\\right)_\\xi,$$\nor\n$$\\mbox{Iteration scheme (2b):}\\;\\;\\triangle U_{n+1} =-\\frac{a}{\\nu}(\\xi U_{n})_\\xi +\\frac{1}{\\nu}\\left( \\frac{U_{n}^2}{2}\\right)_\\xi,\n\\;\\;(n \\geq 1)$$\n$$\\mbox{For}\\;\\;n=1:\\;\\;\\triangle U_{2} =\\underbrace{-\\frac{a}{\\nu}(\\xi U_{1})_\\xi}_{= \\triangle U_1}\n+\\frac{1}{\\nu}\\left(\\frac{U_{1}^2}{2}\\right)_\\xi.$$\nNote that iteration schemes (1) and (2a) coincide with each other at $n=1$.\n\n\\subsection{Estimation of the size of nonlinear terms}\nFor the Burgers equation we can work out the two results to the second-order approximations\nanalytically. After some algebra they are\n \\begin{align*}\n \\mbox{(1)}\\;\\; U & \\approx Ce^{-\\frac{a}{2\\nu}\\xi^2}\n \\left( 1+\\frac{C}{2\\nu} \\int_0^\\xi e^{-\\frac{a}{2\\nu}\\eta^2} d\\eta \\right),\\\\\n \\mbox{(2b)}\\;\\; U & \\approx Ce^{-\\frac{a}{2\\nu}\\xi^2}\n +\\frac{C^2}{2\\nu} \\int_0^\\xi e^{-\\frac{a}{\\nu}\\eta^2} d\\eta,\n \\end{align*}\n where $C \\approx \\sqrt{\\frac{a}{2\\pi\\nu}} M.$\n On this basis we estimate the size of the nonlinear term $N(\\xi).$\n After non-dimensionalisation the second-order term is proportional to the Reynolds number\n $Re=\\int U d\\xi\/ \\nu.$ Separating out the $Re$-dependence, we define $N(\\xi)$ by\n $$U \\propto 1+ Re N(\\xi).$$\n From the above expressions for the 1D Burgers equation we then find\n $$\\mbox{(1)}\\;\\; N= \\frac{1}{4}=0.25,$$\n $$\\mbox{(2b)}\\;N=\\frac{1}{4\\sqrt{2}}\\approx 0.2,$$\n where $N=\\max_\\xi N(\\xi).$\n We conclude that the typical size of nonlinearity $N=O(10^{-1}),$ irrespective of the choice of schemes.\n \n\\subsection{Burgers equations in several dimensions} \n\nThe source-type solution is basically a near-identity function of the Gaussian\nform.\nIt has been seen how the source-type solutions show up in the long-time limit\nin one and two spatial dimensions in \\cite{Ohkitani2020}. Here we will take a look at\ncases in three and higher dimensions. We have by the Cole-Hopf transform\n$$U_i(\\bm{\\xi},\\tau)\n=-2\\nu\\frac{\\partial_{\\xi_i} \\int_{\\mathbb{R}^3} \\psi_0(\\lambda \\bm{\\eta})\n\\exp \\left(-\\dfrac{a}{2\\nu}\\dfrac{|\\bm{\\xi}-\\bm{\\eta}|^2}{1-e^{-2a\\tau}}\\right)\nd\\bm{\\eta}}{\\int_{\\mathbb{R}^3} \\psi_0(\\lambda \\bm{\\eta})\n\\exp\\left(-\\dfrac{a}{2\\nu}\\dfrac{|\\bm{\\xi}-\\bm{\\eta}|^2}{1-e^{-2a\\tau}}\\right) \nd\\bm{\\eta}}.$$\nIn view of the type 2 scale-invariance and differentiating it twice, we find\n$$\\partial_{\\xi_j}\\partial_{\\xi_k} U_i(\\bm{\\xi},\\tau)\n=-2\\nu\\frac{\\partial_{\\xi_i} \\partial_{\\xi_j} \\partial_{\\xi_k}\n \\int_{\\mathbb{R}^3} \\psi_0(\\lambda \\bm{\\eta})\n\\exp \\left(-\\dfrac{a}{2\\nu}\\dfrac{|\\bm{\\xi}-\\bm{\\eta}|^2}{1-e^{-2a\\tau}}\\right)\nd\\bm{\\eta}}{\\int_{\\mathbb{R}^3} \\psi_0(\\lambda \\bm{\\eta})\n\\exp\\left(-\\dfrac{a}{2\\nu}\\dfrac{|\\bm{\\xi}-\\bm{\\eta}|^2}{1-e^{-2a\\tau}}\\right) \nd\\bm{\\eta}}+\\ldots\n$$\n$$\n=-2\\nu\\frac{\\lambda^3\\int_{\\mathbb{R}^3} \\partial_i \\partial_j \\partial_k\\psi_0(\\lambda \\bm{\\eta})\n\\exp \\left(-\\dfrac{a}{2\\nu}\\dfrac{|\\bm{\\xi}-\\bm{\\eta}|^2}{1-e^{-2a\\tau}}\\right)\nd\\bm{\\eta}}{\\int_{\\mathbb{R}^3} \\psi_0(\\lambda \\bm{\\eta}) \n\\exp\\left(-\\dfrac{a}{2\\nu}\\dfrac{|\\bm{\\xi}-\\bm{\\eta}|^2}{1-e^{-2a\\tau}}\\right) \nd\\bm{\\eta}}+\\ldots\n$$\nThe denominator then tends to $K_{ijk}\\exp \\left(-\\frac{a}{2\\nu}|\\bm{\\xi}|^2 \\right)$ as $\\tau \\to \\infty,$\nwhere $K_{ijk}=\\int_{\\mathbb{R}^3} \\partial_i \\partial_j \\partial_k \\psi_0( \\bm{\\eta}) d\\bm{\\eta},\\;\n(i=1,2,3).$\nHence\n$$\\partial_{\\xi_j}\\partial_{\\xi_k} U_i(\\bm{\\xi},\\infty)\n=-2\\nu \\left( \\frac{K_{ijk}\\exp \\left(-\\frac{a}{2\\nu}|\\bm{\\xi}|^2 \\right)}{F_{ijk}(\\bm{\\xi})}\n+\\ldots\\right),$$\nwhere the function $F_{ijk}$ is to be determined such that\n$\\partial_i \\partial_j \\partial_k F_{ijk} \\propto\n \\exp \\left(-\\frac{a}{2\\nu}|\\bm{\\xi}|^2 \\right).$\n We can thus take\n$$\nF_{ijk}(\\bm{\\xi})=-\\frac{K_{ijk}}{2\\nu}\n\\int_0^{\\xi_1} \\exp \\left(-\\frac{a \\xi^2}{2\\nu} \\right) d\\xi\n\\int_0^{\\xi_2} \\exp \\left(-\\frac{a \\eta^2}{2\\nu} \\right) d\\eta\n\\int_0^{\\xi_3} \\exp \\left(-\\frac{a \\zeta^2}{2\\nu} \\right) d\\zeta\n+1.$$\nTherefore we find in three dimensions, say, with $i=1,j=2,k=3,$\n\\begin{equation}\\label{source_Burgers3D}\n\\frac{\\partial^2 U_1}{\\partial \\xi_2 \\partial \\xi_3}\n=K_{123}\n\\exp \\left(-\\frac{a}{2\\nu}(\\xi_1^2+\\xi_2^2+\\xi_3^2)\\right)\n\\frac{1+R(\\xi_1, \\xi_2,\\xi_3)}{(1-R(\\xi_1,\\xi_2,\\xi_3))^3},\n\\end{equation}\nwhere\n$$R(\\xi_1, \\xi_2,\\xi_3)=\\frac{K_{123}}{2\\nu} \n\\int_0^{\\xi_1} \\exp \\left(-\\frac{a \\xi^2}{2\\nu} \\right) d\\xi\n\\int_0^{\\xi_2} \\exp \\left(-\\frac{a \\eta^2}{2\\nu} \\right) d\\eta\n\\int_0^{\\xi_3} \\exp \\left(-\\frac{a \\zeta^2}{2\\nu} \\right) d\\zeta$$\ndenotes the Reynolds number. Because $R$ is small the expression (\\ref{source_Burgers3D}) is near-Gaussian.\nNote that $K_{123}=\\sqrt{\\dfrac{32a^3}{\\pi^3 \\nu}} \\tanh \\dfrac{M_{123}}{16\\nu},$ where\n$M_{123}=\\int \\frac{\\partial^2 U_1}{\\partial \\xi_2 \\partial \\xi_3} d\\bm{\\xi}.$\nWe can also write\n$$\\frac{\\partial^2 U_1}{\\partial \\xi_2 \\partial \\xi_3}\n=-2 \\nu \\frac{\\partial^3}{\\partial \\xi_1 \\partial \\xi_2 \\partial \\xi_3} \n\\log(1-R(\\xi_1, \\xi_2,\\xi_3)),$$\nwhich more directly reflects the Cole-Hopf transform. \n$\\bm{U}=\\nabla_{\\bm{\\xi}} \\phi,\\;\\phi=-2\\nu \\log(1-R(\\bm{\\xi})).$\nSee {\\bf Appendix B} for the general form in $n$-dimensions.\n\n\\section{2D Navier-Stokes equation}\nWe briefly recall the case of the 2D Navier-Stokes equation.\nThe so-called Burgers vortex was introduced originally to represent the reaction of a vortex under\nthe influence of the collective effect of surrounding vortices in the ambient medium.\nWhen we write the steady solution in velocity and vorticity\nusing cylindrical coordinates\n$$\\bm{u}=(u_r, u_\\theta, u_\\phi)=(-a r, v(r), 2 a z),\\;\n\\bm{\\omega}=(0, 0, \\omega(r)),$$\nthe solution takes the following forms:\n$$\\omega(r)=\\frac{a\\Gamma}{2\\pi\\nu}\\exp\\left(-\\frac{a r^2}{2\\nu} \\right),$$\n$$v(r)=\\frac{\\Gamma}{2\\pi r}\\left(1-\\exp \\left(-\\frac{a r^2}{2\\nu}\\right)\\right),$$\nwhere\n$\\Gamma \\equiv \\int_{\\mathbb{R}^2}\\omega_0(\\bm{x})d \\bm{x}$\ndenotes the velocity circulation.\n\nThe scaled form of the vorticity equation in two dimensions reads\n$$\n\\dfrac{\\partial \\Omega}{\\partial \\tau}+ \\bm{U}\\cdot \\nabla \\Omega\n=\\nu \\triangle \\Omega + a \\nabla \\cdot (\\bm{\\xi}\\Omega),$$\nwhere $\\Omega$ satisfies the type 2 scale-invariance.\nIt is known that the self-similar solution under scaling has\nthe mathematically identical form as the Burgers vortex tube above.\nIndeed in the scaled variables the above expression can be written\n$$\\Omega(\\xi)=\\frac{a \\Gamma}{2\\pi\\nu}\\exp\\left(-\\frac{a|\\bm{\\xi}|^2}{2\\nu} \\right),\n\\,\\bm{\\xi}=\\frac{\\bm{x}}{\\sqrt{2at}}.$$\n(See \\textbf{Appendix A(b)} for the other non-Gaussian solution.)\n\nWe recall that \n$\\frac{1}{2at}\\Omega(\\xi)\n=\\frac{\\Gamma}{4\\pi\\nu t}\\exp\\left(-\\frac{|\\bm{x}|^2}{4\\nu t} \\right)$\nis an exact self-similar solution with the following property\n$$\\lim_{t \\to 0} \\Omega(\\cdot)=\\Gamma \n\\delta(\\bm{x}).$$\nIt also satisfies the following asymptotic property, for $\\omega(\\cdot,0) \\in L^1,$ \n$$t^{1-\\frac{1}{p}}\\left\\|\\omega(\\bm{x},t)-\\frac{1}{2at}\\Omega(\\bm{\\xi})\n\\right\\|_{L^p} \\to 0\\;\\;\\mbox{as}\\;\\; t \\to \\infty,$$\nwhere $1 \\leq p \\leq \\infty,$ see e.g. \\cite{GW2005}.\n\n\\section{3D Navier-Stokes equations}\n\nWe will describe two approaches for handling a perturbative treatment for the 3D\nNavier-Stokes equations. First we describe a general framework based on the Green's function.\nSecond we describe the other iterative approach which is specifically suited for calculations\nassociated with the 3D Navier-Stokes problem.\n\n\\subsection{Governing equations}\nWe consider the 3D Navier-Stokes equations written in four different dependent\nvariables. Starting from the vector potential, taking the curl successively\n$\\bm{u}=\\nabla \\times \\bm{\\psi},\\; \\bm{\\omega}=\\nabla \\times \\bm{u},\\;\n\\bm{\\chi}=\\nabla \\times \\bm{\\omega},$ we have\n\\begin{equation}\\label{NS3D}\n\\left\\{\n\\begin{array}{l}\n\\dfrac{\\partial \\bm{\\psi}}{\\partial t}\n=\\dfrac{3}{4\\pi}{\\rm p.v.}\\displaystyle{\\int_{\\mathbb{R}^3}}\n\\dfrac{\\bm{r} \\times( \\nabla \\times \\bm{\\psi} (\\bm{y}))\\,\n\\bm{r} \\cdot (\\nabla \\times \\bm{\\psi} (\\bm{y}))}\n {|\\bm{r}|^5}\\;{\\rm d}\\bm{y}+\\nu\\triangle \\bm{\\psi},\\\\\n\\noalign{\\vskip 0.2cm} \n\\dfrac{\\partial \\bm{u}}{\\partial t}\n+ \\bm{u}\\cdot \\nabla \\bm{u} =-\\nabla p + \\nu\\triangle\\bm{u},\\\\\n\\noalign{\\vskip 0.2cm} \n\\dfrac{\\partial \\bm{\\omega}}{\\partial t}+ \\bm{u}\\cdot \\nabla \\bm{\\omega}\n=\\bm{\\omega} \\cdot \\nabla \\bm{u} + \\nu \\triangle \\bm{\\omega},\\\\\n\\noalign{\\vskip 0.2cm} \n\\dfrac{\\partial \\bm{\\chi}}{\\partial t}\n=\\nabla \\times \\left( \\bm{u}\\times \\bm{\\chi}+2 (\\bm{\\omega}\\cdot\\nabla)\\bm{u}\n\\right)+\\nu \\triangle \\bm{\\chi}, \n\\end{array} \\right.\n\\end{equation}\nwhere $\\bm{r}\\equiv\\bm{x}-\\bm{y}$ and p.v. denotes a principal-value integral. \nWe also have\n$ \\bm{\\omega}=- \\triangle \\bm{\\psi}, \\bm{\\chi}=-\\triangle \\bm{u},$\nbecause of the incompressibility condition. Equation (\\ref{NS3D})$_1$ can be found in \\cite{Ohkitani2015}.\nThe final fourth equation (\\ref{NS3D})$_4$ is obtained by applying a curl on the vorticity equations\n(details to be found in {\\bf Appendix C}). This is suitable for handling inviscid fluids.\nFor the equations for $\\bm{\\chi}$, we may alternatively take the Laplacian of the velocity equations\n(\\ref{NS3D})$_2$ to obtain the following form\n\\begin{equation}\\label{Chi.eq}\n\\frac{\\partial \\bm{\\chi}}{\\partial t}\n=\\triangle( \\bm{u} \\cdot \\nabla \\bm{u}+ \\nabla p) +\\nu \\triangle \\bm{\\chi}\n\\end{equation}\ninstead of the final line (\\ref{NS3D})$_4$.\n\nLikewise the dynamically-scaled 3D Navier-Stokes equations in four different unknowns read\n\\begin{equation}\\label{NS3Dscaled}\n\\left\\{\n\\begin{array}{l}\n\\dfrac{\\partial \\bm{\\Psi}}{\\partial \\tau}\n=\\dfrac{3}{4\\pi} {\\rm p.v.}\\displaystyle{\\int_{\\mathbb{R}^3}}\n\\dfrac{\\bm{\\rho} \\times( \\nabla \\times \\bm{\\Psi} (\\bm{\\eta}))\\,\n\\bm{\\rho} \\cdot (\\nabla \\times \\bm{\\Psi} (\\bm{\\eta}))}\n{|\\bm{\\rho}|^5}\\;{\\rm d}\\bm{\\eta}+\\nu\\triangle \\bm{\\Psi} \n+ a(\\bm{\\xi}\\cdot\\nabla)\\Psi,\\\\\n\\noalign{\\vskip 0.2cm} \n\\dfrac{\\partial \\bm{U}}{\\partial \\tau}\n+ \\bm{U}\\cdot \\nabla \\bm{U} =-\\nabla P + \\nu\\triangle\\bm{U}\n+ a(\\bm{\\xi}\\cdot\\nabla)\\bm{U}\n+a \\bm{U},\\\\\n\\noalign{\\vskip 0.2cm} \n\\dfrac{\\partial \\bm{\\Omega}}{\\partial \\tau}+ \\bm{U}\\cdot \\nabla \\bm{\\Omega}\n=\\bm{\\Omega} \\cdot \\nabla \\bm{U} + \\nu \\triangle \\bm{\\Omega}\n+a(\\bm{\\xi}\\cdot\\nabla)\\bm{\\Omega} +2a \\bm{\\Omega},\\\\\n\\noalign{\\vskip 0.2cm}\n\\dfrac{\\partial \\bm{X}}{\\partial \\tau}\n=\\nabla \\times \\left( \\bm{U}\\times \\bm{X}+2 (\\bm{\\Omega}\\cdot\\nabla)\\bm{U}\n\\right)+\\nu \\triangle \\bm{X}\n+ a\\nabla\\cdot(\\bm{\\xi}\\otimes \\bm{X}),\n\\end{array}\\right.\n\\end{equation}\nwhere $\\bm{\\rho}\\equiv\\bm{\\xi}-\\bm{\\eta}.$\nAlternatively for the final equation\nwe may take a Laplacian of the velocity equations to obtain the following form:\n\n\\begin{equation}\\label{scaled3DNS}\n\\dfrac{\\partial \\bm{X}}{\\partial \\tau}\n=\\triangle \\left(\\bm{U}\\cdot\\nabla\\bm{U}+\\nabla P \\right)\n+\\nu \\triangle \\bm{X}+a\\nabla\\cdot(\\bm{\\xi}\\otimes \\bm{X}).\n\\end{equation}\nIt is to be noted that the coefficient of the linear term increases in number with the order of derivatives\nand with the variable $\\bm{\\chi}$ a divergence is completed in the convective term. \nObserve that type 1 scale-invariance is achieved with $\\bm{\\Psi}$ and type 2 scale-invariance with $\\bm{X}$.\n\\subsection{Successive approximations}\nUsing the Duhamel principle, we convert the scaled Navier-Stokes equations\ninto integral equations\n$$\n\\bm{X}(\\bm{\\xi}, \\tau)=e^{\\nu \\tau \\triangle^*} \\bm{X}_0(\\bm{\\xi})\n+\\int_0^{\\tau} e^{\\nu s \\triangle^*}\\triangle \\left(\\bm{U}\\cdot\\nabla\\bm{U}+\\nabla P \\right)(\\bm{\\xi},\\tau-s)ds.\n$$\nHere $\\triangle^* \\equiv \\triangle +\\frac{a}{\\nu}\\nabla\\cdot(\\bm{\\xi}\\otimes \\cdot),$\nthe action of whose exponential operator is given by\n\\begin{equation}\\label{FP3D}\n\\exp(\\nu \\tau \\triangle^*)f(\\cdot)\n=\\left( \\frac{a}{2\\pi \\nu(1-e^{-2 a \\tau})}\\right)^{3\/2}\n\\int_{\\mathbb{R}^3} e^{3a\\tau}f(e^{a\\tau}\\bm{y})\n\\exp \\left( -\\frac{a}{2\\nu} \\frac{|\\bm{\\xi}-\\bm{y}|^2}{1-e^{-2 a \\tau}}\\right) d\\bm{y}\n\\end{equation}\nfor any function $f$; see (e) below for the derivation.\n\nThe inverse operator associated with\nthe fundamental solution to the Fokker-Planck equation in 3D is defined by\n$$(\\nu \\triangle^*)^{-1} \\equiv -\\int_{0}^{\\infty} ds e^{\\nu s \\triangle^*} =\\int d\\eta g(\\bm{\\xi},\\bm{\\eta}),$$\n where\n $$g(\\bm{\\xi},\\bm{\\eta}) \\equiv \\frac{-1}{(2\\pi\\nu)^{3\/2}} {\\rm f.p.}\\int_{\\sqrt{a}}^{\\infty}\n \\frac{\\sigma^2 d\\sigma}{\\sigma^2-a}e^{-\\frac{1}{2\\nu}|\\sigma \\bm{\\xi} -\\bm{\\eta}\\sqrt{\\sigma^2-a}|^2}.$$\n It can be verified by changing the variable $\\sigma=\\sqrt{\\frac{a}{1-e^{-2a\\tau}}}$\n in the solution to the Fokker-Planck equation (\\ref{FP3D}).\n\nWe consider the steady solution $\\bm{X}(\\bm{\\xi})$\nin the long-time limit of $\\tau \\to \\infty$\n$$\n\\bm{X}(\\bm{\\xi})=\\bm{X}^1(\\bm{\\xi}) + \\lim_{\\tau \\to \\infty}\n\\int_0^{\\tau} e^{\\nu s \\triangle^*}\\triangle \\left(\\bm{U}\\cdot\\nabla\\bm{U}+\\nabla P \\right)(\\bm{\\xi},\\tau-s)ds,\n$$\nwhere $\\bm{X}^1=\\mathbb{P}\\bm{M} G$ denotes the leading-order approximation.\n\nOn the other hand, steady equations are obtained by assuming $\\partial\/\\partial\\tau=0$ in (\\ref{scaled3DNS})\n$$\n\\triangle^* \\bm{X}\\equiv \\triangle \\bm{X}\n+\\frac{a}{\\nu} \\nabla\\cdot(\\bm{\\xi}\\otimes \\bm{X})=-\\frac{1}{\\nu}\n\\triangle \\left(\\bm{U}\\cdot\\nabla\\bm{U}+\\nabla P \\right),\n$$\nor,\n$$\n\\bm{X}=-\\frac{1}{\\nu}\\left(\\bm{U}\\cdot\\nabla\\bm{U}+\\nabla P \\right)\n-\\frac{a}{\\nu}\\triangle^{-1} \\nabla\\cdot(\\bm{\\xi}\\otimes \\bm{X}).\n$$\nThis is yet another form of the steady Navier-Stokes equations after dynamic scaling.\nIt is noted that one of the potential problems associated with the nonlinear\nterm has been eliminated without having a recourse to the Green's function.\nIt is this virtually trivial fact that allows us to set up a simple successive\napproximation.\n\nTo summarise, the steady Navier-Stokes equations after dynamic scaling can be written as\n\\begin{equation}\n\\bm{X}=-\\frac{1}{\\nu}\\mathbb{P} \\left(\\bm{U}\\cdot\\nabla\\bm{U}\\right)\n-\\frac{a}{\\nu}\\triangle^{-1} \\nabla\\cdot(\\bm{\\xi}\\otimes \\bm{X}),\n\\end{equation}\nor, by $\\bm{X}=-\\triangle \\bm{U},$ we can express it solely in terms of $\\bm{X}$ as\n\\begin{equation}\\label{scaled.Chi.eq}\n\\bm{X}=-\\frac{1}{\\nu}\\mathbb{P} \\left(\\triangle^{-1}\\bm{X}\\cdot\\nabla \\triangle^{-1}\\bm{X}\\right)\n-\\frac{a}{\\nu}\\triangle^{-1} \\nabla\\cdot(\\bm{\\xi}\\otimes \\bm{X}).\n\\end{equation}\nThis is the equation that we need to solve.\n\nIn passing we note the following facts before proceeding to the specific results. By the definition of\nscaled variables it is easily seen that for $p \\ge 1$\n $$t^{\\frac{3}{2}\\left(1-\\frac{1}{p}\\right)}\n\\left\\| \\bm{\\chi}(\\bm{x},t)-\\frac{\\bm{X}(\\bm{\\xi})}{(2at)^{3\/2}}\\right\\|_{L^p}\n=\\left\\| \\bm{X}(\\bm{\\xi},\\tau)-\\bm{X}(\\bm{\\xi}) \\right\\|_{L^p}.$$\nThis means that if \n$\\left\\| \\bm{X}(\\bm{\\xi},\\tau)-\\bm{X}(\\bm{\\xi}) \\right\\|_{L^p} \\to 0$ as $\\tau \\to \\infty,$\nwe have\n $$t^{\\frac{3}{2}\\left(1-\\frac{1}{p}\\right)}\n\\left\\| \\bm{\\chi}(\\bm{x},t)-\\frac{\\bm{X}(\\bm{\\xi})}{(2at)^{3\/2}}\n\\right\\|_{L^p} \\to 0\\;\\;\\mbox{as}\\;\\; t \\to \\infty.$$\nThat is about the long-time asymptotics. On the other hand, as time zero asymptotics\nwe have\n$$\\frac{\\bm{X}(\\bm{\\xi})}{(2at)^{3\/2}} \\to \\bm{X}_0*\\delta=\\bm{X}_0(\\cdot)\n\\;\\;\\mbox{as}\\;\\; t \\to 0,$$\nwhere $\\bm{X}_0(\\bm{\\xi})$ is singular, like the Dirac mass.\n\n\\subsection{Leading-order approximation}\nThe first-order (or, leading-order) approximation can be given explicitly because the Gaussian\nfunction is a radial function.\n(See \\textbf{Appendix A(c)} for the other non-Gaussian solution.)\n\nBefore discussing the second-order approximation, we give expressions of the first-order\nsolutions in different variables. In terms of the vorticity curl, the first-order approximation is given by\n$$X_i=M_j\\left( \\frac{a}{2\\pi\\nu}\\right)^{3\/2}\n\\left(\\delta_{ij}-\\partial_i \\partial_j \\triangle^{-1} \\right)\n\\exp\\left(-\\frac{a}{2\\nu}r^2\\right),\\;i=1,2,3,$$\nwhere $r=|\\bm{\\xi}|$ and $M_j=\\int X_j d\\bm{\\xi},\\; i=1,2,3.$\nThroughout this subsection we take $\\frac{a}{2\\nu}=1$ for simplicity. The corresponding\nexpressions for the four different unknowns are ($i=1,2,3$)\n\\begin{eqnarray}\nB_i&=&\\frac{1}{\\pi^{3\/2}}\n\\left(M_i \\triangle^{-2} e^{-r^2} \n- M_j \\partial_i \\partial_j \\triangle^{-3} e^{-r^2} \\right),\\\\\nA_i&=&-\\frac{\\epsilon_{ijk}M_j x_k}{8\\pi^{3\/2}r^3}\n\\left[re^{-r^2}+\\sqrt{\\pi}{\\rm erf}(r)\\left(r^2-\\frac{1}{2}\\right) \\right],\\\\\nU_i&=&-\\frac{1}{\\pi^{3\/2}} \\left(M_i \\triangle^{-1} e^{-r^2} \n- M_j \\partial_i \\partial_j \\triangle^{-2} e^{-r^2} \\right),\\\\\n\\Omega_i&=&-\\frac{\\epsilon_{ijk}M_j x_k}{4\\pi^{3\/2}r^3}\n\\left(2re^{-r^2}-\\sqrt{\\pi}{\\rm erf}(r) \\right),\\\\\nX_i&=&\\frac{1}{\\pi^{3\/2}}\n\\left(M_ie^{-r^2} - M_j \\partial_i \\partial_j \\triangle^{-1} e^{-r^2} \\right),\n\\end{eqnarray}\nwhere\n$\\bm{A}=\\nabla \\times \\bm{B}, \\bm{U}=\\nabla \\times \\bm{A}, \n\\bm{\\Omega}=\\nabla \\times \\bm{U},\\bm{X}=\\nabla \\times \\bm{\\Omega},$\nand ${\\rm erf}(r)\\equiv\\frac{2}{\\sqrt{\\pi}}\\int_0^r e^{-t^2}dt$ denotes the error function. \nBecause all the fields are incompressible we also have\n$\\bm{U}=-\\triangle \\bm{B}, \\bm{\\Omega}=- \\triangle \\bm{A}, \\bm{X}=-\\triangle \\bm{U}.$\n\nNote that $\\triangle^{-n} e^{-r^2}$ for $n=1,2,3$ can be evaluated by quadratures\nand their explicit form are as follows, which can be obtained most conveniently with\nthe assistance of computer algebra. The results are\n\\begin{eqnarray}\n\\triangle^{-1} e^{-r^2}&=&-\\frac{1}{2r}\\int_0^r e^{-s^2}ds\n=-\\frac{\\sqrt{\\pi}}{4r}{\\rm erf}(r),\\\\\n\\triangle^{-2} e^{-r^2}&=&-\\frac{1}{2r}\n\\int_0^r ds \\int_0^s ds' \\int_0^{s'} e^{-s''^2}ds'' \\nonumber\\\\\n&=&-\\frac{e^{-r^2}}{8}\n-\\frac{\\sqrt{\\pi}}{8}\\left(r+\\frac{1}{2r}\\right){\\rm erf}(r),\\\\\n\\triangle^{-3} e^{-r^2}&=&-\\frac{1}{2r}\n\\int_0^r ds \\int_0^s ds' \\int_0^{s'} ds'' \\int_0^{s''} ds''' \n\\int_0^{s'''} ds'''' e^{-s''''^2},\\nonumber\\\\\n&=&-\\frac{1}{384r}\n\\left[(4r^3+10r)e^{-r^2}+4\\sqrt{\\pi}\\left(r^4+3r^2+\\frac{3}{4}\\right)\n{\\rm erf}(r) \\right].\n\\end{eqnarray}\nUsing the above formulas it is instructive to compare a component of\nthe Gaussian function $\\dfrac{1}{\\pi^{3\/2}}e^{-x^2}$ with that of\nvorticity curl\n$$\\chi_1(x,0,0)=\\frac{\\sqrt{\\pi}{\\rm erf}(x)-2xe^{-x^2}}{2\\pi^{3\/2}x^3}.$$\nFigure \\ref{GvsPG} shows how $\\chi_1(\\xi_1,0,0)$ is affected by the incompressible condition (solenoidality),\nin particular the peak value at $x=0$ is reduced by a factor of $2\/3$. We have also confirmed that the numerical\nresult obtained with a Poisson solver agrees with the analytical expression (figures omitted).\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.3,angle=0]{GvsPG.eps}\n\\caption{Comparison of $\\exp(-\\xi^2)$ (dashed) with $\\pi^{3\/2}\\chi_1(\\xi)$ (solid). }\n\\label{GvsPG}\n\\end{figure}\n\n\\subsection{Numerical results for the second-order approximation}\nFor simplicity we make use of the iteration scheme (2b) illustrated in the previous section. The second-order solution\nin this case is given by\n$$\\bm{X}_2=\\bm{X}_1-\\frac{1}{\\nu}\\mathbb{P} \\left(\\triangle^{-1}\\bm{X}_1\\cdot\\nabla \\triangle^{-1}\\bm{X}_1\\right).$$\n\\begin{figure}[ht]\n\\begin{minipage}{0.5\\linewidth}\n\\includegraphics[scale=0.4,angle=0]{NS3D_nu0.5_PG1y.AL40.DH0.05.rn.ps}\n\\caption{$\\bm{X}_1|_2=\\mathbb{P}\\bm{M} G$}\n\\label{NS1st-order}\n\\end{minipage}\n\\begin{minipage}{0.5\\linewidth}\n\\includegraphics[scale=0.4,angle=0]{NS3D_nu0.5_FIXNLy.AL40.DH0.05.1st-it1.rn.ps}\n\\caption{$\\frac{1}{\\nu}\\mathbb{P}\\left(\\triangle^{-1}\\bm{X}_1\\cdot\\nabla \\triangle^{-1}\\bm{X}_1\\right)$}\n\\label{NS2nd-order}\n\\end{minipage}\n\\end{figure}\nIt turns out that the first component of the second-order correction is identically equal to zero.\nFigure \\ref{NS1st-order} shows the second component of the first-order approximation $\\bm{X}_1$ as a function\nof $\\xi_1$. It has a peak at the origin whose height is approximately 0.12.\nHence we show in Figure \\ref{NS2nd-order} the second component of the second-order correction due to nonlinearity\n$\\frac{1}{\\nu}\\mathbb{P} \\left(\\triangle^{-1}\\bm{X}_1\\cdot\\nabla \\triangle^{-1}\\bm{X}_1\\right)$\nas a function of $\\xi_1$.\nIt has double peaks near the origin, but their value is small and is about 0.0015.\nNoting $Re=\\frac{M}{\\nu}=\\frac{1}{1\/2}=2,$ after non-dimensionalisation\nwe can estimate the size of nonlinearity as\n$$ N \\approx \\frac{0.0015}{2 \\times 0.12} \\approx 6 \\times 10^{-3},$$\nActually the maximum value of the nonlinear term in $\\mathbb{R}^3$ is 0.0022, not much different from the above value.\nThus $N$ is at most, $N \\approx 9\\times 10^{-3}$ and we conclude\n$$N=O(10^{-2})\\;\\;\\mbox{for the 3D Navier-Stokes equations.}$$\nIt should be noted that it is much smaller than the value of $N$ for the Burgers equations, whose solutions are known to\nremain regular all time. Because the difference between the Navier-Stokes and Burgers equations is the presence or absence \nof the incompressibility condition, it is incompressibility that makes the value of $N$ for the Navier-Stokes equations\nsmall.\nOn a practical side, this also means that even if we add the second-order correction\nto the first-order term at low Reynolds number, say $Re=1,$ the superposed solution is virtually indistinguishable from\nthe first-order approximation.\n\n\\subsection{Derivations of some formulas}\nWe will derive the basic formulas used above.\nSolving the heat equation\n$$\\frac{\\partial \\bm{\\psi}_1}{\\partial t}=\\nu \\triangle \\bm{\\psi}_1$$\nthe first-order approximation is given by\n$$\\bm{\\psi}_1(\\bm{x},t)\n=\\frac{1}{(4\\pi \\nu t)^{3\/2}}\\int_{\\mathbb{R}^3} \\bm{\\psi}_0(\\bm{y})\n\\exp \\left( -\\frac{|\\bm{x}-\\bm{y}|^2}{4\\nu t}\\right) d \\bm{y}.$$\nAfter applying dynamic scaling\n$$\\bm{\\psi}(\\bm{x},t)=\\bm{\\Psi}(\\bm{\\xi},\\tau),$$\n$$\\bm{\\xi}=\\frac{\\bm{x}}{\\sqrt{2 a(t+t_*)}},\\; \n\\tau=\\frac{1}{2 a}\\log(1+2at),$$\nthe equations for the vector potential read\n$$\\frac{\\partial \\bm{\\Psi}_1}{\\partial \\tau}=a\\bm{\\xi}\\cdot \\nabla \\bm{\\Psi}_1\n+\\nu \\triangle \\bm{\\Psi}_1.$$\nIts solution is given by\n$$\\bm{\\Psi}_1(\\bm{\\xi},\\tau)=e^{-3 a \\tau}\n\\left( \\frac{a}{2\\pi \\nu(1-e^{-2 a \\tau})}\\right)^{3\/2}\n\\int_{\\mathbb{R}^3} \\bm{\\Psi}_0(\\bm{\\eta})\n\\exp \\left( -\\frac{a}{2\\nu} \\frac{|\\bm{\\xi}-\\bm{\\eta}e^{-a \\tau}|^2}\n{1-e^{-2 a \\tau}}\\right) d\\bm{\\eta}$$\n$$=\\left( \\frac{a}{2\\pi \\nu(1-e^{-2 a \\tau})}\\right)^{3\/2}\n\\int_{\\mathbb{R}^3} \\bm{\\Psi}_0(e^{a\\tau}\\bm{y})\n\\exp \\left( -\\frac{a}{2\\nu} \\frac{|\\bm{\\xi}-\\bm{y}|^2}\n{1-e^{-2 a \\tau}}\\right) d\\bm{y}.$$\nTaking a curl with respect to $\\bm{\\xi}$ three times,\nwe find the expressions for the vorticity curl\n$$\\bm{X}_1(\\bm{\\xi},\\tau)\n=\\left( \\frac{a}{2\\pi \\nu(1-e^{-2 a \\tau})}\\right)^{3\/2}\n\\int_{\\mathbb{R}^3} e^{3a\\tau}\\bm{X}_0(e^{a\\tau}\\bm{y})\n\\exp \\left( -\\frac{a}{2\\nu} \\frac{|\\bm{\\xi}-\\bm{y}|^2}\n{1-e^{-2 a \\tau}}\\right) d\\bm{y}.$$\nWe distinguish two cases to proceed.\n\n(i) If the initial data satisfies the similarity condition $\\lambda^3 \\bm{X}_0(\\lambda\\bm{y})=\\bm{X}_0(\\bm{y}),$\nwe have\n$$\\bm{X}_1(\\bm{\\xi},\\tau)\\to\n\\left( \\frac{a}{2\\pi \\nu}\\right)^{3\/2}\n\\int_{\\mathbb{R}^3} \\bm{X}_0(\\bm{y})\n\\exp \\left( -\\frac{a}{2\\nu} |\\bm{\\xi}-\\bm{y}|^2 \\right) d\\bm{y}\n= \\bm{X}_0*G,\n$$\nwhere $G(\\bm{\\xi})\\equiv\\left( \\frac{a}{2\\pi \\nu}\\right)^{3\/2}\n\\exp \\left(-\\frac{a}{2\\nu}|\\bm{\\xi}|^2 \\right).$\n\n(ii) For more general initial data without the similarity assumption, we make use of the formula\n$\\lambda^3 \\bm{X}_0(\\lambda\\bm{y}) \\to \\bm{M}\\delta(\\bm{y})$ where $\\bm{M}=\\int \\bm{X}_0 d\\bm{y}.$\nWe have, noting $\\bm{P}\\bm{X}_0=\\bm{X}_0,$\n\\begin{eqnarray}\n\\bm{X}_1(\\bm{\\xi},\\tau)\n&=&\\left( \\frac{a}{2\\pi \\nu(1-e^{-2 a \\tau})}\\right)^{3\/2}\n\\int_{\\mathbb{R}^3} e^{3a\\tau}(\\bm{P}\\bm{X}_0(e^{a\\tau}\\bm{y}))\n\\exp \\left( -\\frac{a}{2\\nu} \\frac{|\\bm{\\xi}-\\bm{y}|^2}{1-e^{-2 a \\tau}}\\right) d\\bm{y} \\nonumber\\\\\n&=&\\left( \\frac{a}{2\\pi \\nu(1-e^{-2 a \\tau})}\\right)^{3\/2}\n\\int_{\\mathbb{R}^3} e^{3a\\tau}\\bm{X}_0(e^{a\\tau}\\bm{y})\n\\bm{P}\\exp \\left( -\\frac{a}{2\\nu} \\frac{|\\bm{\\xi}-\\bm{y}|^2}{1-e^{-2 a \\tau}}\\right) d\\bm{y} \\nonumber\\\\\n&\\to& \\bm{P}\\bm{M}G\\:\\:\\mbox{as}\\;\\; \\tau \\to \\infty. \\nonumber\n\\end{eqnarray}\nThis is the leading-order approximation for the scaled 3D Navier-Stokes equations.\n(Alternatively we may apply the solenoidal projection to restore the incompressibility\nas the incompressibility is not maintained in the limiting procedure.)\n\n\n\\section{Summary and outlook}\nWe have studied self-similar solutions to the fluid dynamical equations,\nwith particular focus on the so-called source-type solutions.\nAs an illustration of successive approximation schemes we have discussed the 1D Burgers equation\nwhich is exactly soluble. In this case the velocity is the most convenient choice for its analysis.\nWe have introduced a method of quantitatively assessing the size of nonlinearity $N$ using the source-type solutions.\nThe similar analyses have been carried over to higher dimensional Burgers equations.\nFor Burgers equations in any dimensions, we find $N=O(0.1)$.\n\nWe then move on to consider the 2D and 3D Navier-Stokes equations.\nIn two dimensions we review the known results done with the vorticity.\nIn three dimensions the most convenient choice of the unknown is the vorticity curl.\nWe have formulated the dynamically-scaled equations using that variable and set up the successive\napproximation schemes. We have found that the second-order correction, stemming from the nonlinear term,\nis $N \\approx 0.01$, an order of magnitude smaller than that for the Burgers equations.\n\nIt may be in order to give our outlook on the topic.\nThe current approach relies on perturbative treatments.\nIt may be challenging, but worthwhile to study the functional form of the solution by non-perturbative\nmethods for further theoretical developments.\nIt is also of interest to seek a fully non-linear solution by numerical methods. \nIt is noted that this is at least one order of magnitude smaller than $N$ found for the Burgers equations\nwhose solutions are known to remain regular all the time.\nAs an application of the source-type solution, it is useful to characterise the late stage of\nstatistical solutions of the Navier-Stokes equations \\cite{Ohkitani2020}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn the recent past, collaborative and crowdsourcing platforms \\citep{estelles2012towards} have been investigated for their ability to obtain large amounts of user interactions for the annotation of image databases. Particularly, the capacity to outsource simple human intelligence tasks to a crowd population and simultaneously draw from client computing resources for interfacing, are being increasingly appreciated in the imaging community \\citep{mckenna2012strategies,maier2014can,mavandadi2012biogames}. First studies employing collaborative \\citep{haehn2014design,rajchl2016learning} or crowd sourcing platforms \\citep{maier2014can,albarqouni2016aggnet} via web interfaces have been proposed for biomedical image segmentation. Since such interfaces often have limited capacity to interact with image data, weak forms of annotations (\\emph{e.g.} bounding boxes, user scribbles, image-level tags, \\emph{etc.}) have been investigated to reduce the required annotation effort. Recent studies have shown that placing bounding box annotations is approximately 15 times faster than creating pixel-wise manual segmentations \\citep{lin2014microsoft,papandreou2015weakly}.\n\nHowever, in contrast to annotating natural images \\citep{lin2014microsoft,russell2008labelme} or recognising instruments in a surgical video sequence \\citep{maier2014can}, the correct interpretation of medical images requires specialised training and experience \\citep{nodine2000nature,gurari2015collect}, and therefore might pose a challenge for non-expert annotators, leading to incorrectly annotated data \\citep{cheplygina2016early}. Nonetheless, considering the limited resources of available clinical experts and the rapid increase in information of medical imaging data (\\emph{e.g.} through high-resolution, whole-body imaging, \\emph{etc.}) alternative approaches are sought. Particular challenges arise, when trying to develop machine learning based approaches that can be scaled to very large datasets (\\emph{i.e.} population studies). Many of the currently available approaches require large amounts of labelled training data to deal with the variability of anatomy and potential pathologies.\n\n\\subsection{Related Work}\nTo reduce the annotation effort, many well-known studies propose segmentation methods employing simple forms of user annotations to obtain voxel-wise segmentations \\citep{boykov2000interactive,rother2004grabcut,rajchl2017deepcut,koch2017multi}. While adjusting hyperparameters can be considered an interaction, in this study we concentrate on simplified forms of pictorial input \\citep{olabarriaga2001interaction}, called weak annotations (WA). Such annotations have been extensively used in the literature, particularly in the context of medical object segmentation problems. \\cite{boykov2000interactive} used user-provided scribbles (SC) or brush strokes as input and hard constraints to an interactive graphical segmentation problem. Similarly, \\cite{baxter2015optimization}, \\cite{baxter2017directed} and \\cite{rajchl2012fast} expressed this problem in a spatially continuous setting by using prior region ordering constraints and exploiting parallelism via GPU computing. The GrabCut algorithm \\citep{rother2004grabcut} employs rectangular regions (RR) as bounding boxes to both compute a colour appearance model and spatially constrain the search for an object. These spatial constraints further allow to reduce the computational effort \\citep{pitiot2004expert}. The segmentation platform ITK-SNAP\\footnote{\\emph{http:\/\/www.itksnap.org\/}} \\citep{yushkevich2006user} combines RR with SC and employs a pre-segmentation (PS) to initialise an active contour model. \n\nWhile the above object segmentation methods concentrate on how to accurately compute segmentations based on WA, recent studies have examined how to efficiently acquire the required annotations. Collaborative annotation platforms such as LabelMe\\footnote{\\emph{http:\/\/labelme.csail.mit.edu\/}} \\citep{russell2008labelme} or \\citep{lin2014microsoft} were proposed to distribute the effort of placing image annotations to a crowd of users. Such crowdsourcing approaches have been successfully used in proof-reading connectomic maps \\citep{haehn2014design}, identification of surgical tools in laparoscopic videos \\citep{maier2014can}, polyps from computed tomography (CT) colonography images \\citep{mckenna2012strategies} or the identification of the fetal brain \\citep{rajchl2016learning}.\nHowever, most studies concentrate on tasks that require little expertise of the crowd, as the objects to identify are either known from everyday life \\citep{russell2008labelme,lin2014microsoft} or foreign to background context \\citep{maier2014can}. \\cite{russell2008labelme} and \\cite{lin2014microsoft} concentrated on object recognition tasks in natural images, the latter constrained to objects \"easily recognizable by a 4 year old\". \\cite{maier2014can} asked users to identify a foreign surgical object in a video scene and \\cite{haehn2014design} provided an automated pre-segmentation to be corrected by users. \\cite{mckenna2012strategies} compensated for the lack of expertise in reading the colonography images by improving image rendering, implementing a training module and using a large number of redundant users (\\emph{i.e.} 20 knowledge workers per task). \nExpertise in the interpretation of medical images is largely acquired through massive amounts of case-reading experience \\citep{nodine2000nature} and it has been shown that novices performed with lower accuracy than an average expert in tasks such as screening mammograms for breast cancer \\citep{nodine1999experience,nodine2000nature}. In contrast to \\emph{diagnostic interpretation} of medical images, automated segmentation pipelines merely require the \\emph{identification} of anatomical structures (\\emph{i.e.} it requires less expertise to identify the liver in a CT image than a lesion in the liver). \n\n\\subsection{Contributions}\nIn this study, we examine types of commonly employed WA and investigate the impact of reducing the annotation frequency (\\emph{i.e.} only annotating every $k$-th slice in a volume) and the expertise on the segmentation accuracy. For this purpose, we employ a well-known graphical segmentation method and provide weak image annotations as initialisation and constraint to the segmentation problem at hand.\nWe address the problem of liver segmentation from a database of abdominal CT images and corresponding \\emph{non-redundant annotations}. It is of great importance for both planning for laparoscopic surgery and computed-assisted diagnosis \\citep{wolz2012multi} and requires extensive manual annotation effort because of its high spatial resolution and large field of view. \nFurther, we propose and evaluate how a \\emph{weakly labelled atlas} can be used for the detection and removal of incorrect annotations and achieve similar accuracy using weak annotations as a state-of-the-art fully-supervised automated segmentation method \\citep{wolz2012multi}. \n\n\\section{Methods} \nTo study their impact on accuracy, we simulate user annotations from expert manual segmentations $M$, subject to different expertise levels and annotation frequency:\n\n\\subsubsection*{Expertise} \nWe assume that the task of placing an annotation itself is defined well enough to be handled by a pool of general users with any experience level. However, the correct identification of anatomical structures might pose a challenge to non-expert users. We define expertise as the rate of correctly identifying an object of interest in a 2D image slice extracted from a 3D volume. If an error occurs, the user annotates the wrong object or rates that the object is not visible in this slice. We define the error rate (ERR $\\in [0,1]$), i.e. the frequency of misidentification, as a measure of expertise.\nAn annotation error is simulated by computing an annotation from another organ (\\emph{e.g.} the kidney, instead of the liver) or by setting the slice to background (\\emph{i.e.} the organ is not visible in this slice). \n\n\\subsubsection*{Annotation Rate} \nWe collect an equal amount of annotations from each of the three slice directions $d \\in D$ at an annotation rate (AR $\\in [0,1]$). When computing the AR, we annotate every $k$-th slice, where $k = AR^{-1}$. Note, that each slice was annotated at most \\emph{once}, \\emph{i.e.} annotations are \\emph{non-redundant}. \n\n\\begin{figure*}[!h]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{ann_types}\n\\caption{Weak annotation types (top left to bottom right): image, expert manual segmentation (blue), scribbles (SC, green), binary decision making (BD, magenta), rectangular bounding box regions (RR, cyan) and merging of pre-segmentations (PS, yellow). }\n\\label{fig:ann_types}\n\\end{figure*}\n\n\n\\subsection{Annotation strategies}\nFor all experiments, we simulate following forms of weak annotations from expert manual segmentations $M$:\n\n\\subsubsection*{Brush strokes or scribbles (SC)}\nSimilarly to many interactive segmentation methods \\citep{boykov2000interactive,rajchl2012fast,baxter2015optimization}, the user is asked to place a scribble or brush stroke into the object. We simulate placing scribble annotations by the iterative morphological erosion of the manual segmentation $M$ until a maximum of desired scribble size is reached. An example of such a generated SC label is depicted in Fig. \\ref{fig:ann_types} (green).\n\n\\subsubsection*{Binary decision making (BD)}\nThe image is split into $N_{\\textrm{BD}} = ds^2$ equally sized rectangular sub-regions, with the same number of splits ($ds$) per image dimension. For this type of weak annotation, a user is tasked to make a series of binary decisions on which sub-regions contain the object, such that all of the object is contained. We compute these weak annotations (BD) such that $\\forall \\mbox{BD} \\, \\cap \\, M \\neq \\emptyset$. Fig. \\ref{fig:ann_types} shows a BD (magenta) generated from $M$ (blue) for $ds = 4$. \n\n\\subsubsection*{Rectangular (bounding box) regions (RR)}\nSimilarly to \\citep{rother2004grabcut}, the user is asked to draw a tight rectangular region around the object. We compute a bounding box based on the maximum extent of $M$ within the respective image slice. An example RR (cyan) computed from $M$ (blue) is shown in Fig. \\ref{fig:ann_types}.\n\n\\subsubsection*{Merging pre-segmentations (PS)}\nInspired by recent work in \\citep{haehn2014design}, a user merges regions computed from an automated pre-segmentation method. We use a multi-region max flow intensity segmentation with the Potts energy, according to \\citep{yuan2010continuous}: \n\\begin{equation}\nE(u) = \\sum\\limits_{\\forall L} \\int\\limits_{\\Omega}(D_L(x)u_L(x)+ \\alpha_{\\textrm{Potts}} |\\nabla u_L(x)|)dx \\, , \n\\label{eq:potts_energy}\n\\end{equation} \n\\begin{equation}\ns.t. \\, u_L(x) \\geq 0 \\mbox{ and } \\sum\\limits_{\\forall L}u_L(x) = 1\n\\end{equation}\nto obtain piecewise constant regions. The data fidelity term for each label $L = 1,\\ldots,N_L$ is defined as the intensity L1-distance \n\\begin{equation}\nD_L(x)=|I(x)-l_L| \\, , \n\\label{eq:potts_dt}\n\\end{equation}\nwhere $l_L$ denotes the $L$-th most frequent intensity according to the histogram of the image volume. For all experiments, we fix $N_L$ = 16. The GPU-accelerated solver was provided with the ASETS library \\citep{rajchl2016hierarchical}. After convergence, a discrete label map is calculated voxel-wise as $\\argmax_l u_L(x)$.\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{potts}\n\\caption{Example pre-segmentation (PS) results on an abdominal CT volume: Top row (from left to right): CT slice images in transverse, coronal and sagittal direction. Bottom row: Corresponding PS segmentation labels after $\\argmax_l u_L(x)$, when minimising \\eqref{eq:potts_energy}, using \\eqref{eq:potts_dt}, $N_L = 16$ and $\\alpha_{Potts} = 0.05$.}\n\\label{fig:potts}\n\\end{figure*}\n\n\nTo obtain connected individual segments, the obtained segmentation is subsequently partitioned via 4-connected component analysis. Given such pre-segmentation (PS), the user is tasked to merge subregions, such that they contain the object of interest (\\emph{i.e.} the liver). We simulate the merging similar to BD, such that $\\forall \\mbox{PS} \\, \\cap \\, M \\neq \\emptyset$. A simulated PS annotation is shown in Fig. \\ref{fig:ann_types} in yellow and the corresponding $M$ in blue. \n\n\\subsection{Annotations as Segmentation Priors}\n\\label{sec:fuse_ann}\nTo be employed as priors in a volume segmentation problem, the annotations from individual slice directions $d \\in D$ need to be consolidated to account for voxels $x$ located on intersecting slices: \\\\\nThe binary SC annotations from all slice directions $d \\in D$ are combined to a volume annotation $SC_{Vol}$\n\\begin{equation}\nSC_{Vol}(x) \\, = \\, \\cup_{d=1}^D \\, SC_d(x) \\, \\, ,\n\\end{equation}\nand employed as foreground samples $S_{FG} = SC_{Vol}$. \\\\\n\nAll binary annotations $A_d \\in \\{BD, RR, PS\\}$ and all unannotated slices $U_d$ are similarly combined to volumes to establish $S_{BG}$. Note, that $A_d(x) = 1$ denotes the user rated the location $x$ as \"foreground\" and $U_d(x) = 1$ denotes, that the user has not seen this location.\n\\begin{equation}\nA_{Vol}(x) = \\, \\cup_{d=1}^D \\, A_d(x) \\,; \\, \\, \\, \\, U_{Vol}(x) = \\, \\cap_{d=1}^D \\, U_d(x) \\,;\n\\end{equation}\nThe background samples $S_{BG}$ are computed as all voxels $x$ that are outside $A_{Vol}$ and \\emph{are} annotated:\n\\begin{equation}\nS_{BG}(x) \\, = \\, \\neg \\, A_{Vol}(x) \\, \\, \\cap \\,\\, \\, \\neg \\, U_{Vol}(x) . \n\\end{equation}\n\nThe resulting samples $S_{FG}$ and $S_{BG}$ can then be used to compute intensity models or to enforce spatial constraints. For all experiments, we employ SC annotations as priors for foreground voxels and \\{BD, RR, PS\\} annotations as priors for background voxels. For each of these three combinations (\\emph{e.g.} SC and BD, \\emph{etc.}), we calculate $S_{FG}$ and $S_{BG}$ for each volume image to be segmented.\n\n\n\n\n\\subsection{Segmentation Problem}\nThe method employed to obtain a segmentation can be considered as a black box to be replaced by any specialised pipeline that suits a specific problem. For our experiments, we employ a well-known interactive flow maximisation \\citep{boykov2000interactive,rajchl2012fast} approach to compute image segmentations from the input annotations $A$, subject to a certain $AR$ and $ERR$. For this purpose, we use the continuous max-flow solver \\citep{yuan2010study,rajchl2016hierarchical} supporting GPU acceleration and allowing us to tackle the computational load for our experiments. We find a solution by minimising an energy $E(u)$ defined for the labelling or indicator function $u$ at each voxel location $x$ in the image $I$, $ \\mbox{ s.t. } u(x) \\in [ 0, 1 ]$ as, \n\n\\begin{equation}\nE(u) = \\int\\limits_{\\Omega}(D_s(x)u(x) + D_t(x)(1-u(x))+ \\alpha|\\nabla u(x)|)dx \\, , \\\\\n\\label{eq:binary_e}\n\\end{equation}\nHere, the data fidelity terms $D_{s,t}(x)$ are defined as the negative log-likelihood of the probabilities $\\omega_{1,2}$, computed from normalised intensity histograms of the foreground (FG) and background (BG) region, respectively,\n\\begin{equation}\nD_s(x) \\, = \\, -log(\\omega_1 (I(x))) \\, \\mbox{ and } \\, \nD_t(x) \\, = \\, -log(\\omega_2 (I(x))) \\, , \\\\\n\\label{eq:ll_data_term}\n\\end{equation}\nas described in \\citep{boykov2001interactive}. Additionally, we employ a soft spatial constraint by setting the cost for regions annotated as FG and BG, to a minimum:\n\n\\begin{align}\nD_s(x) = 0, \\, \\, \\forall x \\in \\mbox{FG}; \\, \\, D_t(x) = 0, \\, \\, \\forall x \\in \\mbox{BG};\n\\label{eq:soft_constraints}\n\\end{align}\n\nConsolidated volume annotations (see Section \\ref{sec:fuse_ann}) are used to compute samples $S_{FG}$ and $S_{BG}$ of FG and BG, respectively. $S_{FG}$ and $S_{BG}$ are subsequently employed to compute $\\omega_{1,2}$ in \\eqref{eq:ll_data_term} and as spatial constraints in \\eqref{eq:soft_constraints}.\nAfter optimisation of the energy in \\eqref{eq:binary_e}, the resulting continuous labelling function $u$ is thresholded at 0.5 to obtain a discrete segmentation result for the FG, as described in \\citep{yuan2010study}.\n\n\\subsection{Outlier Detection \\& Removal}\n\\label{sec:outlier_detection}\nWe propose a method for quality assessment to mitigate the impact of annotation errors on the accuracy of the segmentation results (\\emph{e.g.} when using databases labelled by crowds with low expertise). Note, that contrary to other studies \\citep{mckenna2012strategies,lin2014microsoft}, we do \\emph{not} require redundant annotations for outlier detection. Instead, we propose to make use of redundant information in the \\emph{flawed} and \\emph{weakly labelled} atlas database and retrieve similar image slices and their annotations in the fashion of multi-atlas segmentation pipelines \\citep{wolz2012multi,aljabar2009multi}.\n\nIf spatial variability is accounted for (\\emph{e.g.} through registration), we can retrieve slices from other atlas volumes and use their annotations to compute an agreement measure to rate a given annotation. For this purpose, we borrow from the concept of the SIMPLE method \\citep{langerak2010label}, where an iteratively refined agreement is used to assess the quality of individual atlases in a multi-atlas segmentation approach.\n\n\\subsubsection*{Weakly Labelled Atlas as Quality Reference}\nWe assume that $S$ subjects $s_i=\\{s_1,\\ldots,s_S\\}$ have been weakly (and potentially erroneously) annotated and aim to automatically detect the slices of each subject $s_i$ that have an annotation of insufficient quality (\\emph{e.g.} the wrong organ has been annotated or the organ was present, but not detected). \nIn the following, the $j$-th slice of subject $s_i$ in direction $d$ is denoted by $v_{i}^{j,d}$. For each slice in the database $v_{i}^{j,d}$, we first find a subset of the most similar spatially corresponding images $v_{q}^{j,d}$ of the subjects $s_q$ in the \\emph{weakly labelled atlas} using a global similarity measure, such as the sum of squared differences. \nWe then calculate a consensus segmentation $\\bar{O_1}$ from the annotations of these anatomically similar image slices using mean label fusion. For each of these selected atlas annotations, the overlap between the annotation and the estimated consensus segmentation is calculated with an accuracy metric.\n\nFor this purpose, we use the Dice similarity coefficient (DSC) between the regions $A$ and $B$ as a measure of overlap:\n\\begin{equation}\nDSC = \\frac{2 |A| \\cap |B|}{|A| + |B|}\n\\label{eq:dsc}\n\\end{equation} \n\nUsing the mean regional overlap $\\mu_\\textrm{DSC}^1$ between the atlas annotations and the consensus segmentation $\\bar{O_1}$, we can discard potentially inaccurately annotated atlas slices if their DSC with $\\bar{O_1}$ is less than this average $\\mu_\\textrm{DSC}^1$. Following \\citep{langerak2010label}, we calculate another fusion estimate $\\bar{O_2}$ using the reduced subset of both anatomically similar and reasonably accurate annotations and calculate another mean DSC, $\\mu_\\textrm{DSC}^2$, and reject the annotations corresponding to $v_{i}^{j,d}$ if its DSC with $\\bar{O_2}$ is less than $\\mu_\\textrm{DSC}^2$.\n\nThis procedure is repeated for each annotation in the database. Note that the \\emph{weakly labelled atlas} can be built from the database itself so that no external\/additional input is required. An illustration of the approach is provided in Fig. \\ref{fig:outlier_detection}.\n\n\\begin{algorithm}[h]\n \\KwData{weak annotation for $v_{i}^{j,d}$: $wa_{i}^{j,d}$\\;\n corresponding WAs in the weakly labelled atlas $Q$: $wa_{q}^{j,d}$, $q \\in Q$}\n \\KwResult{$v_{i}^{j,d}$ is outlier: yes\/no}\n $Q^1 \\leftarrow \\{ p \\in Q: |Q^1|=N_\\textrm{similar}$ and $\\sum_{q\\in Q^{1}}||v_{i}^{j,d}-v_{q}^{j,d}|| \\rightarrow min$ \\}\\;\n i = 1\\;\n \\While{ i $\\le$ N$_\\textrm{iterations}$}{\n $\\bar{O_i} \\leftarrow$ MajorityVote$(wa_{q}^{j,d} ~ \\forall q\\in Q^{i}$)\\;\n $\\mu_i \\leftarrow $ Average( Dice( $\\bar{O_i}, wa_{q}^{j,d}$) ~ $\\forall q\\in Q^{i}$)\\;\n $Q^{i+1} \\leftarrow \\{ p \\in Q^{i}:$ Dice($\\bar{O_i}, wa_{q}^{j,d}) \\ge \\mu_i\\}$\\;\n$i \\leftarrow i+1$\n}\n\\eIf{Dice($wa_{i}^{j,d}, Q^{N_\\textrm{iterations}}$) $\\ge \\mu_{N_\\textrm{iterations}}$}\n {\n return yes\\;\n }{\n return no\\;\n }\n\\caption{Outlier detection using a weakly labelled, flawed atlas database.}\n\\end{algorithm}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{outlier_detection}\n\\caption{Schematic illustration of the outlier detection and removal approach.}\n\\label{fig:outlier_detection}\n\\end{figure}\n\n\n\\section{Experiments}\n\\subsubsection*{Image Database}\nThe image database used in the experimental setup consists of 150 (114 \\mars, 36 \\female) abdominal volume CT images with corresponding manual segmentations from expert raters. Available labelled anatomical regions include the liver, spleen, pancreas and the kidneys. All scans were acquired at the Nagoya University Hospital with a TOSHIBA Aquilion 64 scanner and obtained under typical clinical protocols. The volume images were acquired at an in-plane resolution of 512 x 512 voxels (spacing 0.55 to 0.82 mm) and contain between 238 and 1061 slices (spacing 0.4 to 0.8 mm).\n\n\n\\subsubsection*{Pre-processing \\& Generation of Weak Annotations}\nPrior to the experiments, all volume image data were affinely registered using the NiftiReg library \\citep{modat2010fast} (default parameters) to a random subject to spatially normalize the images and to account for variability in size. Weak annotations are generated for each slice in each direction $d$ in all volume images of the database, subject to the annotation rate $AR = \\{1, 0.5, 0.33, 0.25, 0.1, 0.05, 0.01\\}$ and the error rate $ERR = \\{0, 0.05, 0.1, 0.25, 0.5\\}$.\n\n\\subsubsection*{Liver Segmentation with Weak Annotations}\nThe weak annotations are fused to compute $S_{FG}$ and $S_{BG}$ (see \\ref{sec:fuse_ann}) to subsequently compute the data terms $D_{s,t}(x)$ in \\eqref{eq:ll_data_term} and the soft constraints in \\eqref{eq:soft_constraints}. A continuous max-flow segmentation \\citep{yuan2010study}, minimizing \\eqref{eq:binary_e} is computed to obtain a segmentation result.\nThe regularisation parameter $\\alpha = 4$ in \\eqref{eq:binary_e} and the parameters $\\alpha_{Potts} = 0.05$ and $N_L = 16$ in \\eqref{eq:potts_energy} were determined heuristically on a single independent dataset. For the outlier detection, $N_{iterations}$ was set to 2. All experiments were performed on an Ubuntu 14.04 desktop machine with a Tesla 40c (NVIDIA Corp., Santa Clara, CA) with 12 GB of memory.\n\n\\subsubsection*{Experimental Setup}\n20 consecutively acquired subject images are used as a subset to examine the impact of AR, ERR and type of weak annotation on the mean segmentation accuracy. An average DSC is reported as a measure of accuracy between the obtained segmentations and the expert segmentations $M$ for all the parameter combinations of ERR, AR and all examined annotation types (SC in combination with \\{BD, RR, PS\\}), resulting in 2100 single volume segmentations results.\n\nFurther, the proposed outlier detection (see Section \\ref{sec:outlier_detection}) is employed using annotations from all 150 subjects. The annotations after outlier removal are used for repeated segmentations, resulting in additional 2100 segmentations. A study on atlas selection \\citep{aljabar2009multi} suggests an optimal subset size for brain segmentation to be 20. We increased $N_{similar}$ of the globally similar atlases $Q^1$ to 30, to account for the variation in abdominal soft tissue organs, such as the liver.\n\nA series of paired T-tests is computed to determine significant changes in accuracy at a $p = 0.05$ level between resulting DSC before and after outlier detection.\n\n\\section{Results}\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{vis_res_slices_lbls}\n\\caption{Example segmentation results (from top left to bottom right): expert manual segmentation (magenta), segmentation with $< 0.8$ (blue), $\\approx 0.85$ (cyan), $\\approx 0.9$ (orange) and $\\approx 0.95$ accuracy according to DSC. The colour coding is chosen to reflect those of accuracy matrices in Fig. ~\\ref{fig:results_mean}.}\n\\label{fig:results_visual}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{vis_res_90}\n\\caption{Surface rendering of example segmentation results. The colour coding are chosen to reflect those of accuracy matrices in Fig. ~\\ref{fig:results_mean}.}\n\\label{fig:results_visual_surf}\n\\end{figure}\n\n\\subsubsection*{Segmentation Accuracy}\nFig. \\ref{fig:results_visual} depicts example segmentations of transverse slices on a single subject as comparative visual results with obtained DSC ranges. Visual inspection suggests that a DSC \\textgreater 0.9 can be considered an acceptable segmentation result. A DSC of lower than 0.8 can be considered a segmentation failure. Mean accuracy results of all examined methods are shown in Fig. \\ref{fig:results_mean}. \nThe main contribution to a decrease in accuracy was observed to be high error rates. Without outlier correction, acceptable segmentation results could be obtained with all annotation types, down to an AR of 25\\%. Using rectangular regions, this accuracy can still be obtained when annotating 1\\% of available slices, when the ERR is simultaneously less than 5\\%. In a densely annotated database (\\emph{i.e.} AR = 100\\%) more than 10\\% of erroneous annotations lead to segmentation failure. This is particularly interesting for medical image analysis studies considering a crowdsourcing approach with \\emph{non-redundant annotations} of non-experts. This effect still persists to a degree, after outlier correction at the highest tested ERR of 50\\%.\n\n\\subsubsection*{Performance after Outlier Correction}\nThe mean accuracy improves after the proposed outlier correction, however slight decreases in accuracy are observed at lower error rates. This is mainly due to the decreased number of available annotations after correction. The differences in mean DSC after outlier removal range from $-0.05$ to $+0.94$ (BD), $-0.0006$ to $+0.94$ (RR) and $-0.02$ to $+0.92$ (PS). Statistically significant changes are visualised in Figure \\ref{fig:results_mean} (bottom row) and numerical ranges reported in Tab. \\ref{tab:acc_diff}. \n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{results_mean}\n\\caption{Top Row: Accuracy results for each annotation type, subject to AR and ERR, without outlier detection. Middle Row: Accuracy results with proposed outlier detection. Bottom Row: Results of paired T-tests between top and middle row (p \\textless 0.05). Increased and decreased mean accuracy are depicted in red and blue, respectively. Black elements show no significant difference.}\n\\label{fig:results_mean}\n\\end{figure*}\n\n\n\\begin{table}[!h]\n\\label{tab:acc_diff}\n\\centering\n\\caption{Minimal and maximal changes in DSC accuracy after outlier removal, for all tested ERR and AR and each annotation type.}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\nANN Type & \\textbf{BD} & \\textbf{RR} & \\textbf{PS} \\\\\n\\hline\nincr. $N$ (DSC) & 8 (+0.94) & 18 (+0.94) & 10 (+0.92) \\\\\ndecr. $N$ (DSC) & 14 (-0.05) & 3 (-0.0006) & 5 (-0.04) \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table*}[!h]\n\\label{tab:acc_diff}\n\\centering\n\\caption{Mean segmentation accuracy with decreasing annotation rate (AR) and ERR = 0. All results as DSC [\\%].}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|}\n\\hline\n& \\multicolumn{7}{c|}{Annotation Rate (AR) [\\%]} \\\\ \\hline\nType & 100 & 50 & 33 & 25 & 10 & 5 & 1 \\\\\n\\hline\nBD & \\, 94.3 (0.8) & \\, 94.3 (1.2) & \\, 94.0 (1.4) & \\, 93.3 (2.2) & \\, 88.7 (3.7) & \\, 87.0 (4.0) & \\, 86.9 (3.7) \\\\\nRR & \\, 94.6 (0.8) & \\, 94.5 (0.8) & \\, 94.4 (0.8) & \\, 94.3 (0.8) & \\, 94.1 (0.9) & \\, 93.2 (1.2) & \\, 92.1 (1.9) \\\\\nPS & \\, 92.1 (1.4) & \\, 92.7 (1.4) & \\, 92.9 (1.4) & \\, 92.8 (1.8) & \\, 89.6 (4.0) & \\, 88.2 (4.8) & \\, 88.8 (4.1) \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\section{Discussion}\nIn this study, we tested types of weak annotations to be used for liver segmentation from abdominal CTs. We examined the effects of different expertise levels and annotation rates of crowd populations for their impact on the segmentation accuracy outcome and proposed a method to remove potential incorrect annotations. In the conducted experiments each of the slices was annotated at most \\emph{once}, reducing the effort associated with the acquisition of redundant annotations.\n\n\\subsubsection*{Segmentation Accuracy}\nWhile the max-flow segmentation was employed without any problem-specific adaptions, it yields comparably high accuracy to a state-of-the-art hierarchical multi-atlas approach described in \\citep{wolz2012multi}, where a mean DSC of 94.4\\% was reported for the segmentation of the liver. Comparable accuracy was obtained when employing RR annotations at AR = 10\\% and no errors present or for BD at AR = 50\\% and ERR = 10\\%. After the proposed outlier correction, using RR at an AR of 25\\%, errors of up to 25\\% yielded similarly high accuracy - a scenario realistic enough to be obtained by a non-expert crowd. Both BD and PS annotation types performed similarly robust to RR at higher annotation rates and yielded acceptable accuracy at AR down to 25\\% and ERR of up to 25\\% without outlier correction. \n\n\\subsubsection*{Impact of Expertise and Annotation Rate}\nAn expected decrease in accuracy with both higher ERR and lower AR is observed for all annotation types. We report more robust behaviour of the RR annotations and at a wide range of ERR and at an AR of down to 5\\%. For BD and PS annotations, AR of less than 25\\% yielded insufficient accuracy, even at the highest expertise levels (\\emph{i.e.} ERR = 0). For all annotation types, the presence of errors has a larger impact at high AR, suggesting that the total amount of incorrectly annotated image slices is related with segmentation failure, rather than its rate. Without correction, high ERR were tolerated in annotation rates of 1-10\\%, however lead to segmentation failure (\\emph{i.e.} DSC \\textless 0.8) at higher AR. This suggests that an increased number of annotations is not beneficial if performed at an high error rate. \n\n\\subsubsection*{Outlier Correction}\nThe proposed \\emph{weakly labelled atlas}-based outlier detection approach performed well, yielding maximal improvements of \\textgreater 0.9 DSC in accuracy, which is particularly observed in presence of high error rates (see Fig. \\ref{fig:results_mean}). Its application allows to obtain high quality (DSC \\textgreater 0.9) segmentations at the maximum tested ERR of 50\\%. At lower ERR, small decreases in accuracy are found. These are associated with a decrease in AR due to outlier removal. This effect can be seen when no annotation errors were present in the atlas prior to outlier correction (\\emph{i.e.} ERR = 0\\%). Fig. \\ref{fig:results_mean} nicely illustrates the existence of an upper accuracy bound (\\emph{i.e.} where ERR = 0\\%) and the comparable performance at higher ERR after correction. \n\n\n\n\\subsection*{Conclusions}\nWe tested forms of weak annotations to be used in medical image segmentation problems and examined the effects of expertise and frequency of annotations in their impact on accuracy outcome. Resulting segmentation accuracy was comparable to state-of-the-art performance of a fully-supervised segmentation method and the proposed outlier correction using a \\emph{weakly labelled atlas} was able to largely improve the accuracy outcome for all examined types of weak annotations. The robust performance of this approach suggests that weak annotations from non-expert crowd populations could be used obtain accurate liver segmentations and the general approach can be readily adapted to other organ segmentation problems.\n\n\n\\section*{Acknowledgements}\nWe gratefully acknowledge the support of NVIDIA Corporation with the donation of a Tesla K40 GPU used for this research. This work was supported by Wellcome Trust and EPSRC IEH award [102431] for the iFIND project and the Developing Human Connectome Project, which is funded through a Synergy Grant by the European Research Council (ERC) under the European Union's Seventh Framework Programme (FP\/2007-2013) \/ ERC Grant Agreement number 319456. \n\n\n\\bibliographystyle{model2-names}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Useful facts}\n\\label{sec:useful-facts}\n\n\nIn this section,\nwe jot down basic definitions and facts\nrelated Stieltjes transform that we will be using\nthroughout the paper.\n\nLet $Q$ be a bounded nonnegative measure on $\\mathbb{R}$.\nThe Stieltjes transform of $Q$ is defined at $z \\in \\mathbb{C}^{+}$\nby\n\\[\n m_Q(z) = \\int_{\\mathbb{R}} \\frac{1}{x - z} \\, \\mathrm{d}Q(x).\n\\]\n\n\n\n\\begin{fact}\n \\label{fact:lim_stiletjes_im}\n Let $m$ be the Stieltjes transform of bounded measure $Q$ on $\\mathbb{R}_{\\ge 0}$.\n Let $z \\in \\mathbb{C}^{+}$ with $\\Re(z) < 0$.\n Then, $\\Im(m(z)) \\to 0$ as $\\Im(z) \\to 0$.\n\\end{fact}\n\\begin{proof}\n Let $z = x + i y$ with $x < 0$ and $y > 0$.\n Since $m$ is a Stieltjes transform of $Q$,\n we have\n \\[\n \\Im(m(z)) \n = \\Im \\left( \\int \\frac{1}{r - z} \\, \\mathrm{d}Q(r) \\right)\n = \\Im \\left( \\int \\frac{1}{r - (x + i y)} \\, \\mathrm{d}Q(r) \\right)\n = \\int \\frac{-y}{(r - x)^2 + y^2} \\, \\mathrm{d}Q(r).\n \\]\n Thus,\n we can bound\n \\[\n | \\Im(m(z)) |\n \\le \\frac{y}{x^2} \\int \\mathrm{d}Q(r).\n \\]\n Since $Q$ is a bounded measure, by letting $y \\to 0$, one has $\\Im(m(z)) \\to 0$ \n as $\\Im(z) \\to 0$.\n\\end{proof}\n\n\nWe will be interested in the Stieltjes transforms of spectral measures.\nThe spectral distribution of a symmetric matrix ${\\mathbf{A}} \\in \\mathbb{C}^{p \\times p}$\nwith eigenvalues $\\lambda_1({\\mathbf{A}}), \\dots, \\lambda_p({\\mathbf{A}})$\nis the probability distribution that places a point mass of $\\tfrac{1}{p}$ at each\neigenvalue\n\\[\n F_\\mA(\\lambda) = \\tfrac{1}{p} \\sum_{i=1}^{p} \\mathbbm{1}\\{ \\lambda_i \\le \\lambda \\}.\n\\]\nThe matrices of interest for us will be the population covariance matrix \n${\\bm{\\Sigma}} \\in \\mathbb{C}^{p \\times p}$\nand the sample covariance matrix \n$\\tfrac{1}{n} {\\mathbf{X}}^\\ctransp {\\mathbf{X}}$ where ${\\mathbf{X}} \\in \\mathbb{C}^{n \\times p}$ is the random design matrix.\n\nIf the Stieltjes transform \nof spectrum of the sample covariance matrix $\\tfrac{1}{n} \\mX^\\ctransp \\mX$\nis \n\\begin{equation}\n \\label{eq:stieltjes-transform}\n m(z) = \\tfrac{1}{p} \\tr[(\\tfrac{1}{n} \\mX^\\ctransp \\mX - z \\mI_p)^{-1}],\n\\end{equation}\nthen the so-called \\emph{companion} Stieltjes transform \n\\begin{equation}\n \\label{eq:companion-stieltjes-transform}\n v(z) = \\tfrac{1}{n} \\tr[(\\tfrac{1}{n} \\mX \\mX^\\ctransp - z \\mI_n)^{-1}] \n\\end{equation}\nis the Stieltjes transform of $\\tfrac{1}{n} \\mX \\mX^\\ctransp$\n(and hence the prefix).\nThe reason it is useful is that\nit is often easier to work with the companion Stieltjes transform\nthan the Stieltjes transform.\nThe following fact relates the companion Stieltjes transform\nto the Stieltjes transform.\n\n\\begin{fact}\n\\label{fact:stieltjes-companion-stieltjes-relation}\nThe companion Stieltjes transform $v(z)$ \ncan be expressed in terms of the Stieltjes transform $m(z)$ at $z \\in \\mathbb{C}^{+}$ as\n\\begin{equation}\n \\label{eq:stieltjes-companion-stieltjes-relation}\n v(z) = \\frac{p}{n} m(z) + \\frac{1}{z} \\left(\\frac{p}{n} - 1\\right).\n\\end{equation}\n\\end{fact}\n\\begin{proof}\n Let $(\\lambda_i)_{i = 1}^r$ be the nonzero eigenvalues of \n $\\tfrac{1}{n} {\\mathbf{X}}^\\ctransp {\\mathbf{X}}$\n (which are also the nonzero eigenvalues of $\\tfrac{1}{n} {\\mathbf{X}} {\\mathbf{X}}^\\ctransp$).\n Define $\\Lambda(z) = \\sum_{i=1}^r \\tfrac{1}{\\lambda_i - z}$.\n From \\cref{eq:stieltjes-transform,eq:companion-stieltjes-transform},\n note that we can write\n \\begin{align}\n m(z) = \\frac{\\Lambda(z)}{p} - \\frac{(p - r)}{pz}, \\quad\n v(z) = \\frac{\\Lambda(z)}{n} - \\frac{(n - r)}{nz}.\n \\end{align}\nCombining these equations proves the claim.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proofs in \\cref{thm:sketched-pseudoinverse}}\n\\label{sec:proofs-main-results}\nWe first present the remaining details of the proof that extends the argument to positive semidefinite $\\mA$, and then we present the alternative proof sketch based on Jacobi's formula.\n\n\\subsection{Proof of \\cref{thm:sketched-pseudoinverse} for positive semidefinite $\\mA$}\n\\label{sec:proof:thm:sketched-pseudoinverse}\n\n\\begin{proof}\nWe begin by proving the equivalence \\cref{eq:thm:sketched-pseudoinverse} and then show that the limit as $\\lambda \\to 0$ is well-behaved when we multiply by $\\mA^{1\/2}$ to obtain \\cref{eq:thm:sketched-pseudoinverse-A-half}.\n\nLet $\\mA_\\delta \\defeq \\mA + \\delta \\mI_p$, $\\mU \\defeq \\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q} \\mS^\\ctransp$, and $\\mV \\defeq \\inv{\\mA + \\mu \\mI_p}$. By the Woodbury matrix identity, we have the following two identities:\n\\begin{gather}\n \\mS \\biginv{\\mS^\\ctransp \\mA_\\delta \\mS + \\lambda \\mI_q} \\mS^\\ctransp\n = \\mU - \\delta \\mU \\inv{\\mI_p + \\delta \\mU} \\mU, \\\\\n \\inv{\\mA_\\delta + \\lambda \\mI_p}\n = \\mV - \\delta \\mV \\inv{\\mI_p + \\delta \\mV} \\mV.\n\\end{gather}\nIf either $\\lambda \\neq 0$ or $\\limsup \\tfrac{q}{p} < \\liminf r(\\mA)$, then we can conclude that (see, e.g., \\cite{bai_silverstein_1998}) that $\\bignorm[\\rm op]{\\biginv{\\mS^\\ctransp \\mA_\\delta \\mS + \\lambda \\mI_q}}$ is almost surely uniformly bounded and that $\\mu$ is bounded away from zero (see \\cref{rem:joint-signs-lambda-mu}). Thus, since $\\norm[\\rm op]{\\mS}$ is also almost surely bounded asymptotically, $\\norm[\\rm op]{\\mU}$ and $\\norm[\\rm op]{\\mV}$ are asymptotically bounded by constants $C_\\mU$ and $C_\\mV$, respectively. Therefore, for $\\delta < \\tfrac{1}{2}\\min\\set{C_\\mU, C_\\mV}$, we have the following bound on the trace functional difference:\n\\begin{multline}\n \\limsup \\big| \\tr \\bigbracket{\\mTheta \\bigparen{\\mS \\biginv{\\mS^\\ctransp \\mA_\\delta \\mS + \\lambda \\mI_q} \\mS^\\ctransp - \\inv{\\mA_\\delta + \\lambda \\mI_p}}}\n - \\tr \\bigbracket{\\mTheta \\bigparen{\\mU - \\mV}}\n \\big| \\\\\n \\leq \\tfrac{\\delta}{2} \\norm[\\tr]{\\mTheta} \\bigparen{C_\\mU^2 + C_\\mV^2}.\n\\end{multline}\nThus, as $\\delta \\searrow 0$, the trace functionals converge uniformly over $p$. We can therefore apply the Moore--Osgood Theorem to interchange limits, such that almost surely\n\\begin{align}\n \\lim_{p \\to \\infty} \\big| \\tr \\bigbracket{\\mTheta \\bigparen{\\mU - \\mV}}\n \\big| &= \n \\lim_{\\delta \\searrow 0} \\lim_{p \\to \\infty} \n \\big|\n \\tr \\bigbracket{\\mTheta \\bigparen{\\mS \\biginv{\\mS^\\ctransp \\mA_\\delta \\mS + \\lambda \\mI_q} \\mS^\\ctransp - \\inv{\\mA_\\delta + \\lambda \\mI_p}}}\n \\big| \\\\\n &= 0.\n\\end{align}\n\nTo prove the equivalence in \\cref{eq:thm:sketched-pseudoinverse-A-half}, we can apply the equivalence in \\cref{eq:thm:sketched-pseudoinverse} proved above unless $\\lambda = 0$ and $\\limsup \\tfrac{q}{p} \\geq \\liminf r(\\mA)$. We need only consider $\\limsup \\lambda_0 < 0$, so it suffices to consider $\\liminf \\tfrac{q}{p} > \\limsup r(\\mA)$ (see \\cref{rem:mu0-lambda0-vs-alpha}).\nThe condition $\\limsup \\lambda_0 < 0$ implies that there exists $c_\\lambda > 0$ such that $\\lambda_{\\min}^{+}(\\mS^\\ctransp \\mA \\mS) > c_\\lambda$. Therefore, $\\bignorm[\\rm op]{\\mA^{1\/2} \\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q}}$ is almost surely uniformly bounded in $p$ for all $\\lambda \\in D_\\lambda$, where $D_\\lambda = \\bigset{z \\in \\complexset \\colon |z| < \\tfrac{c_\\lambda}{2}}$. We now need to bound $\\bignorm[\\rm op]{\\mA^{1\/2} \\inv{\\mA + \\mu \\mI_p}}$. \nFrom the definition of $\\mu_0$ in \\cref{eq:mu0-lambda0-fps},\nwe observe that\n\\begin{align}\n \\frac{p}{q}\n \\frac{r(\\mA) \\lambda_{\\max}(\\mA)^2}{ (\\lambda_{\\max}(\\mA) + \\mu_0)^2} \n \\leq 1 \\leq \n \\frac{p}{q}\n \\frac{r(\\mA) \\lambda_{\\min}^{+}(\\mA)^2}{ (\\lambda_{\\min}^{+}(\\mA) + \\mu_0)^2},\n\\end{align}\nfrom which we can conclude for the case that $\\tfrac{q}{p} > r(\\mA)$ and $\\lambda_{\\min}^{+}(\\mA) > 0$,\nwe can bound\n\\begin{equation}\n \\label{eq:mu-0-upper-lower-bounds}\n \\paren{\\tfrac{p r(\\mA)}{q} - 1} \\lambda_{\\max}(\\mA) < \n \\paren{\\sqrt{\\tfrac{p r(\\mA)}{q}} - 1} \\lambda_{\\max}(\\mA)\n \\leq \\mu_0 \\leq \n \\paren{\\sqrt{\\tfrac{p r(\\mA)}{q}} - 1} \\lambda_{\\min}^{+}(\\mA) < 0.\n\\end{equation}\nSince $\\liminf \\tfrac{q}{p} > \\limsup r(\\mA)$ and $\\liminf \\lambda_{\\min}^{+}(\\mA) > 0$, we therefore must have $\\limsup \\mu_0 < 0$.\nDefine the set $D_{\\mu} = \\bigset{z \\in \\complexset \\colon |z| < \\tfrac{-\\limsup \\mu_0}{2}}$. Since $-\\liminf \\lambda_{\\min}^{+}(\\mA) \\leq \\mu_0$, for all $\\mu \\in D_{\\mu}$, we must have the bound\n\\begin{equation}\n \\bignorm[\\rm op]{\\mA^{1\/2} \\inv{\\mA + \\mu \\mI_p}} \\leq \\frac{2\\bignorm[\\rm op]{\\mA^{1\/2}}}{-\\limsup \\mu_0}.\n\\end{equation}\nWe also know from \\cref{eq:sketched-modified-lambda} that\n\\begin{equation}\n |\\lambda| = |\\mu| \\big| 1 - \\tfrac{1}{q} \\tr \\bigbracket{\\mA \\inv{\\mA + \\mu \\mI_p}} \\big|.\n\\end{equation}\nOne can confirm that the second factor on the right-hand side is uniformly lower bounded away from 0 for $\\mu \\in D_\\mu$\nusing the first bound in \\cref{eq:mu-0-upper-lower-bounds}. Let $D_p = \\set{\\lambda : \\mu(\\lambda) \\in D_\\mu}$ be the inverse image of $D_\\mu$ under the map $\\lambda \\mapsto \\mu$ for each $p$. By the above arguments, the set $D = D_\\lambda \\cap \\limsup D_p$ is an open set over which the functions \n\\begin{equation}\n f_p(\\lambda) = \\big|\\tr \\bigparen{\\bigbracket{\\mTheta \\mA^{1\/2} \\mS \\big( \\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q \\big)^{-1} \\mS^\\ctransp} - \\mA^{1\/2} (\\mA + \\mu \\mI_q )^{-1}} \\big|\n\\end{equation}\nconverge uniformly as $\\lambda \\to 0$ over $p$. By Montel's theorem, these functions form a normal family. Since $f_p(\\lambda) \\to 0$ pointwise for $\\lambda \\neq 0$, this implies that $f_p(0) \\to 0$.\n\n\\end{proof}\n\n\\subsection{Alternative proof sketch for \\cref{thm:sketched-pseudoinverse} via Jacobi's formula}\n\\label{sec:proofs-main-results-jacobi}\n\nBelow we provide an alternate strategy for proving \\Cref{thm:sketched-pseudoinverse}\nvia Jacobi's formula.\nWe only sketch the outline for the proof \nand do not attempt to make it completely rigorous.\nBut nonetheless we think the idea behind it is interesting and decided to include it, in hopes that it may be useful, perhaps for proving the asymptotically free sketching result we propose in \\cref{conj:general-free-sketching}.\nIt is in fact how we first discovered the statement of \\Cref{thm:sketched-pseudoinverse} ourselves\nand only in the retrospect proved it using the shorter argument\nthat uses the Woodbury matrix identity\npresented in \\Cref{sec:proof:thm:sketched-pseudoinverse}\n\n\\begin{proof}[Proof sketch]\n We note that without loss of generality, we can consider positive semidefinite $\\mTheta$, since we can always replace $\\mTheta$ with $\\frac{1}{2} (\\mTheta + \\mTheta^\\ctransp)$ and evaluate separately $\\mTheta_+$ and $\\mTheta_-$, where the latter refer to the restriction of symmetric $\\mTheta$ to the eigenspaces corresponding to positive and negative eigenvalues, respectively.\n \n Define $\\mB_t \\defeq \\mA + t \\mTheta$ and $\\mB_t^\\mS \\defeq \\mS^\\ctransp \\mB_t \\mS$. Observe that $\\log \\abs{\\mB_t + \\lambda \\mI}$ is a monotonic function of $\\lambda$ that ranges from $-\\infty$ to $\\infty$ for $\\lambda \\in (-\\lambda_{\\mathrm{min}}(\\mB_t), \\infty)$, where $\\lambda_{\\mathrm{min}}(\\mB_t)$ is the smallest eigenvalue of $\\mB_t$; therefore there exist\\footnote{The existence of $\\mu$ is not unique since this relation can hold for any $D$.} $D \\in \\reals$ and $\\mu$ such that for $\\lambda > -\\lambda_{\\mathrm{min}}(\\mB_t^\\mS)$, \n \\begin{align}\n \\tfrac{1}{p} \\log \\abs{\\mB_t^\\mS + \\lambda \\mI_q} = \\tfrac{1}{p} \\log \\abs{\\mB_t + \\mu(\\lambda) \\mI_p} + D + o(1).\n \\end{align}\n Here a function $f(t) \\in o(g(t))$ if $\\lim_{t \\to 0} \\big|\\tfrac{f(t)}{g(t)}\\big| = 0$.\n Now, using Jacobi's formula, the fundamental theorem of calculus, and Leibniz' integral rule,\n \\begin{align}\n &\\tfrac{1}{p} \\tr \\bracket{\\mTheta \\paren{\\mS \\inv{\\mB_0^\\mS + \\lambda \\mI_q}\\mS^\\ctransp - \\inv{\\mB_0 + \\mu(\\lambda) \\mI_p}}} \\\\\n &= \\tfrac{1}{p} \\tfrac{\\partial}{\\partial t} \\paren{ \\log \\abs{\\mB_t^\\mS + \\lambda \\mI_q} - \\log \\abs{\\mB_t + \\mu(\\lambda) \\mI_p}} \\Big|_{t=0} \\\\\n &= \\tfrac{\\partial}{\\partial t}\n \\left. \\paren{\\int_{\\lambda_0(t)}^\\lambda \\tfrac{1}{p} \\tr\\bracket{\\inv{\\mB_t^\\mS + u \\mI_q}} - \\tfrac{1}{p} \\tr\\bracket{\\inv{\\mB_t + \\mu(u) \\mI_p}} \\tfrac{\\partial \\mu(u)}{\\partial u} du + C} \\right|_{t=0} \\\\\n &= \\int_{\\lambda_0(0)}^\\lambda \\tfrac{\\partial}{\\partial t}\n \\Bigg(\n \\underbrace{\\tfrac{1}{p} \\tr\\bracket{\\inv{\\mB_t^\\mS + u \\mI_q}}}_{\\alpha m_t^\\mS(-u)} - \\underbrace{\\tfrac{1}{p} \\tr\\bracket{\\inv{\\mB_t + \\mu(u) \\mI_p}}}_{m_t(-\\mu(u))} \\tfrac{\\partial \\mu(u)}{\\partial u} \n \\Bigg)\n \\Bigg|_{t=0} du.\n \\label{eq:jacobi-trace-functionals}\n \\end{align}\n Here $\\lambda_0(t)$ and $C$ are chosen\\footnote{We assume that such a $\\lambda_0(t)$ exists, but a guarantee of its existence needs to be formally proved and is currently the non-rigorous part in this proof sketch.} such that for a given $t$, the integrand is equal to zero when $u = \\lambda_0(0)$, which simplifies the application of Leibniz' rule.\n Thus, by the dominated convergence theorem, we need only show that this derivative converges to 0 for any fixed $u$.\n By the differentiation rule of asymptotic equivalence, we know that we need only consider $m_t^\\mS(-u)$ asymptotically.\n Applying the result of \\cite{rubio_mestre_2011}, as $p \\to \\infty$ with $\\alpha = \\tfrac{q}{p}$, $m_t^\\mS(-u)$ almost surely converges to the solution to\n \\begin{align}\n \\label{eq:lem:sketch:jacobi:rubio}\n \\frac{1}{m_t^\\mS(-u)} - u = \\tfrac{1}{q} \\tr \\bracket{\\mB_t \\inv{\\mI_p + m_t^\\mS(-u) \\mB_t}},\n \\end{align}\n which we can manipulate to obtain\n \\begin{align}\n \\frac{1}{m_t^\\mS(-u)} - u = \\frac{1}{\\alpha m_t^\\mS(-u)} \\paren{1 - \\frac{1}{ m_t^\\mS(-u)} m_t \\paren{- \\tfrac{1}{m_t^\\mS(-u)}}}.\n \\end{align}\n Let $\\rho_t(u) = \\tfrac{1}{m_t^\\mS(-u))}$. The above becomes\n \\begin{align}\n \\alpha(\\rho_t(u) - u) = \\rho_t(u) \\paren{1 - \\rho_t(u) m_t(- \\rho_t(u))}.\n \\end{align}\n Let us compute the partial derivatives with respect to $t$ and $u$ of $\\rho_t(u)$. Let $z = -\\rho_t(u)$ be the argument of $m_t(z)$ for the purposes of computing partial derivatives.\n Taking partial derivatives with respect to $t$,\n we obtain\n \\begin{equation}\n \\begin{split}\n \\alpha \\frac{\\partial \\rho_t(u)}{\\partial t} \n &= \\frac{\\partial \\rho_t(u)}{\\partial t} \\paren{1 - \\rho_t(u) m_t(- \\rho_t(u))} \\\\\n & \\quad + \\rho_t(u) \\paren{- \\frac{\\partial \\rho_t(u)}{\\partial t} m_t(- \\rho_t(u))\n - \\rho_t(u) \\frac{\\partial m_t(- \\rho_t(u))}{\\partial t} + \\rho_t(u) \\frac{\\partial m_t(- \\rho_t(u))}{\\partial z} \\frac{\\partial \\rho_t(u)}{\\partial t}}.\n \\end{split}\n \\end{equation}\n This yields\n \\begin{equation}\n \\frac{\\partial \\rho_t(u)}{\\partial t} \n = \\ddfrac{\\rho_t(u)^2 \\frac{\\partial m_t(- \\rho_t(u))}{\\partial t}}{1 - \\alpha - 2 \\rho_t(u) m_t(- \\rho_t(u)) + \\rho_t(u)^2 \\frac{\\partial m_t(- \\rho_t(u))}{\\partial z}}.\n \\end{equation}\n Taking partial derivatives with respect to $u$,\n we obtain\n \\begin{equation}\n \\begin{split}\n \\alpha \\frac{\\partial \\rho_t(u)}{\\partial u} - \\alpha \n &= \\frac{\\partial \\rho_t(u)}{\\partial u} \\paren{1 - \\rho_t(u) m_t(- \\rho_t(u))} \\\\\n & \\quad + \\rho_t(u) \\paren{- \\frac{\\partial \\rho_t(u)}{\\partial u} m_t(- \\rho_t(u))\n + \\rho_t(u) \\frac{\\partial m_t(- \\rho_t(u))}{\\partial z} \\frac{\\partial \\rho_t(u)}{\\partial u}}.\n \\end{split}\n \\end{equation}\n This yields\n \\begin{gather}\n \\frac{\\partial \\rho_t(u)}{\\partial u} \n = \\ddfrac{-\\alpha}{1 - \\alpha - 2 \\rho_t(u) m_t(- \\rho_t(u)) + \\rho_t(u)^2 \\frac{\\partial m_t(- \\rho_t(u))}{\\partial z}}.\n \\end{gather}\n Now note that by the chain rule, \n \\begin{align}\n \\frac{\\partial \\rho_t(u)}{\\partial t} \n = - \\frac{\\partial m_t^\\mS(-u)}{\\partial t} \\rho_t(u)^2,\n \\end{align}\n and therefore \n \\begin{align}\n \\alpha \\frac{\\partial m_t^\\mS(-u)}{\\partial t} \n = \\frac{\\partial m_t(- \\rho_t(u))}{\\partial t} \\frac{\\partial \\rho_t(u)}{\\partial u}.\n \\end{align}\n Thus, for $\\mu(u) = \\rho_0(u)$, the difference of trace functionals in \\cref{eq:jacobi-trace-functionals} converges to 0. This coincides precisely with \\cref{thm:sketched-pseudoinverse}, as $\\mu(u) = \\rho_0(u) = \\tfrac{1}{\\widetilde{v}(u)}$.\n\\end{proof}\n\\section{Proof of \\Cref{cor:basic-ridge-asympequi-in-r}}\n\n\\label{sec:proof:cor:basic-ridge-asympequi-in-r}\n\nAs a preliminary that we will need later, through a standard argument,\nwe will first show that\n$\\Im(c(z)) \\to 0$ as $\\Im(z) \\to 0$\nin \\cref{eq:basic-ridge-asympequi-in-r}\nfor $z \\in \\mathbb{C}^{+}$ with $\\Re(z) < 0$.\nTo proceed, \ndenote $\\tfrac{1}{p} \\tr\\bracket{{\\bm{\\Sigma}} (c(z) {\\bm{\\Sigma}} - z {\\mathbf{I}}_p)^{-1}}$ by $d(z)$.\nFrom the last part of \\cref{lem:basic-ridge-asympequi},\n$d(z)$ is a Stieltjes transform of a certain positive measure on $\\mathbb{R}_{\\ge 0}$\nwith total mass $\\tfrac{1}{p} \\tr[{\\bm{\\Sigma}}]$.\nSince the operator norm of ${\\bm{\\Sigma}}$ is uniformly bounded in $p$,\nwe have that $\\tfrac{1}{p} \\tr[{\\bm{\\Sigma}}]$ is bounded above by some constant independent of $p$.\nCombining this with \\cref{fact:lim_stiletjes_im},\nwe have that $\\Im(d(z)) \\to 0$ as $\\Im(z) \\to 0$\nfor $z \\in \\mathbb{C}^{+}$ with $\\Re(z) < 0$.\nNow manipulating \\cref{eq:basic-ridge-fp-in-c},\nwe can write\n\\begin{equation}\n \\label{eq:c-in-d}\n c(z)\n = \\frac{1}{1 + \\tfrac{p}{n} d(z)}.\n\\end{equation}\nThus,\nwe can conclude that $\\Im(c(z)^{-1}) \\to 0$ as $\\Im(z) \\to 0$\nfor $z \\in \\mathbb{C}^{+}$ with $\\Re(z) < 0$.\nThis in turn implies that $\\Im(c(z)) \\to 0$\nfor $z \\in \\mathbb{C}^{+}$ with $\\Re(z) < 0$.\n\nWe now begin the proof.\n\n\\begin{proof}\nWe start by considering $z \\in \\complexset^+$.\nTo obtain \\cref{eq:basic-ridge-asympequi-in-r}, we multiply both sides of \\cref{eq:basic-ridge-asympequi-in-c} by $z$:\n\\begin{align}\n z \\big( \\tfrac{1}{n} {\\mathbf{X}}^\\ctransp {\\mathbf{X}} - z {\\mathbf{I}}_p \\big)^{-1}\n &\\simeq z \\inv{c(z) {\\bm{\\Sigma}} -z {\\mathbf{I}}_p} \\\\\n &= \\tfrac{z}{c(z)} \\biginv{\\mSigma - \\tfrac{z}{c(z)} \\mI_p} \\label{eq:resolvent-symmetric-pre}.\n\\end{align}\nWe will let $\\zeta = \\tfrac{z}{c(z)}$ shortly. \nFirst let \n$m(z) = \\tfrac{1}{p} \\tr \\bigbracket{\\inv{c(z) \\mSigma - z \\mI_p}}$.\nBy an additional application of \\cref{lem:basic-ridge-asympequi}, \n$m(z)$ is asymptotically equal to $\\tfrac{1}{p} \\tr \\bigbracket{\\inv{\\tfrac{1}{n} \\mX^\\ctransp \\mX - z {\\mathbf{I}}}}$, the Stieltjes transform of the spectrum of $\\tfrac{1}{n} \\mX^\\ctransp \\mX$. \nNow note that we can write \\cref{eq:basic-ridge-fp-in-c} in terms of $m(z)$ as\n\\[\n \\tfrac{1}{c(z)} - 1 = \\tfrac{p}{n} m(z).\n\\]\nWe can manipulate the equation in the display above into the following form:\n\\begin{align}\n - \\frac{c(z)}{z} = \\tfrac{p}{n} m(z) + \\frac{1}{z} \\left(\\tfrac{p}{n} - 1\\right).\n\\end{align}\nFrom the relationship between Stieltjes and the companion Stieltjes transforms in \n\\cref{fact:stieltjes-companion-stieltjes-relation},\nthis means that $-\\tfrac{c(z)}{z}$ is asymptotically equal to $v(z) = \\tfrac{1}{n} \\tr \\bracket{\\inv{\\tfrac{1}{n} \\mX \\mX^\\ctransp - z {\\mathbf{I}}}}$, the companion Stieltjes transform\nof the spectrum of $\\tfrac{1}{n} {\\mathbf{X}}^\\ctransp {\\mathbf{X}}$.\nThus, letting $\\zeta = \\tfrac{z}{c(z)}$ in \\cref{eq:resolvent-symmetric-pre}, \nwe have that\n\\[\n z \\big( \\tfrac{1}{n} {\\mathbf{X}}^\\ctransp {\\mathbf{X}} - z {\\mathbf{I}}_p \\big)^{-1} \n \\simeq \\zeta ({\\bm{\\Sigma}} - \\zeta {\\mathbf{I}}_p)^{-1},\n\\]\nand that asymptotically,\n$\\zeta = -\\tfrac{1}{v(z)}$ is the unique solution in $\\complexset^+$ to \\cref{eq:basic-ridge-fp-in-r} for $z \\in \\complexset^+$.\nMoreover, through analytic continuation,\none can extend this relationship to the real line\noutside the support of the spectrum of $\\tfrac{1}{n} {\\mathbf{X}} {\\mathbf{X}}^\\ctransp$\nwhere by the similar argument as for $c(z)$ above,\nboth $v(z)$ and $\\zeta$ are real.\n\nIt remains to determine the interval for which the analytic continuation coincides with a unique solution to \\cref{eq:basic-ridge-fp-in-r} for a given $z$. \nLet $z_0 \\in \\reals$ denote the most negative zero of $v$. Then for all $z < z_0$, $\\zeta \\in \\reals$ is well-defined, asymptotically being a solution to\n\\begin{align}\n \\label{eq:proof:zeta-fp-in-r}\n z - \\zeta = -\\zeta \\tfrac{1}{n} \\tr \\bracket{\\mSigma \\inv{\\mSigma - \\zeta \\mI_p}},\n\\end{align}\nwhich is an algebraic manipulation of \\cref{eq:basic-ridge-fp-in-c}. However, as we will now show, the solution to this equation is not in general unique, so we will show that the most negative solution for $\\zeta$ is the correct analytic continuation of the corresponding solution in $\\complexset^+$.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/zeta_z_proof_illustration.pdf}\n \\caption{\\textbf{Left:} Numerical illustration of the solutions to \\cref{eq:proof:zeta-fp-in-r} for $\\mSigma = \\mI$ and $\\tfrac{p}{n} = \\tfrac{1}{2}$. \n The right-hand side of \\cref{eq:proof:zeta-fp-in-r} is a fixed function of $\\zeta$ (blue, solid), \n but the left-hand-side is a line with slope $-1$ shifted by $z$ (orange to green, dashed). \n Solutions are the most negative intersections of the curves (circles), and not the most positive intersections (x's). The greatest possible value of $z$ yielding an intersection, $z = z_0$ (triangle), gives $\\zeta = \\zeta_0$ (dotted). For this example, we know that $z_0 = (1 - \\sqrt{\\tfrac{p}{n}})^2 \\approx 0.0858$ since the spectrum of $\\tfrac{1}{n} \\mX \\mX^\\ctransp$ follows the Marchenko--Pastur distribution.\n \\textbf{Right:} Illustration of the convergence of $z_0$ to $\\lambda_{\\min}^{+}(\\tfrac{1}{n} \\mX \\mX^\\ctransp)$. For $\\mSigma = \\mI$, $p = 500$, $n = 1000$, we draw a random $\\tfrac{1}{n} \\mX \\mX^\\ctransp$ and compute its eigenvalues. To simulate increasing the dimensionality of the matrix while keeping $\\lambda_{\\min}^{+}(\\tfrac{1}{n} \\mX \\mX^\\ctransp)$ fixed, we then take a subsample of size $n_s$ of the eigenvalues, comprised of $\\lambda_{\\min}^{+}(\\tfrac{1}{n} \\mX \\mX^\\ctransp)$ and $n_s - 1$ other eigenvalues chosen uniformly at random. We then plot $v(z)$ (solid) using this subsample. For any finite $n_s$, $z_0$ (dashed) will always lie between $0$ and $\\lambda_{\\min}^{+}(\\tfrac{1}{n} \\mX \\mX^\\ctransp)$, but $z_0$ approaches $\\lambda_{\\min}^{+}(\\tfrac{1}{n} \\mX \\mX^\\ctransp)$ as $n_s$ tends to infinity.\n }\n \\label{fig:zeta_z_proof_illustration}\n\\end{figure}\n\nConsider the two sides of \\cref{eq:proof:zeta-fp-in-r}. The left-hand side is linear in $\\zeta$, and the right-hand side is concave for $\\zeta < \\lambda_{\\min}^{+}(\\mSigma)$. To see this, observe that\n\\begin{align}\n \\tfrac{\\partial^2}{\\partial \\zeta^2} \\paren{-\\zeta \\tfrac{1}{n} \\tr \\bracket{\\mSigma \\inv{\\mSigma - \\zeta \\mI_p}}}\n &= \\tfrac{\\partial}{\\partial \\zeta} \\paren{- \\tfrac{1}{n} \\tr \\bracket{\\mSigma \\inv{\\mSigma - \\zeta \\mI_p}} - \\zeta \\tfrac{1}{n} \\tr \\bracket{\\mSigma \\paren{\\mSigma - \\zeta \\mI_p}^{-2}}} \\\\\n &= \\tfrac{\\partial}{\\partial \\zeta} \\paren{- \\tfrac{1}{n} \\tr \\bracket{\\mSigma^2 \\paren{\\mSigma - \\zeta \\mI_p}^{-2}}} \\\\\n &= -\\tfrac{2}{n} \\tr \\bracket{\\mSigma^2 \\paren{\\mSigma - \\zeta \\mI_p}^{-3}} < 0.\n\\end{align}\nA linear function and a concave function can intersect at zero, one, or two points. If at one point, this must occur at the unique point $(z_1, \\zeta_1)$, $\\zeta_1 < \\lambda_{\\min}^{+}(\\mSigma)$ for which the derivatives of each side of \\cref{eq:proof:zeta-fp-in-r} coincide, satisfying\n\\begin{align}\n 1 = \\tfrac{1}{n} \\tr \\bracket{ \\mSigma^2 \\paren{\\mSigma - \\zeta_1 \\mI_p}^{-2} }.\n\\end{align}\nThis right-hand side of this equation sweeps the range $(0, \\infty)$ for $\\zeta_1 \\in (-\\infty, \\lambda_{\\min}^{+}(\\mSigma))$, so such a $(z_1, \\zeta_1)$ always exists.\nFurthermore, since the solutions $\\zeta$ are continuous as a function $z$, the analytic continuation of the complex solution to the reals of the map $z \\mapsto \\zeta$ with domain $(-\\infty, z_1)$ must have image of either $(-\\infty, \\zeta_1)$ or $(\\zeta_1, \\lambda_{\\min}^{+}(\\mSigma))$.\nThe correct image must be $(-\\infty, \\zeta_1)$, which we illustrate in \\cref{fig:zeta_z_proof_illustration} (left). \n\nTo see why this must be the correct image, consider $z = x + i \\varepsilon$ for a fixed $\\varepsilon > 0$ with $x$ very negative.\nRewriting \\cref{eq:proof:zeta-fp-in-r}, we have the form of \\cref{eq:basic-ridge-fp-in-r}:\n\\begin{align}\n \\label{eq:proof:zeta-fp-all-on-right-hand-side}\n z = \\zeta \\paren{1 - \\tfrac{1}{n} \\tr \\bracket{\\mSigma \\inv{\\mSigma - \\zeta \\mI_p}}}.\n\\end{align}\nWe begin by considering the behavior of the trace term.\nLet $\\zeta = \\chi + i \\xi$, and suppose that $\\chi < \\tfrac{x}{2}$, which means that $\\chi$ is also very negative. The trace is a sum of terms of the form\n\\begin{align}\n \\frac{\\sigma}{\\sigma - \\zeta} = \\frac{\\sigma (\\sigma - \\chi + i \\xi)}{(\\sigma - \\chi)^2 + \\xi^2}.\n\\end{align}\nLet $g(\\zeta)$ and $h(\\zeta)$ denote the real and imaginary parts of $\\tfrac{1}{n} \\tr \\bracket{\\mSigma \\inv{\\mSigma - \\zeta \\mI_p}}$.\nFor $x$ (and therefore $\\chi$) sufficiently negative, this gives us the simple bounds\n\\begin{align}\n \\left| g(\\zeta) \\right| \n &\\leq \\frac{\\tfrac{p}{n} \\sigma_{\\max}(\\mSigma)}{- \\chi} \n \\leq \\frac{2 \\tfrac{p}{n} \\sigma_{\\max}(\\mSigma)}{- x}, \\\\\n \\left| h(\\zeta) \\right| \n &\\leq \\frac{\\tfrac{p}{n} \\sigma_{\\max}(\\mSigma) \\xi}{\\chi^2}\n \\leq \\frac{4 \\tfrac{p}{n} \\sigma_{\\max}(\\mSigma) \\xi}{x^2}.\n\\end{align}\nWe therefore have by \\cref{eq:proof:zeta-fp-all-on-right-hand-side} that\n\\begin{align}\n \\chi = \\frac{x (1 - g(\\zeta)) + \\varepsilon h(\\zeta)}{(1 - g(\\zeta))^2 + h(\\zeta)^2}, \\quad\n \\xi = \\frac{\\varepsilon (1 - g(\\zeta)) - x h(\\zeta)}{(1 - g(\\zeta))^2 + h(\\zeta)^2}.\n\\end{align}\nBy our bounds on $g$ and $h$, we can conclude that for sufficiently negative $x$, there exists $a > 0$ and $0 < b < 1$ such that $ |\\xi| \\leq a \\varepsilon + b |\\xi| $, implying that $|\\xi| \\leq \\tfrac{a \\varepsilon}{1 - b}$, and therefore $|\\xi|$ is bounded. Since $|\\xi|$ is bounded, $|h(\\zeta)|$ has an upper bound of the form $\\tfrac{1}{x^2}$, so for any $c \\in (\\tfrac{1}{2}, 1)$ and sufficiently negative $x$, we have the bound $\\chi \\leq c x$. Therefore, we can confirm that our supposition that $\\chi < \\tfrac{x}{2}$ leads to the unique solution with $\\xi > 0$, since for any $c' \\in (0, 1)$ we similarly have $\\xi > c' \\varepsilon > 0$ for sufficiently negative $x$. One can similarly argue that for solutions with $\\chi \\to \\lambda_{\\min}^{+}(\\mSigma)$, it must be that $\\xi < 0$, which is the solution in the wrong half-plane. By continuity of $z \\mapsto \\zeta$, identifying these extreme cases is sufficient to identify the correct image.\nTherefore, for real-valued $z < z_1$, the correct $\\zeta$ is the most negative solution, which is the unique $\\zeta < \\zeta_1$, and $\\zeta$ is undefined for $z > z_1$.\n\nLastly, we argue that asymptotically, $z_0 = z_1$. In the case $n < p$, this is straightforward, as the most negative zero of $v$ must lie between the two most negative distinct eigenvalues of $\\frac{1}{n} \\mX \\mX^\\ctransp$. This is because there is a pole at each distinct eigenvalue, so the entire range $(-\\infty, \\infty)$ (including crossing $0$) is mapped to by $v$ between each successive pair of distinct eigenvalues. When $n < p$, there is not a point mass at $0$, so these two most negative eigenvalues must converge to the same value as the discrete eigenvalue distribution converges to a continuous distribution, and this value marks the beginning of the continuous support of the spectrum of $\\frac{1}{n} \\mX \\mX^\\ctransp$, so $z_0 \\to \\lambda_{\\min}(\\frac{1}{n} \\mX \\mX^\\ctransp)$. Moreover, $\\zeta$, being asymptotically equal to $\\tfrac{1}{v}$, is undefined only on the support of the limiting spectrum and continuous elsewhere; therefore by the argument in the previous paragraph, the solution to \\cref{eq:proof:zeta-fp-in-r} does not exist for $z > z_1$, and it must be that $\\lambda_{\\min}(\\frac{1}{n} \\mX \\mX^\\ctransp) \\to z_1$.\n\nFor $n > p$, we apply similar reasoning; however, we must take care to consider the point mass of the spectrum at 0. This means that $z_0 \\in (0, \\lambda_{\\min}^{+}(\\frac{1}{n} \\mX \\mX^\\ctransp))$, because like before, the first zero must lie between the two most negative distinct eigenvalues, as we illustrate in \\cref{fig:zeta_z_proof_illustration} (right).\nHowever, asymptotically, it must be that $z_0 \\to z_1 = \\lambda_{\\min}^{+}(\\frac{1}{n} \\mX \\mX^\\ctransp)$. \nThis is most easily seen by a contradiction argument. Suppose\nwe have $z_0 < z_1 - \\varepsilon$ for some $\\varepsilon > 0$.\nBecause $\\tfrac{1}{v}$ has a pole at $z_0$, $-\\zeta = \\tfrac{1}{v} \\to \\infty$ as $z \\searrow z_0$. In particular, this means that $\\tfrac{1}{v}$ is discontinuous at $z_0$, tending to $\\infty$ from the right. \nMeanwhile, as argued above, $\\lambda_{\\min}^{+}(\\frac{1}{n} \\mX \\mX^\\ctransp) \\to z_1$, and we know that for $z < z_1$, $\\zeta < \\zeta_1 \\in (-\\infty, \\lambda_{\\min}^{+}(\\mSigma))$.\nThis is a contradiction, because on the one hand $\\zeta$ is upper bounded by $\\lambda_{\\min}^{+}(\\mSigma)$ for any $z \\in (z_0, z_1)$, but on the other hand $\\tfrac{1}{v}$ can be made arbitrarily large by taking $z \\to z_0^+$.\nTherefore, we must have, asymptotically, that $z_0 = z_1 = \\lambda_{\\min}^{+}(\\frac{1}{n} \\mX \\mX^\\ctransp)$. For this reason, in the theorem statement, we denote $\\zeta_0 = \\zeta_1$.\n\\end{proof}\n\\section{Proofs in \\Cref{sec:properties}}\n\\label{sec:proofs-properties}\n\nWe collect the proofs of the various properties of the equivalences obtained in our paper.\n\n\n\\subsection{Proof of \\cref{rem:mu0-lambda0-vs-alpha}}\n\\label{sec:mu0-lambda0-vs-alpha}\n\n\\begin{proof}\nRecall from \\cref{eq:lambda0-mu0-in-alpha} that\nfor $\\alpha \\in (0, \\infty)$,\n\\begin{equation}\n \\label{eq:lambda0-with-mu0}\n \\lambda_0(\\alpha)\n = \\mu_0\n \\paren{ 1 - \\tfrac{1}{\\alpha} \\tfrac{1}{p} \n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu_0(\\alpha) {\\mathbf{I}})^{-1} } }.\n\\end{equation}\nFrom the statement of \\cref{rem:mu0-lambda0-vs-alpha},\n$\\lim_{\\alpha \\to \\infty} \\mu_0(\\alpha) = - \\lambda_{\\min}^{+}({\\mathbf{A}})$.\nWe will argue below that\n\\begin{equation}\n \\label{eq:mu0-by-alpha-lim}\n \\lim_{\\alpha \\to \\infty}\n \\frac\n {\\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu_0(\\alpha) {\\mathbf{I}})^{-1} }}\n {\\alpha}\n = 0,\n\\end{equation}\nwhich combined with \\cref{eq:lambda0-with-mu0} provides the desired result.\n\nObserve that the limit on the left-hand side of \\cref{eq:mu0-by-alpha-lim} \nis in the indeterminate $\\infty\/\\infty$ form because\n$\\lim_{\\alpha \\to \\infty} \\mu(\\alpha) = - \\lambda_{\\min}^{+}({\\mathbf{A}})$\nand thus\n$\\lim_{\\alpha \\to \\infty} \n\\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu(\\alpha) {\\mathbf{I}})^{-1} } = \\infty$.\nTo evaluate the limit, we will appeal to L'H{\\^o}pital's rule.\nThe derivative of the denominator with respect to $\\alpha$ is 1,\nwhile the derivative of the numerator with respect to $\\alpha$ is\n\\begin{equation}\n \\label{eq:deriv-numer-wrt-alpha}\n \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu_0(\\alpha) {\\mathbf{I}})^{-2} }\n \\frac{\\partial \\mu_0(\\alpha)}{\\partial \\alpha}.\n\\end{equation}\nImplicitly differentiating \\cref{eq:mu0-fp-in-alpha}\nwith respect to $\\alpha$, we have\n\\begin{equation}\n \\label{eq:deriv-mu0-wrt-alpha-relation}\n 1 \n = \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu_0(\\alpha) {\\mathbf{I}})^{-3} }\n \\frac{\\partial \\mu_0(\\alpha)}{\\partial \\alpha}.\n\\end{equation}\nSubstituting for $\\tfrac{\\partial \\mu_0(\\alpha)}{\\partial \\alpha}$\nfrom \\cref{eq:deriv-mu0-wrt-alpha-relation}\ninto \\cref{eq:deriv-numer-wrt-alpha},\nwe can write the derivative of the numerator as\n\\begin{align}\n \\frac\n {\n \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu_0(\\alpha) {\\mathbf{I}})^{-2} }\n }\n {\n \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu_0(\\alpha) {\\mathbf{I}})^{-3}}\n }.\n\\end{align}\nAs $\\alpha \\to \\infty$ and $\\mu_0(\\alpha) \\to - \\lambda_{\\min}^{+}({\\mathbf{A}})$,\nthe limit of the quantity in the display above becomes\n\\begin{align}\n \\lim_{\\alpha \\to \\infty}\n \\frac\n {\\lambda_{\\min}^{+}({\\mathbf{A}})}\n {(\\lambda_{\\min}^{+}({\\mathbf{A}}) + \\mu_0(\\alpha))^2}\n \\cdot\n \\frac\n {(\\lambda_{\\min}^{+}({\\mathbf{A}}) + \\mu(\\alpha))^3}{(\\lambda_{\\min}({\\mathbf{A}})^{+})^2}\n = \n \\lim_{\\alpha \\to \\infty}\n 1 + \\frac{\\mu_0(\\alpha)}{\\lambda_{\\min}^{+}}\n = 1 - 1\n = 0.\n\\end{align}\nThus, we can conclude that \\cref{eq:mu0-by-alpha-lim} holds,\nand the statement then follows.\nThe remaining claims follow by similar calculations.\n\\end{proof}\n\n\\subsection{Proof of \\cref{rem:mu0-lambda0-signs}}\n\\label{sec:mu0-lambda0-signs}\n\n\n\\begin{proof}\nWe start by noting that \n\\begin{align}\n \\lim_{x \\searrow 0}\n \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + x {\\mathbf{I}})^{-2} }\n &=\n \\lim_{x \\searrow 0}\n \\tfrac{1}{p}\n \\sum_{i = 1}^{p}\n \\frac{\\lambda_i^2({\\mathbf{A}})}{(\\lambda_i({\\mathbf{A}}) + x)^2}\n \\\\\n &=\n \\lim_{x \\searrow 0}\n \\tfrac{1}{p}\n \\sum_{i = 1}^{p}\n \\frac{\\lambda_i({\\mathbf{A}})}{\\lambda_i({\\mathbf{A}}) + x}\n =\n \\lim_{x \\searrow 0}\n \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + x {\\mathbf{I}})^{-1} } \\\\\n &=\n \\tfrac{1}{p} \\sum_{i=1}^{p}\n \\mathbbm{1}\\{ \\lambda_i({\\mathbf{A}}) > 0 \\} \n = r({\\mathbf{A}}).\n\\end{align}\nNow, write the first equation in \\cref{eq:mu0-lambda0-fps} \nin terms of $\\alpha$ as\n\\[\n \\alpha = \\tfrac{1}{p} \\tr \\bracket{\\mA^2 \\paren{\\mA + \\mu_0 \\mI}^{-2}}.\n\\]\nThus, when $\\alpha = r({\\mathbf{A}})$, we have $\\mu_0 = 0$\nas the solution to the first equation of \\cref{eq:mu0-lambda0-fps}.\nBecause $\\mu \\mapsto \\tfrac{1}{p} \\tr[{\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-2}]$\nis monotonically decreasing in $\\mu$,\nif $\\alpha < r({\\mathbf{A}})$, we have $\\mu_0 > 0$,\nwhile if $\\alpha > r({\\mathbf{A}})$, we have $\\mu_0 < 0$.\n\nNext we argue about sign pattern of $\\lambda_0$.\nWhen $\\alpha > r({\\mathbf{A}})$,\nwe have\n\\begin{align}\n \\alpha\n = \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}})^{-2} }\n = \\tfrac{1}{p}\n \\sum_{i = 1}^{p}\n \\frac{\\lambda_i^2({\\mathbf{A}})}{(\\lambda_i({\\mathbf{A}}) + \\mu_0)^2}\n &\\overset{(a)}{>}\n \\tfrac{1}{p}\n \\sum_{i = 1}^{p}\n \\frac{\\lambda_i({\\mathbf{A}})}{\\lambda_i({\\mathbf{A}}) + \\mu_0} \\\\\n &=\n \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}})^{-1} },\n\\end{align}\nwhere the inequality $(a)$ follows because $\\mu_0 < 0$.\nFrom \\cref{eq:mu0-lambda0-fps},\nit thus follows that $\\lambda_0 < 0$.\nSimilarly, when $\\alpha < r({\\mathbf{A}})$,\nnote that\n\\begin{align}\n \\alpha\n = \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}})^{-2} }\n = \\tfrac{1}{p}\n \\sum_{i = 1}^{p}\n \\frac{\\lambda_i^2({\\mathbf{A}})}{(\\lambda_i({\\mathbf{A}}) + \\mu_0)^2}\n &\\overset{(b)}{<}\n \\tfrac{1}{p}\n \\sum_{i = 1}^{p}\n \\frac{\\lambda_i({\\mathbf{A}})}{\\lambda_i({\\mathbf{A}}) + \\mu_0} \\\\\n &=\n \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}})^{-1} },\n\\end{align}\nwhere inequality $(b)$ follows from the fact that\n\\[\n 0\n <\n \\left(\\frac{\\lambda_{\\min}^{+}({\\mathbf{A}})}{\\lambda_{\\min}^{+}({\\mathbf{A}}) + \\mu_0}\\right)^2\n <\n \\frac{\\lambda_{\\min}^{+}({\\mathbf{A}})}{\\lambda_{\\min}^{+}({\\mathbf{A}}) + \\mu_0}\n < 1,\n\\]\nsince $\\mu_0 > 0$ in this case\nand $\\lambda_{\\min}^{+}({\\mathbf{A}}) > 0$.\nFrom \\cref{eq:mu0-lambda0-fps},\nit thus again follows that $\\lambda_0 < 0$.\nThis completes the proof.\n\\end{proof}\n\n\n\n\n\\subsection{Proof of \\cref{prop:monotonicities-lambda-alpha}}\n\\label{sec:monotonicies-lambda-alpha}\n\n\\begin{proof}\nThe claims follow from simple derivative calculations.\nWe split into two cases, one with respect to $\\lambda$,\nand the other with respect to $\\alpha$.\n\n\\subsubsection{Monotonicity with respect to $\\lambda$}\n\nFor a fixed $\\alpha$,\nimplicitly differentiating the fixed-point equation \\cref{eq:sketched-modified-lambda}\nwith respect to $\\lambda$, we obtain\n\\begin{equation}\n \\label{eq:fp-differentiation-lambda}\n 1 =\n \\frac{\\partial \\mu}{\\partial \\lambda}\n - \\left( \\tfrac{1}{q} \\tr \\bracket{{\\mathbf{A}} \\inv{{\\mathbf{A}} + \\mu {\\mathbf{I}}} } \n - \\mu \\tfrac{1}{q}\\tr \\bracket{ {\\mathbf{A}} \\paren{{\\mathbf{A}} + \\mu {\\mathbf{I}}}^{-2} }\n \\right) \n \\frac{\\partial \\mu}{\\partial \\lambda}.\n\\end{equation}\nNote the following algebraic simplification:\n\\begin{align}\n {\\mathbf{A}} \\paren{ {\\mathbf{A}} + \\mu {\\mathbf{I}} }^{-1}\n - \\mu {\\mathbf{A}} \\paren{ {\\mathbf{A}} + \\mu {\\mathbf{I}} }^{-2}\n &= {\\mathbf{A}} \\paren{ {\\mathbf{A}} + \\mu {\\mathbf{I}} }^{-1}\n \\paren{ {\\mathbf{I}} - \\mu \\paren{ {\\mathbf{A}} + \\mu {\\mathbf{I}} }^{-1} } \\nonumber \\\\\n &= {\\mathbf{A}} \\paren{ {\\mathbf{A}} + \\mu {\\mathbf{I}} }^{-1} {\\mathbf{A}} \\paren{ {\\mathbf{A}} + \\mu {\\mathbf{I}} }^{-1}\n = {\\mathbf{A}}^2 \\paren{ {\\mathbf{A}} + \\mu {\\mathbf{I}} }^{-2} \\label{eq:matrix-diff-manip}.\n\\end{align}\nSubstituting \\cref{eq:matrix-diff-manip} into \\cref{eq:fp-differentiation-lambda},\nwe have\n\\begin{equation}\n \\label{eq:mu-deriv-lambda}\n \\frac{\\partial \\mu}{\\partial \\lambda}\n = \\frac\n {1}\n {\n 1 - \\tfrac{1}{q} \\tr \\bracket{ {\\mathbf{A}}^2 \\paren{ {\\mathbf{A}} + \\mu {\\mathbf{I}} }^{-2} }}.\n\\end{equation}\nObserve that $\\mu \\mapsto \\tfrac{1}{q} \\tr[{\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-2}]$\nis monotonically decreasing function of $\\mu$\nover $(\\mu_0, \\infty)$\nand because $1 = \\tfrac{1}{q} \\tr[{\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}})^{-2}]$\nfrom the first equation in \\cref{eq:mu0-lambda0-fps},\nthe denominator of \\cref{eq:mu-deriv-lambda} is positive over $(\\mu_0, \\infty)$.\nConsequently, $\\tfrac{\\partial \\mu}{\\partial \\lambda}$ is positive,\nand $\\mu$ is a monotonically increasing function of $\\lambda$.\nFinally, note that as $\\lambda \\to \\lambda_0^{+}$, $\\mu(\\lambda) \\to \\mu_0$,\nand as $\\lambda \\to \\infty$, $\\mu(\\lambda) \\to \\infty$.\nThis completes the proof of the first part.\n\n\\subsubsection{Monotonicity with respect to $\\alpha$}\n\nWe begin by writing \\cref{eq:sketched-modified-lambda} \nin $\\alpha$ as\n\\begin{equation}\n \\label{eq:sketched-modified-lambda-in-alpha}\n \\lambda\n = \\mu\n \\paren{ 1 - \\tfrac{1}{\\alpha} \\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1} } }.\n\\end{equation}\nFor a fixed $\\lambda$,\nimplicitly differentiating \\cref{eq:sketched-modified-lambda}\nwith respect to $\\alpha$, we have\n\\begin{equation}\n \\label{eq:fp-differentiation-alpha}\n 0\n =\n \\frac{\\partial \\mu}{\\partial \\alpha}\n +\n \\frac{\\mu}{\\alpha^2}\n \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1} }\n -\n \\frac{1}{\\alpha}\n \\left(\n \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1} }\n - \\mu \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-2} }\n \\right)\n \\frac{\\partial \\mu}{\\partial \\alpha}.\n\\end{equation}\nSolving for $\\tfrac{\\partial \\mu}{\\partial \\alpha}$,\nwe obtain\n\\begin{equation}\n \\label{eq:mu-deriv-alpha-1}\n \\frac{\\partial \\mu}{\\partial \\alpha}\n = \n \\frac\n {- \\tfrac{1}{\\alpha^2} \\mu \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1}} }\n {1 - \\tfrac{1}{\\alpha} \\tfrac{1}{p}\n \\left( \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1} \n - \\mu {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-2} } \\right)}.\n\\end{equation}\nSimilar to the part above,\nsubstituting the relation \\cref{eq:matrix-diff-manip} into \\cref{eq:fp-differentiation-alpha}\nand simplifying yields\n\\begin{equation}\n \\label{eq:mu-deriv-alpha-2}\n \\frac{\\partial \\mu}{\\partial \\alpha}\n = \n \\frac\n {- \\tfrac{1}{\\alpha} \\mu \\tfrac{1}{q}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1}} }\n {1 - \\tfrac{1}{q} \\tr \\bracket{ {\\mathbf{A}}^2 \\paren{ {\\mathbf{A}} + \\mu {\\mathbf{I}} }^{-2} }}.\n\\end{equation}\n\n\nBecause the denominator of \\cref{eq:mu-deriv-alpha-2} is positive\nfrom \\cref{eq:mu0-lambda0-fps}\nas argued above\nand $\\tr[{\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1}]$ is positive\nfor $\\mu \\in (\\mu_0, \\infty)$,\nthe sign of $\\frac{\\partial \\mu}{\\partial \\alpha}$\nis opposite the sign of $\\mu$.\nBecause when $\\lambda \\ge 0$, $\\mu \\ge 0$\n(from the first part of \\cref{rem:joint-signs-lambda-mu}),\nin this case, $\\frac{\\partial \\mu}{\\partial \\alpha}$\nis negative, and $\\mu$ is monotonically decreasing in $\\alpha$.\nWhen $\\lambda < 0$,\nfor $\\alpha \\le r({\\mathbf{A}})$,\nwe have $\\mu(\\lambda) \\ge 0$ \n(from the second part of \\cref{rem:joint-signs-lambda-mu}).\nThus, over $(0, r({\\mathbf{A}}))$, $\\mu$ is monotonically decreasing in $\\alpha$.\nOn the other hand, for $\\alpha > r({\\mathbf{A}})$,\n$\\mu(\\lambda) < 0$ \n(since $\\mathrm{sign}(\\mu(\\lambda)) \n= \\mathrm{sign}(\\lambda)$ and $\\lambda < 0$),\nand consequently, $\\mu$ is monotonically increasing in $\\alpha$\nover $(r({\\mathbf{A}}), \\infty)$.\n\nFinally, to obtain the limit of $\\mu(\\alpha)$\nas $\\alpha \\to 0$,\nwe write \\cref{eq:sketched-modified-lambda-in-alpha} as\n\\[\n \\lambda \\alpha\n = \\mu \\alpha\n - \\mu \\tfrac{1}{p}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1} }.\n\\]\nNow, for any $\\lambda \\in (\\lambda_0, \\infty)$,\n$\\lim_{\\alpha \\to 0^{+}} \\lambda \\alpha = 0$.\nThus, we have\n\\[\n \\lim_{\\alpha \\to 0^{+}}\n \\mu(\\alpha)\n = \\lim_{\\alpha \\to 0^{+}}\n f^{-1}(\\alpha),\n\\]\nwhere $f(x) = \\tfrac{1}{p} \\tr[{\\mathbf{A}} ({\\mathbf{A}} + x {\\mathbf{I}})^{-1}]$.\nObserve that function $f$ is strictly decreasing\nover $(\\mu_0, \\infty)$,\nand $\\lim_{x \\to \\infty} f(x) = 0$.\nHence, the function $f^{-1}$ is strictly decreasing\nand $\\lim_{\\alpha \\to 0^{+}} f^{-1}(\\alpha) = \\infty$.\nThis provides us with the first limit.\nTo obtain the limit of $\\mu(\\alpha)$ as $\\alpha \\to \\infty$,\nwrite from \\cref{eq:sketched-modified-lambda}\n\\[\n \\mu\n = \\lambda \n + \\tfrac{1}{\\alpha} \\tfrac{1}{p}\n \\tr \\bracket{ \\mu {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1} }.\n\\]\nObserve that $\\tfrac{1}{p} \\tr[\\mu {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1}]$\nis bounded for $\\mu \\in (\\mu_0, \\infty)$.\nThus, taking the limit $\\alpha \\to \\infty$,\nwe conclude that $\\lim_{\\alpha \\to \\infty} \\mu(\\alpha) = \\lambda$.\nThis finishes the second part, and completes the proof.\n\\end{proof}\n\n\n\n\n\\subsection{Proof of \\cref{rem:joint-signs-lambda-mu}}\n\\label{sec:joint-signs-lambda-mu}\n\n\\begin{proof}\nWe start by writing \\cref{eq:sketched-modified-lambda}\nin terms of $\\alpha$ as\n\\[\n \\lambda = \\mu\n \\paren{1 - \\tfrac{1}{\\alpha} \\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1} } }.\n\\]\nFor the subsequent argument, it will help to rearrange the terms in the equation in display above to arrive at\nthe following equivalent equation:\n\\begin{equation}\n \\label{eq:sketched-modified-lambda-rewriting}\n 1 - \\frac{\\lambda}{\\mu}\n = \\tfrac{1}{\\alpha} \\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1} }.\n\\end{equation}\nWe consider two separate cases depending on $\\lambda \\ge 0$ and $\\lambda < 0$.\n\n\\textbf{Case $\\lambda \\ge 0$}:\nFix $\\alpha > 0$.\nObserve that the left side of \\cref{eq:sketched-modified-lambda-rewriting}\nis an increasing function of $\\mu$,\nand the right side of \\cref{eq:sketched-modified-lambda-rewriting}\nis a decreasing function of $\\mu$.\nAs $\\mu$ varies from $0^{+}$ to $\\infty$, \nthe right hand side decreases from $\\tfrac{r(A)}{\\alpha}$ to $0$,\nwhile the left hand side increases from $-\\infty$ to $1$.\nSince $1 > 0$,\nthere is a unique intersection for $\\mu \\ge 0$.\n\n\\textbf{Case $\\lambda < 0$}:\nFix $\\alpha \\le r({\\mathbf{A}})$.\nFor this subcase,\nfrom \\Cref{rem:mu0-lambda0-signs},\n$\\mu_0 \\ge 0$.\nThus, there is a unique intersection for $\\mu \\ge 0$.\nFix now $\\alpha > r({\\mathbf{A}})$.\nFor this subcase, the term in the parenthesis of \\cref{eq:sketched-modified-lambda}\nis positive. Thus, $\\mathrm{sign}(\\mu) = \\mathrm{sign}(\\lambda)$.\n\nThis completes all the three cases, and finishes the proof.\n\\end{proof}\n\n\n\\subsection{Proof of \\cref{rem:concavity-mu-in-lambda}}\n\\label{sec:concavity-mu-in-lambda}\n\n\\begin{proof}\nRecall that $\\mu_0 > - \\lambda_{\\min}^{+}({\\mathbf{A}})$.\nFor $x \\in (\\mu_0, \\infty)$,\nobserve that\n\\[\n \\frac{\\partial}{\\partial x}\n \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + x {\\mathbf{I}})^{-1} }\n = - \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + x {\\mathbf{I}})^{-2} }\n < 0,\n\\]\n\\[\n \\frac{\\partial^2}{\\partial x^2}\n \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + x {\\mathbf{I}})^{-1} }\n = 2 \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + x {\\mathbf{I}})^{-3} } \n > 0.\n\\]\nThus, the function\n\\[\n x \\mapsto\n \\tfrac{1}{q}\n \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + x {\\mathbf{I}})^{-1} }\n\\]\nis strictly decreasing and convex over $(\\mu_0, \\infty)$,\nand consequently\nthe function\n\\[\n x \\mapsto\n \\tfrac{1}{q}\n \\tr \\bracket{ x {\\mathbf{A}} ({\\mathbf{A}} + x {\\mathbf{I}})^{-1} }\n = \n \\tfrac{1}{q}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{I}} - {\\mathbf{A}} ({\\mathbf{A}} + x {\\mathbf{I}})^{-1}) }\n =\n \\tfrac{1}{q}\n \\tr[ {\\mathbf{A}} ]\n - \\tfrac{1}{q}\n \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + x {\\mathbf{I}})^{-1} }\n\\]\nis strictly increasing and concave over $(\\mu_0, \\infty)$.\nHence, the function $f$ \n(appearing in the right-hand side of \\cref{eq:sketched-modified-lambda} in $\\mu$)\ndefined by\n\\begin{equation}\n \\label{eq:sketched-modified-lambda-rhs}\n f(x)\n =\n x \n - x \\tfrac{1}{q}\n \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + x {\\mathbf{I}})^{-1} }\n =\n x\n \\paren{ 1 - \\tfrac{1}{q} \\tr \\bracket{ {\\mathbf{A}} ({\\mathbf{A}} + x {\\mathbf{I}})^{-1} } }\n\\end{equation}\nis strictly increasing and convex over $(\\mu_0, \\infty)$.\n\nNow, observe from \\cref{eq:sketched-modified-lambda} that\nfor a given $\\lambda$,\n$\\mu(\\lambda) = f^{-1}(\\lambda)$,\nwhere $f$ is as defined in \\cref{eq:sketched-modified-lambda-rhs}.\nBecause inverse of a strictly increasing, continuous, and convex function\nis strictly increasing, continuous, and concave \n(see, e.g., Proposition 3 of \\cite{hiriart_urruty-martinez_legaz_2003}),\nwe conclude that $\\lambda \\mapsto \\mu(\\lambda)$\nwhere $\\mu(\\lambda)$ solves \\cref{eq:sketched-modified-lambda}\nis concave in $\\lambda$ over $(\\lambda_0, \\infty)$.\nWe remark that, more directly,\nwe can also compute the second derivative of $\\mu(\\lambda)$ \nwith respect to $\\lambda$.\nFrom \\cref{eq:mu-deriv-lambda}, we have\n\\begin{equation}\n \\label{eq:mu-deriv-lambda-in-alpha}\n \\frac{\\partial \\mu}{\\partial \\lambda}\n = \\frac\n {1}\n {\n 1 - \\tfrac{1}{\\alpha} \\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}}^2 \\paren{ {\\mathbf{A}} + \\mu {\\mathbf{I}} }^{-2} }}.\n\\end{equation}\nTaking partial derivative of \\cref{eq:mu-deriv-lambda-in-alpha}\nwith respect to $\\lambda$, we get\n\\[\n \\frac{\\partial^2 \\mu}{\\partial \\lambda^2}\n = \n \\frac\n {\n -2 \\tfrac{1}{\\alpha} \\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-3} }\n }\n {\n \\left( \n 1 - \\tfrac{1}{\\alpha} \\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-2} } \n \\right)^2\n }\n \\frac{\\partial \\mu}{\\partial \\lambda}\n =\n \\frac\n {\n -2 \\tfrac{1}{\\alpha} \\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-3} }\n }\n {\n \\left( \n 1 - \\tfrac{1}{\\alpha} \\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-2} } \n \\right)^3\n }\n < 0,\n\\]\nfrom which the concavity claim follows.\n\nUsing the concavity of $\\mu$ in $\\lambda$,\nwe can write for \n$\\lambda, \\widetilde{\\lambda} \\in (\\lambda_0, \\infty)$,\n\\begin{equation}\n \\label{eq:concavity-bound-mu-lambda-1}\n \\mu(\\lambda)\n \\le \\mu(\\widetilde{\\lambda})\n + \\frac{\\partial \\mu}{\\partial \\lambda} \n \\mathrel{\\Big |}_{\\lambda = \\widetilde{\\lambda}} \n (\\lambda - \\widetilde{\\lambda}).\n\\end{equation}\nNow, from \\cref{eq:sketched-modified-lambda},\nfor any $\\widetilde{\\lambda} \\in (\\lambda_0, \\infty)$, \nwe have \n\\begin{equation}\n \\label{eq:concavity-bound-mu-lambda-2}\n \\mu(\\widetilde{\\lambda}) - \\widetilde{\\lambda}\n = \\tfrac{1}{q}\n \\tr \\bracket{ \\mu(\\widetilde{\\lambda}) {\\mathbf{A}} ({\\mathbf{A}} + \\mu(\\widetilde{\\lambda}) {\\mathbf{I}})^{-1} }\n = \\tfrac{1}{\\alpha} \\tfrac{1}{p}\n \\tr \\bracket{ \\mu(\\widetilde{\\lambda}) {\\mathbf{A}} ({\\mathbf{A}} + \\mu(\\widetilde{\\lambda}) {\\mathbf{I}})^{-1} }.\n\\end{equation}\nSubstituting in \\cref{eq:concavity-bound-mu-lambda-2}\nin \\cref{eq:concavity-bound-mu-lambda-1} yields\n\\begin{equation}\n \\label{eq:concavity-bound-mu-lambda-3}\n \\mu(\\lambda)\n \\le\n \\frac{\\partial \\mu}{\\partial \\lambda} \n \\mathrel{\\Big |}_{\\lambda = \\widetilde{\\lambda}} \n \\lambda \n +\n \\tfrac{1}{\\alpha}\n \\tfrac{1}{p}\n \\tr\n \\bracket{ \\mu(\\widetilde{\\lambda}) {\\mathbf{A}} ({\\mathbf{A}} + \\mu(\\widetilde{\\lambda}) {\\mathbf{I}})^{-1} }.\n\\end{equation}\nFrom \\Cref{prop:monotonicities-lambda-alpha},\n$\\lambda \\mapsto \\mu(\\lambda)$ is monotonically increasing in $\\lambda$\nand $\\lim_{\\lambda \\to \\infty} \\mu(\\lambda) = \\infty$.\nIn addition, $\\mu \\mapsto \\tr[{\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-2}]$\nis monotonically decreasing in $\\mu$\nand $\\lim_{\\mu \\to \\infty} \\tr[{\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-2}] = 0$,\nwhile \n$\\mu \\mapsto \\tr[\\mu {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1}]$\nis monotonically increasing in $\\mu$,\nand \n$\n\\lim_{\\mu \\to \\infty} \\tr[\\mu {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1}] = \\tr[ {\\mathbf{A}} ].\n$\nThus, \nfrom \\cref{eq:mu-deriv-lambda-in-alpha},\nchoosing $\\widetilde{\\lambda}$ large enough\nso that $\\mu(\\widetilde{\\lambda})$ is large enough,\nfor any $\\epsilon > 0$, we can write\n\\begin{equation}\n \\label{eq:deriv-asymp-mu-lambda}\n \\frac{\\partial \\mu}{\\partial \\lambda} \n \\mathrel{\\Big |}_{\\lambda = \\widetilde{\\lambda}}\n = \n \\frac\n {1}\n {\n 1 - \\tfrac{1}{\\alpha} \n \\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu(\\widetilde{\\lambda}) {\\mathbf{I}})^{-2} } }\n \\le\n 1 \n +\n \\epsilon,\n\\end{equation}\n\\begin{equation}\n \\label{eq:concavity-intercept-asymp}\n \\tfrac{1}{\\alpha}\n \\tfrac{1}{p}\n \\tr \\bracket{ \\mu(\\widetilde{\\lambda}) {\\mathbf{A}} ({\\mathbf{A}} + \\mu(\\widetilde{\\lambda}) {\\mathbf{I}})^{-1} }\n \\le\n \\tfrac{1}{\\alpha} \\tfrac{1}{p} \\tr \\bracket{ {\\mathbf{A}} } + \\epsilon.\n\\end{equation}\nCombining \n\\cref{eq:concavity-bound-mu-lambda-2,eq:deriv-asymp-mu-lambda,eq:concavity-intercept-asymp},\none then has\n\\[\n \\mu(\\lambda)\n \\le\n (1 + \\epsilon)\n \\lambda \n + \n \\tfrac{1}{\\alpha}\n \\tfrac{1}{p}\n \\tr[{\\mathbf{A}}]\n + \\epsilon.\n\\]\nSince the inequality holds for any arbitrary $\\epsilon$, \nthe desired upper bound on $\\mu(\\lambda)$ follows.\nFor the lower bound,\nobserve from \\cref{eq:concavity-bound-mu-lambda-2} that\nfor any $\\lambda \\in (\\lambda_0, \\infty)$\n\\[\n \\mu(\\lambda)\n = \\lambda + \\tfrac{1}{q} \\tr \\bracket{ \\mu(\\lambda) {\\mathbf{A}} \n ({\\mathbf{A}} + \\mu(\\lambda) {\\mathbf{I}})^{-1} }.\n\\]\nFrom \\Cref{rem:joint-signs-lambda-mu},\n$\\mu(\\lambda) \\ge 0$ either when $\\lambda \\ge 0$,\nor when $\\alpha \\le r({\\mathbf{A}})$.\nIn either of the cases,\nthe term $\\tfrac{1}{q} \\tr[\\mu(\\lambda) {\\mathbf{A}} ({\\mathbf{A}} + \\mu(\\lambda) {\\mathbf{I}})^{-1}]$\nis positive,\nand thus $\\mu(\\lambda) \\ge \\lambda$.\nFinally, the limit as $\\lambda \\to \\infty$\nfollows simply by noting that\n$\\mu(\\lambda) \\to \\infty$ and \n$\\tr[\\mu {\\mathbf{A}} ({\\mathbf{A}} + \\mu {\\mathbf{I}})^{-1}] \\to \\tr[{\\mathbf{A}}]$\nas $\\lambda \\to \\infty$.\nThis finishes the proof.\n\\end{proof}\n\n\n\n\\subsection{Proof of \\cref{rem:alt-mu-prime}}\n\n\\begin{proof}\nWe begin by rewriting \\cref{eq:mu-prime} using \\cref{eq:thm:sketched-pseudoinverse}:\n\\begin{align}\n \\mu' = \\frac{\\frac{\\mu^3}{q} \\tr \\bracket{\\mPsi \\paren{\\mA + \\mu \\mI}^{-2} }}{\\mu \\paren{ 1 - \\tfrac{1}{q} \\tr \\bracket{\\mA \\inv{\\mA + \\mu \\mI_p}}} + \\frac{\\mu^2}{q} \\tr \\bracket{\\mA \\paren{\\mA + \\mu \\mI}^{-2}} }.\n\\end{align}\nAfter dividing both the numerator and denominator by $\\mu$, we note that the denominator has a form which has already been simplified in \\Cref{sec:monotonicies-lambda-alpha}, and immediately obtain the factorization in terms of $\\tfrac{\\partial \\mu}{\\partial \\lambda}$.\n\\end{proof}\n\\section*{Acknowledgments}\nWe are grateful to Arun Kumar Kuchibhotla, Alessandro Rinaldo, Yuting Wei, Jin-Hong Du, \nand other members of the Operational Overparameterized Statistics (OOPS) Working Group \nat Carnegie Mellon University for helpful conversations.\nWe are also grateful to Edgar Dobriban,\nas well as participants of the ONR MURI on Foundations of Deep Learning\nfor useful discussions and feedback on this work.\n\nThis work was sponsored by Office of Naval Research MURI grant N00014-20-1-2787. \nDL, HJ, and RGB were also supported by NSF grants CCF-1911094, IIS-1838177, and IIS-1730574; ONR grants N00014-18-12571 and N00014-20-1-2534; AFOSR grant FA9550-22-1-0060; and a Vannevar Bush Faculty Fellowship, ONR grant N00014-18-1-2047.\n\n\n\n\\clearpage\n\\input{citations.bbl}\n\n\\clearpage\n\\setcounter{section}{0}\n\\setcounter{equation}{0}\n\\setcounter{figure}{0}\n\\renewcommand{\\thesection}{SM\\arabic{section}}\n\\renewcommand{\\theequation}{SM\\arabic{section}.\\arabic{equation}}\n\\renewcommand{\\thefigure}{SM\\arabic{figure}}\n\n\n\\thispagestyle{empty}\n\\begin{center}\n\\noindent\\textcolor{header1}{\\textbf{SUPPLEMENTARY MATERIALS: \\textsf{Asymptotics of the Sketched Pseudoinverse}}}\n\\color{gray}\\rule{\\textwidth}{4pt}\n\\end{center}\n\\medskip\n\nThis document serves as a supplement to the paper\n``Asymptotics of the Sketched Pseudoinverse.'' \nThe contents of this supplement are organized as follows.\nIn \\Cref{sec:useful-facts},\nwe collect some useful facts regarding Stieltjes transforms\nthat are used in some of the proofs in later sections.\nIn \\Cref{sec:proof:cor:basic-ridge-asympequi-in-r},\nwe provide a detailed proof for \\Cref{cor:basic-ridge-asympequi-in-r}.\nIn \\Cref{sec:proofs-main-results},\nwe provide proof and an alternative proof sketch for \\cref{thm:sketched-pseudoinverse}.\nFinally, in \\Cref{sec:proofs-properties},\nwe provide proofs of various properties regarding our main equivalences\nmentioned \\Cref{sec:properties} in the main paper.\n\n\n\\input{appendix\/appendix_facts}\n\n\\input{appendix\/proofs_prelim}\n\n\\input{appendix\/proofs_main}\n\n\\input{appendix\/proofs_properties}\n\n\n\n\n\n\n\\end{document}\n\n\n\n\\section{Discussion and extensions}\n\\label{sec:discussion}\n\nIn this paper, we have provided a detailed look at the asymptotic effects of i.i.d.\\ sketching on matrix inverses.\nWe have provided an extension of existing asymptotic equivalence results to real-valued regularization (including negative) and used this result to obtain both first- and second-order asymptotic equivalences for the sketched regularized pseudoinverse. \n\n\n\nOur work is far from a complete characterization of sketching. We now list some natural extensions to our results.\n\n\\paragraph{Relaxing assumptions, strengthening conclusions}\nAs mentioned in \\Cref{sec:main_results},\nwe make minimal assumptions on the base matrix ${\\mathbf{A}}$.\nIn particular, we do not assume that the empirical spectral\ndistribution of ${\\mathbf{A}}$ converges to any fixed limit.\nThe assumption that the maximum and minimum eigenvalue of ${\\mathbf{A}}$\nbe bounded away from $0$ and $\\infty$ can be weakened.\nIn particular, one can let some eigenvalues to escape\nto $\\infty$,\nand have some eigenvalues to decay to $0$,\nprovided certain functionals of the eigenvalues\nremain bounded.\nOur assumptions on the sketching matrix ${\\mathbf{S}}$ are also weak.\nWe do not assume any distributional structure on its entries\nand only require bounded moments of order $8 + \\delta$\nfor some $\\delta > 0$.\nUsing a truncation strategy,\none can push this to only requiring moments of order $4 + \\delta$\nfor some $\\delta > 0$ for almost sure equivalences up to order $2$\nthat we show in this paper.\nFinally, while our asymptotic results give practically relevant insights for finite systems, we lack a precise characterization for non-asymptotic settings. In particular, the rate of convergence depends on a number of factors including the choice of $\\lambda$ and the higher order moments of the elements of $\\mS$.\n\n\n\\paragraph{Generalized sketching}\nOur assumption that the elements of the matrix $\\mS$ are i.i.d.\\ draws from some distribution limits its application in practical settings on two key fronts:\nthe effect of a rotationally invariant sketch is isotropic regularization, and there is unnecessary distortion of the spectrum of $\\mA$ for $q \\to p$. We now discuss how to extend our framework to extend to more general classes of sketches that more closely align with those used in practice.\n\nIn practice we may desire to use generalized non-isotropic ridge regularization, to perform Bayes-optimal regression \n(see, e.g., Chapter 3 of \\cite{van-wieringen_2015})\nor to avoid multiple descent \\cite{mel2021regression,yilmaz2022descent}, or we may find ourselves using non-isotropic sketching matrices, such as in adaptive sketching~\\cite{lacotte2019adaptive} where the sketching matrix depends on the data. We can cover these cases with the following extension of \\cref{thm:sketched-pseudoinverse}. \n\\begin{corollary}\n [Non-isotropic sketching equivalence]\n \\label{cor:sketched-pseudoinverse-noniso}\n Assume the setting of \\cref{thm:sketched-pseudoinverse}.\n Let $\\mW$ be an invertible $p \\times p$ positive semidefinite matrix,\n either deterministic or random but independent of $\\mS$ with $\\limsup \\norm[\\mathrm{op}]{\\mW} < \\infty$. \n Let $\\widetilde{\\mS} = \\mW^{1\/2} \\mS$.\n Then for each $\\lambda > - \\liminf \\lambda_{\\min}^{+}(\\widetilde{\\mS}^\\top \\mA \\widetilde{\\mS})$\n as $p, q \\to \\infty$ such that\n $0 < \\liminf \\tfrac{q}{p} \\le \\limsup \\tfrac{q}{p} < \\infty$,\n \\begin{align}\n \\widetilde{\\mS}\n \\biginv{\\widetilde{\\mS}^\\top \\mA \\widetilde{\\mS} + \\lambda \\mI_q }\n \\widetilde{\\mS}^\\top\n \\simeq\n \\biginv{\\mA + \\mu \\mW^{-1} },\n \\end{align}\n where $\\mu$ most positive solution to\n \\begin{align}\n \\lambda = \\mu \\paren{ 1 - \\tfrac{1}{q} \\tr \\bracket{\\mA \\inv{\\mA + \\mu \\mW^{-1}}}}.\n \\end{align}\n\\end{corollary}\n\n\\begin{proof}\nThe proof uses simple algebraic manipulations.\nObserve that, since the operator norm is sub-multiplicative,\nand $\\| {\\mathbf{W}} \\|_{\\mathrm{op}}$, $\\| {\\mathbf{A}} \\|_{\\mathrm{op}}$\nare uniformly bounded in $p$,\n$\\| {\\mathbf{W}}^{1\/2} {\\mathbf{A}} {\\mathbf{W}}^{1\/2} \\|_{\\mathrm{op}}$ is also uniformly bounded $p$.\nUsing \\cref{thm:sketched-pseudoinverse},\nwe then have that\n\\[\n {\\mathbf{S}}\n \\biginv{\n {\\mathbf{S}}^\\top {\\mathbf{W}}^{1\/2} {\\mathbf{A}} {\\mathbf{W}}^{1\/2} {\\mathbf{S}}\n + \\lambda {\\mathbf{I}}_q\n }\n {\\mathbf{S}}^\\top\n \\simeq\n \\biginv{\n {\\mathbf{W}}^{1\/2} {\\mathbf{A}} {\\mathbf{W}}^{1\/2} + \\mu {\\mathbf{I}}_q\n }.\n\\]\nRight and left multiplying both sides by ${\\mathbf{W}}^{1\/2}$,\nand writing $\\widetilde{{\\mathbf{S}}} = {\\mathbf{W}}^{1\/2} {\\mathbf{S}}$,\nwe get\n\\[\n \\widetilde{{\\mathbf{S}}}\n \\biginv{\n \\widetilde{{\\mathbf{S}}}^\\top\n {\\mathbf{A}}\n \\widetilde{{\\mathbf{S}}}\n + \\lambda {\\mathbf{I}}_q\n }\n \\widetilde{{\\mathbf{S}}}^\\top\n \\simeq\n {\\mathbf{W}}^{1\/2}\n \\biginv{\n {\\mathbf{W}}^{1\/2} {\\mathbf{A}} {\\mathbf{W}}^{1\/2}\n + \\mu {\\mathbf{I}}_p\n }\n {\\mathbf{W}}^{1\/2}\n =\n \\biginv{{\\mathbf{A}} + \\mu {\\mathbf{W}}^{-1} }\n\\]\nas desired, completing the proof.\n\\end{proof}\n\nBecause non-isotropic sketching can be used to induce generalized ridge regularization, this can be exploited adaptively to induce a wide range of structure-promoting regularization via iteratively reweighted least squares, in a manner similar to adaptive dropout methods (see \\cite{lejeune2021flipside} and references therein). Additionally, this result shows that methods applying ridge regularization to adaptive sketching methods, using for example $\\mW = \\mA$ as in \\cite{lacotte2019adaptive}, are not equivalent to ridge regression but instead generalized ridge regression. \n\n\\paragraph{Free sketching}\nEven among isotropic sketches, there can be a wide range of behavior beyond i.i.d.\\ sketches. We suspect that a more general result holds for \\emph{free} sketching matrices (a notion from free probability that generalizes independence of random variables; see \\cite{mingo2017free} for an introductory text). We make the following conjecture without proof. However, we believe that our alternative proof sketch for \\cref{thm:sketched-pseudoinverse} via Jacobi's formula in the supplementary material may provide a strategy for proving this result.\n\\begin{conjecture}[General free sketching]\n \\label{conj:general-free-sketching}\n Let $\\mS \\in \\complexset^{p \\times q}$ be a norm-preserving sketch such that $\\mS \\mS^\\ctransp$ and $\\mA$ converge almost surely to operators that are free with respect to the average trace $\\tfrac{1}{p} \\tr [\\cdot]$. Then there exists a monotonic mapping $\\lambda \\mapsto \\gamma$ such~that\n \\begin{align}\n \\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q} \\mS^\\ctransp \\simeq \\biginv{\\mA + \\gamma \\mI_p}.\n \\end{align}\n\\end{conjecture}\n\nA particularly important sketching matrix that fits this broader definition is the orthogonal sketch. Unlike the i.i.d.\\ sketch, an orthogonal sketch does not distort the spectrum near $q = p$ and so has less induced regularization. Assuming \\cref{conj:general-free-sketching} holds, we obtain the following.\n\n\\begin{conjecture}[Orthogonal sketching]\n\\label{conj:orthonormal-sketch}\nFor $q \\leq p$ with $\\lim \\tfrac{q}{p} = \\alpha$, let $\\sqrt{\\tfrac{q}{p}} \\mQ \\in \\complexset^{p \\times q}$ be a Haar-distributed matrix with orthonormal columns, and let $\\mA \\in \\complexset^{p \\times p}$ be positive semidefinite with eigenvalues converging to a bounded limiting spectral measure. Then\n\\begin{align}\n \\mQ \\big( \\mQ^\\ctransp \\mA \\mQ + \\lambda \\mI_q \\big)^{-1} \\mQ^\\ctransp \\simeq \\big( \\mA + \\gamma \\mI_p \\big)^{-1},\n\\end{align}\nwhere $\\gamma$ is the most positive solution to\n\\begin{align}\n \\label{eq:ortho-gamma}\n \\tfrac{1}{p} \\tr \\bracket{\\inv{\\mA + \\gamma \\mI_p}} (\\gamma - \\alpha \\lambda) = 1 - \\alpha.\n\\end{align}\nFurthermore, for $\\mu > 0$ from \\cref{thm:sketched-pseudoinverse} applied to the same $(\\mA, \\alpha, \\lambda)$, we have $\\gamma < \\mu$.\n\\end{conjecture}\n\\begin{proof}\nFirstly, $\\mQ \\mQ^\\ctransp$ and $\\mA$ are almost surely asymptotically free~\\cite[Theorem 4.9]{mingo2017free}. Assuming the equivalence in \\cref{conj:general-free-sketching} and inverting the steps of the proof of \\cref{thm:sketched-pseudoinverse}, we know that we can determine the relation between $\\gamma$ and $(\\mA, \\alpha, \\lambda)$ if we can find $c(z)$ analogously to \\cref{lem:basic-ridge-asympequi} such that \n\\begin{align}\n\\label{eq:proof:ortho-sketch-c}\n(\\mA^{1\/2} \\mQ \\mQ^\\transp \\mA^{1\/2} - z \\mI_p)^{-1} \\simeq (c(z) \\mA - z \\mI_p)^{-1}.\n\\end{align}\nLet $a$ and $b$ denote the limiting distributions of $\\mA$ and $\\mQ \\mQ^\\ctransp$, respectively. Because $\\mA \\mQ \\mQ^\\ctransp$ has the same eigenvalues as $\\mA^{1\/2} \\mQ \\mQ^\\transp \\mA^{1\/2}$, we need only determine how the free product $ab$ depends on $a$. The general strategy is to use the chain of invertible transformations\n\\begin{align}\n G_a(z) = \\tfrac{1}{p} \\tr\\bracket{(z - a)^{-1}} \n \\;\n \\longleftrightarrow\n \\;\n M_a(z) = \\frac{1}{z} G_a\\paren{\\frac{1}{z}} - 1\n \\;\n \\longleftrightarrow\n \\;\n S_a(z) = \\frac{1 + z}{z} M_a^{-1}(z),\n\\end{align}\nwhich are the Cauchy transform (negative of the Stieltjes transform), moment generating series, and $S$-transform of $a$, respectively. Then we exploit the property of free products that $S_{ab}(z) = S_a(z) S_b(z)$, or equivalently $M_{ab}^{-1}(z) = \\tfrac{1 + z}{z} M_a^{-1}(z) M_b^{-1}(z)$.\n\nWe first determine by the simple structure of $b$ that it has $M_b(z) = \\tfrac{\\alpha z}{\\alpha - z}$, which has inverse $M_b^{-1}(z) = \\tfrac{\\alpha z}{\\alpha + z}$. Thus, $M_{ab}^{-1}(z) = \\tfrac{\\alpha(1 + z)}{\\alpha + z} M_a^{-1}(z)$. At the same time, taking the trace of \\cref{eq:proof:ortho-sketch-c}, we observe that we can also express this relation in terms of $c$ as $M_{ab}(z) = M_a(z c(\\tfrac{1}{z}))$. If we define the function $C \\colon z \\mapsto z c(\\tfrac{1}{z})$, we can write $M_{ab} = M_a \\circ C$, where $\\circ$ denotes function composition, which implies that $C^{-1} = M_{ab}^{-1} \\circ M_a$. Now we have \n\\begin{align}\n C^{-1}(z) = \\frac{\\alpha z (1 + M_a(z))}{(\\alpha + M_a(z)}.\n\\end{align}\nReferring to notation from \\cref{cor:basic-ridge-asympequi-in-r}, we can define $\\zeta \\colon z \\mapsto \\tfrac{z}{c(z)}$, such that $\\zeta = \\tfrac{1}{\\cdot} \\circ C \\circ \\tfrac{1}{\\cdot}$. Writing the above equation in terms of $\\zeta$ and $G_a$, we obtain\n\\begin{align}\n \\zeta^{-1}(z) = \\frac{z G_a(z) + \\alpha - 1}{\\alpha G_a(z)} \\implies G_a(\\zeta(z)) (\\alpha z - \\zeta(z)) = \\alpha - 1.\n\\end{align}\nWe now let $\\lambda = -z$, $\\gamma = -\\zeta(z)$, and recall that $-G_a(-\\gamma) = \\tfrac{1}{p} \\tr \\bracket{(a + \\gamma)^{-1}}$, and we obtain the stated equation in \\cref{eq:ortho-gamma}.\nTo see that $\\gamma < \\mu$, observe that we can write \\cref{eq:sketched-modified-lambda} and \\cref{eq:ortho-gamma} as \n\\begin{align}\n \\tfrac{\\mu}{p} \\tr \\bracket{\\big(\\mA + \\mu \\mI_p \\big)^{-1}} &= 1 - \\alpha + \\frac{\\alpha \\lambda}{\\mu} \\\\\n \\tfrac{\\gamma}{p} \\tr \\bracket{\\big(\\mA + \\gamma \\mI_p \\big)^{-1}} &= 1 - \\alpha + \\alpha \\lambda \\tfrac{1}{p} \\tr \\bracket{\\big(\\mA + \\gamma \\mI_p \\big)^{-1}}.\n\\end{align}\nThe left-hand sides of these two equations are the same increasing function of $\\mu$ and $\\gamma$, respectively, while the right-hand sides are decreasing functions, with the function of $\\mu$ being strictly greater than the function of $\\gamma$, since $\\tfrac{1}{p} \\tr \\bracket{\\big(\\mA + \\mu \\mI_p \\big)^{-1}} < \\tfrac{1}{\\mu}$ for $\\mu > 0$. This means that the intersection with the decreasing function for $\\gamma$ must occur for a smaller value than the intersection for $\\mu$, proving the claim.\n\\end{proof}\n\n\nIn the statement, $\\gamma < \\mu$ means that the orthogonal sketch has less effective regularization than the i.i.d.\\ sketch. For settings in which we desire to solve a linear system with as little distortion as possible, we therefore would much prefer an orthogonal sketch to an i.i.d.\\ sketch, especially for $q \\approx p$.\n\n\n\\begin{figure}[t]\n \\label{fig:practical-concentration}\n \\centering\n \\includegraphics[width=6in]{figures\/practical_concentration.pdf}\n \\caption{\n Empirical density histograms over 20 trials demonstrating the concentration of diagonal elements of $\\mS \\inv{\\mS^\\transp \\mA \\mS + \\lambda \\mI} \\mS^\\transp$ for $\\mA$ as in \\cref{fig:empirical-concentration} with $q \\approx 0.8 p$, $\\lambda = 1$ and several normalized sketches $\\mS$ commonly used in practice. We also plot the diagonals of the i.i.d.\\ sketching equivalence $\\inv{\\mA + \\mu \\mI}$ (black, dotted) and the conjectured orthogonal sketching equivalence $\\inv{\\mA + \\gamma \\mI}$ from \\cref{conj:orthonormal-sketch} (red, dashed), where $\\mu \\approx 1.63$ and $\\gamma \\approx 1.17$.}\n\\end{figure}\n\nIn \\cref{fig:practical-concentration}, we repeat the experiment from \\cref{fig:empirical-concentration} for a variety of normalized non-i.i.d.\\ sketches used frequently in practice. Both CountSketch \\cite{charikar2002frequent} and the Fast Johnson--Lindenstrauss Transform (FJLT) \\cite{ailon2009fast} behave similarly to i.i.d.\\ sketching, with the FJLT slightly over-regularizing. As predicted by \\cref{cor:sketched-pseudoinverse-noniso}, adaptive sketching with $\\mW = \\mA$ \\cite{lacotte2019adaptive} behaves very differently from the other sketches, showing only two point masses instead of three since $\\mA^{-1}$ is not well-defined for its eigenvalues of 0. Lastly, the Subsampled Randomized Hadamard Transform (SRHT) \\cite{tropp2011hadamard} is an orthogonal version of the FJLT, and our experiment elucidates the effect of zero padding on the Hadamard transform of the SRHT. The Hadamard transform is defined only for powers of 2, so for other dimensions, the common approach is to simply zero-pad the data to the nearest power of 2. However, from this experiment we can see that this zero-padding can have a significant impact on the effective regularization; for $p$ slightly smaller than a power of 2, the SRHT performs almost identically to an orthogonal sketch. However, for $p$ slightly larger than a power of 2, there is significant effective regularization induced, even though the sketch is still norm-preserving.\n\nOur proposed framework of first- and second-order equivalence promises to provide a principled means of comparison of different sketching techniques. Once $\\gamma$ from \\cref{conj:general-free-sketching} can be determined for a given sketch (which should depend on its spectral properties), an analogous result to \\cref{lem:second-order-sketch} should directly follow to yield inflation with a factor of $\\gamma'$. Armed with both $\\gamma$ and $\\gamma'$ for a collection of sketches, we can compare them using these bias and variance-style comparisons and make principled choices analogously to classical estimation techniques.\n\n\\paragraph{Future work}\n\nAs alluded to in the introduction, \nthe first- and second-order equivalences developed in this work can be used directly to analyze the asymptotics of the predicted values and quadratic errors of sketched ridge regression. \nWe leave a detailed analysis of sketched ridge regression for a companion paper,\nin which we use the results in this work to study both primal (observation-side) and dual (feature-side) sketching of the data matrix, \nas well as joint primal and dual sketching. \nWe believe that our results can also be combined with the techniques in \\cite{liao2021hessian} who obtain deterministic equivalents for the Hessian of generalized linear models,\nenabling precise asymptotics for the implicit regularization due to sketching in nonlinear prediction models such as classification with logistic regression.\n\n\n\n\\section{Introduction}\n\n\nIn large-scale data processing systems, \\emph{sketching} or \\emph{random projections} play an essential role in making computation efficient and tractable. The basic idea is to replace high-dimensional data by relatively low-dimensional random linear projections of the data such that distances are preserved.\nIt is well-known that sketching can significantly reduce the size of the data without harming statistical performance, while providing a dramatic computational advantage \\cite{pmlr-v80-aghazadeh18a,gower2015randomized,lacotte2019adaptive,wang_lee_mahdavi_kolar_srebro_2017}. \nFor a summary of \nresults on the applications of \nsketching in optimization and \nnumerical linear algebra, we refer the reader to \\cite{mahoney2011randomized,woodruff2014sketching}. \n\nIn this work, we present a different kind of result than the usual sketching guarantee. Typically, sketching is guaranteed to preserve the output or statistical performance of computational methods with an error term that vanishes for sufficiently large sketch sizes \\cite{avron2017faster, bakshi2020robust,clarkson2014sketching, ivkin2019communication, pilanci2016iterative, woodruff2021very}. In contrast, we characterize the precise way in which the solution to a computational problem changes when operating on a sketched version of data instead of the original data, showing that sketching induces a specific type of regularization.\n\nOur primary contribution is a statement about the effect of sketching on the (regularized) pseudoinverse of a matrix. An informal statement of our result is as follows. Here the notation $\\mA \\simeq \\mB$ for two matrices $\\mA$ and $\\mB$ indicates an asymptotic first-order equivalence, which we define in \\Cref{sec:preliminaries}, and $\\lambda_{\\min}^{+}(\\mA)$ is the smallest nonzero eigenvalue of a matrix $\\mA$. We refer to $\\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q} \\mS^\\ctransp$ as the sketched (regularized) pseudoinverse of $\\mA$, because when $\\mS$ has orthonormal columns, the pseudoinverse of $\\mS \\mS^\\ctransp \\mA \\mS \\mS^\\ctransp$ is equal to $\\mS (\\mS^\\ctransp \\mA \\mS)^{-1} \\mS^\\ctransp$. This expression is related to the Nystr\\\"om approximation of the inverse of $\\mA$.\n\n\\begin{inftheorem}[\\Cref{thm:sketched-pseudoinverse}, informal]\n \\label{thm:sketched-pseudoinverse-informal}\n Given a positive semidefinite matrix $\\mA \\in \\complexset^{p \\times p}$, random sketching matrix $\\mS \\in \\complexset^{p \\times q}$, for any $\\lambda > -\\lambda_{\\min}^{+}(\\mS^\\ctransp \\mA \\mS)$, there exists $\\mu \\in \\reals$ such that\n \\begin{align}\n \\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q} \\mS^\\ctransp \\simeq \\inv{\\mA + \\mu \\mI_p}.\n \\end{align}\n\\end{inftheorem}\n\nThe general implication of this result is that when we do computation using the sketched version of a matrix, there is a sense in which it is as if we were using ridge regularization. More precisely, when we solve (regularized) linear systems on a sketched version of the data and apply this solution to the sketched data, it is equivalent in a first-order sense to solving a regularized linear system in the original space. To see this, consider for example a least squares problem $\\min_\\vbeta \\norm[2]{\\vy - \\mX \\vbeta}^2$. The first-order optimality condition is $\\mX^\\ctransp \\mX \\vbeta = \\mX^\\ctransp \\vy$, and if we replace $\\mX$ by a sketch $\\mX \\mS$, we have the solution in the sketched domain $\\widehat{\\vbeta}_\\mS = (\\mS^\\ctransp \\mX^\\ctransp \\mX \\mS)^{-1} \\mS^\\ctransp \\mX^\\ctransp \\vy$. If we then apply this solution to a new sketched data point $\\mS^\\ctransp \\vx$, we obtain the prediction $\\widehat{y} = \\vx^\\ctransp \\mS (\\mS^\\ctransp \\mX^\\ctransp \\mX \\mS)^{-1} \\mS^\\ctransp \\mX^\\ctransp \\vy$. By our result, this is asymptotically equivalent to making the prediction $\\widehat{y} = \\vx^\\ctransp (\\mX^\\ctransp \\mX + \\mu \\mI)^{-1} \\mX^\\ctransp \\vy$---that is, as if we had solved the original least squares problem using some regularization $\\mu$.\n\n\\subsection*{Summary of Contributions}\n\nBelow we summarize the main contributions of the paper.\n\n\\begin{enumerate}\n \\item \\textbf{Real-valued equivalence.} We extend previous results from random matrix theory \\cite{rubio_mestre_2011} to real-valued regularization, explicitly characterizing\n the behaviour of the associated fixed-point\n equation extended from the complex half-plane to the reals, allowing\n for consideration of negative regularization. This result includes what is to the best of our knowledge the first characterization of the limiting smallest nonzero eigenvalue of arbitrary sample covariance matrices, which may be of independent interest.\n \\item \\textbf{First-order equivalence.} Applying the real-valued equivalence, we obtain a first-order equivalence for the ridge-regularized sketched pseudoinverse.\n \\item \\textbf{Second-order equivalence.} Using the calculus of asymptotic equivalents,\n we also obtain a second-order equivalence\n for ridge-regularized sketched pseudoinverse.\n \\item \\textbf{Equivalence properties.} We provide a thorough investigation of the theoretical properties of the equivalence relationship, such as how the induced regularization depends on the original applied regularization, sketch size, and matrix rank.\n \\item \\textbf{Free sketching conjecture.} Finally, we extend the scope of our results\n for first-order equivalence of the sketched pseudo-inverse to general asymptotically free sketching in the form of a conjecture and specialize to orthogonal sketching matrices.\n\\end{enumerate}\n\n\\subsection*{Related work}\n\nThe existence of an implicit regularization effect of sketching or random projections has been known for some time \\cite{derezinski2021determinantal,leamer1976bayesian, rudi2015less, thanei_heinze_meinshausen_2017}. %\nWhile prior works have demonstrated clear theoretical and empirical statistical advantages of sketching, our understanding of the precise nature of this implicit regularization has been largely limited to quantities such as error bounds. We provide, in contrast, a precise asymptotic characterization of the solution obtained by a sketching-based solver, not only enabling the understanding of the statistical performance of sketching-based methods, but also opening the door for exploiting the specific regularization induced by sketching in future algorithms.\n\nOur results in this work provide a general extension \nof a few results appearing in recent works that have revealed explicit characterizations of the implicit regularization effects induced by random subsampling. To the best of our knowledge, the first such result was presented by \\cite{lejeune_javadi_baraniuk_2020}, who showed that ensembles of (unregularized) ordinary least squares predictors on randomly subsampled observations and features converge in an $\\ell_2$ metric to an (optimal) ridge regression solution in the proportional asymptotics regime. \nThis result was limited in several aspects: \na) it required a strong isotropic Gaussian data assumption;\nb) it required the subsampled data to have more observations than features;\nc) it considered only unregularized base learners in the ensemble;\nd) it required an ensemble of infinite size to show \nthe ridge regression equivalence;\ne) it provided only a marginal guarantee of convergence \nover the data distribution \nrather than a single-instance convergence guarantee;\nand \nf) it did not provide the relationship between the subsampling ratio and the amount of induced ridge regularization. In addition, the proof relied on rote computation of expectations of matrix quantities, providing limited insight into the underlying mathematical principles at work. The result we present in this work in \\cref{thm:sketched-pseudoinverse} addresses all of these issues.\n\nAround the same time, \\cite{pmlr-v108-mutny20a} showed the remarkably simple result that the expected value of the pseudoinverse of any positive definite matrix sampled by a determinantal point process (DPP) is equal to a resolvent of the matrix. Similarly to the result by \\cite{lejeune_javadi_baraniuk_2020}, this result demonstrated that when random subsampling is applied in techniques without any regularization, the resulting solution is as if a regularized technique was used on the original data. This result provided a simple form of the argument of the induced resolvent as a solution to a matrix trace equation, which is analogous the results we present in this work for sketching. However, the proof technique relied on convenient algebraic properties of the DPP, again providing limited insight into what happens for more commonplace and computationally simple low-dimensional projection techniques such as random sketching. Despite this theoretical limitation, the same authors later empirically demonstrated that the same effects occur when using i.i.d.\\ Gaussian and Rademacher sketches \\cite{derezinski_surrogate_2020}, suggesting the broader result that we present in this work. In addition to our result in \\cref{thm:sketched-pseudoinverse} applying to i.i.d.\\ sketching matrices rather than DPP subsampling, our result also differs from this work in that we provide a single-instance equivalent ridge regularization in the asymptotic regime, rather than an expectation over the random subsamplings.\n\nOur results also echo the finite-sample results of \\cite{pmlr-v134-derezinski21a}, who showed that the unregularized inverse of a particular sketched matrix form has a merely multiplicative bias for sketch size minimally larger than the rank of the original matrix. This is captured by \\cref{cor:basic-ridge-asympequi-in-r} in our work when $z \\to 0$, combined with \\cref{rem:mu-prime-to-0} in which we observe that there is asymptotically no spectral distortion in the range of the original matrix for sketches larger than the rank.\n\n\n\n\n\\subsection*{Organization}\n\nThe rest of the paper is structured as follows.\nIn \\Cref{sec:preliminaries},\nwe start with some preliminaries\non the language of asymptotic equivalence\nof random matrices that we will use to state our results.\nIn \\Cref{sec:real-valued-equivalence},\nwe extend a previous result on asymptotic equivalence\nfor a ridge regularized resolvent\nto include real-valued negative regularization\nand provide a precise limiting lower limit\nof the permitted negative regularization.\nIn \\Cref{sec:main_results},\nwe provide our main results\nabout the first- and second-order\nequivalence of the sketched pseudoinverse.\nThen, in \\Cref{sec:properties}, we explore properties of the equivalence and present illustrative examples.\nFinally, in \\Cref{sec:discussion},\nwe conclude by giving\nvarious extensions and providing\na general conjecture on the asymptotic\nbehaviour of sketched pseudoinverse\nfor a broad family of sketching matrices\nusing the insights obtained from the proof\nof our main result and experimentally compare sketches commonly used in practice to our theory.\nOur code for generating all figures can be found at \\url{https:\/\/github.com\/dlej\/sketched-pseudoinverse}.\n\n\n\n\\subsection*{Notation}\nWe denote the real line by $\\mathbb{R}$\nand the complex plane by $\\mathbb{C}$.\nFor a complex number $z = x + iy$,\n$\\Re(z)$ denotes its real part $x$,\n$\\Im(z)$ denotes its imaginary part $y$,\nand $\\overline{z} = x - iy$ denotes its conjugate.\nWe use $\\mathbb{R}_{\\ge 0}$ and $\\mathbb{R}_{> 0}$ to be denote \nthe set of non-negative and positive real numbers, respectively;\nsimilarly, $\\mathbb{R}_{\\le 0}$ and $\\mathbb{R}_{< 0}$ respectively denote\nthe set of non-positive and negative real numbers.\nWe use $\\mathbb{C}^{+} = \\{ z \\in \\mathbb{C} : \\Im(z) > 0 \\}$ to denote \nthe upper half of the complex plane\nand $\\mathbb{C}^{-} = \\{ z \\in \\mathbb{C} : \\Im(z) < 0 \\}$ to denote \nthe lower half of the complex plane. \n\nWe denote vectors in lowercase bold letters (e.g., $\\vy$)\nand matrices in uppercase bold letters (e.g., $\\mX$).\nFor a vector $\\vy$, \n$\\| \\vy \\|_2$ denotes its $\\ell_2$ norm.\nFor a rectangular matrix ${\\mathbf{S}} \\in \\mathbb{C}^{p \\times q}$, \n${\\mathbf{S}}^\\ctransp \\in \\mathbb{C}^{q \\times p}$\ndenotes its conjugate or Hermitian transpose\n(such that $[\\mS^\\ctransp]_{ij} = \\overline{[\\mS]_{ji}}$),\n$\\norm[\\tr]{\\mS}$ denotes its trace norm (or nuclear norm),\nthat is $\\norm[\\tr]{\\mS} = \\tr\\bracket{(\\mS^\\ctransp \\mS)^{1\/2}}$,\nand $\\| {\\mathbf{S}} \\|_{\\rm op}$ denotes the operator norm\nwith respect to the $\\ell_2$ vector norm\n(which is also its spectral norm).\nFor a square matrix ${\\mathbf{A}} \\in \\mathbb{C}^{p \\times p}$,\n$\\tr[{\\mathbf{A}}]$ denotes its trace,\n$\\mathrm{rank}(\\mA)$ denotes its rank,\n$r(\\mA) = \\frac{1}{p} \\mathrm{rank}(\\mA)$\ndenotes its relative rank,\nand ${\\mathbf{A}}^{-1} \\in \\mathbb{C}^{p \\times p}$ denotes its inverse,\nif it is invertible.\nFor a positive semidefinite matrix ${\\mathbf{A}} \\in \\mathbb{C}^{p \\times p}$,\n${\\mathbf{A}}^{1\/2} \\in \\mathbb{C}^{p \\times p}$ denotes its positive semidefinite \nprincipal square root,\n$\\lambda_{\\min}({\\mathbf{A}})$ denotes its smallest eigenvalue, \nand $\\lambda_{\\min}^{+}({\\mathbf{A}})$ denotes its smallest positive eigenvalue.\n\nA sequence $x_n$ converging to $x_{\\infty}$ from the left \nand right is denoted by $x \\nearrow x_{\\infty}$ and $x \\searrow x_{\\infty}$, respectively.\nWe denote almost sure convergence by $\\xrightarrow{\\text{a.s.}}$.\n\n\n\n\n\n\n\n\n\n\\section{Main results}\n\\label{sec:main_results}\n\nOne way to think about \\cref{cor:basic-ridge-asympequi-in-r}\nis that the data matrix ${\\mathbf{X}} = {\\mathbf{Z}} \\mSigma^{1\/2}$ is a sketched version of the (square root) covariance matrix $\\mSigma^{1\/2}$,\nwhere ${\\mathbf{Z}}$ acts as a sketching matrix. The sketching is done by ``nature'' in the form of the $n$ observations,\nrather than by the statistician, but is otherwise mathematically identical to sketching. Using this insight, along with the Woodbury identity,\nwe can adapt the random matrix resolvent equivalence in \\cref{cor:basic-ridge-asympequi-in-r} to a sketched (regularized) pseudoinverse equivalence. To emphasize the shift in perspective, we denote the dimensionality of the sketched data as $q$ (replacing $n$), replace $\\mSigma$ with $\\mA$, and absorb the normalization by $\\tfrac{1}{q}$ (replacing $\\tfrac{1}{n}$) into the sketching matrix $\\mS$ (replacing $\\mZ$), so that the sketching transformation is norm-preserving (see \\Cref{rem:norm-preserving-sketch} for more details).\n\n\n\n\\subsection{First-order equivalence}\n\nOur first result provides a first-order equivalence for the sketched regularized pseudoinverse.\nBy first-order equivalence,\nwe refer to equivalence for matrices that involve \nthe \\emph{first} power of the ridge resolvent.\nWe also present a second-order equivalence\nfor matrices that involve the \\emph{second} power\nof the ridge resolvent in \\Cref{sec:second-order-sketch-equi}.\n\nIn preparation for the statements to follow,\nrecall that $r({\\mathbf{A}}) = \\frac{1}{p} \\sum_{i=1}^{p} \\mathbbm{1}\\{ \\lambda_i({\\mathbf{A}}) > 0 \\}$,\nor in other words, the normalized number of non-zero eigenvalues of ${\\mathbf{A}}$.\nNote that $0 \\le r({\\mathbf{A}}) \\le 1$.\n\n\\begin{theorem}\n [Isotropic sketching equivalence]\n \\label{thm:sketched-pseudoinverse}\n Let $\\mA \\in \\complexset^{p \\times p}$ be a positive semidefinite\n matrix such that $\\| \\mA \\|_{\\rm op}$ is uniformly bounded in $p$\n and $\\liminf \\lambda_{\\min}^{+}({\\mathbf{A}}) > 0$.\n Let $\\sqrt{q}\\mS \\in \\complexset^{p \\times q}$ be a random matrix\n consisting of i.i.d.\\ random variables that have mean 0, variance 1, and finite $8 + \\delta$ moment for some $\\delta > 0$. \n Let $\\lambda_0, \\mu_0 \\in \\reals$ be the unique solutions, satisfying $\\mu_0 > - \\lambda_{\\min}^{+}(\\mA)$, to the system of equations\n \\begin{align}\n \\label{eq:mu0-lambda0-fps}\n 1 = \\tfrac{1}{q} \\tr \\bracket{\\mA^2 \\paren{\\mA + \\mu_0 \\mI_p}^{-2}}, \\quad\n \\lambda_0 = \\mu_0 \\paren{1 - \\tfrac{1}{q} \\tr \\bracket{\\mA \\inv{\\mA + \\mu_0 {\\mathbf{I}}_p}}}.\n \\end{align}\n Then, \n as $q, p \\to \\infty$ \n such that\n $0 < \\liminf \\tfrac{q}{p} \\le \\limsup \\tfrac{q}{p} < \\infty$, \n the following asymptotic equivalences hold:\n \\begin{enumerate}[topsep=1em,parsep=0pt,label=(\\roman*)]\n \\item for any $\\lambda > \\limsup \\lambda_0$, \n we have\n \\end{enumerate}\n \\vspace{-\\abovedisplayskip}\n \\begin{align}\n \\label{eq:thm:sketched-pseudoinverse-A-half}\n \\mA^{1\/2} \\mS \\big( \\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q\n \\big)^{-1} \\mS^\\ctransp\n \\simeq \\mA^{1\/2} \\inv{\\mA + \\mu \\mI_p};\n \\end{align}\n \\begin{enumerate}[resume*]\n \\item if furthermore either $\\lambda \\neq 0$ or $\\limsup \\tfrac{q}{p} < \\liminf r(\\mA)$,\n we have\n \\end{enumerate}\n \\vspace{-\\abovedisplayskip}\n \\begin{align}\n \\label{eq:thm:sketched-pseudoinverse}\n \\mS \\big( \\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q\n \\big)^{-1} \\mS^\\ctransp\n \\simeq \\inv{\\mA + \\mu \\mI_p},\n \\end{align}\n where $\\mu$\n is the unique solution in $(\\mu_0, \\infty)$ to the fixed point equation\n \\begin{align}\n \\label{eq:sketched-modified-lambda}\n \\lambda = \\mu \\paren{ 1 - \\tfrac{1}{q} \\tr \\bracket{\\mA \\inv{\\mA + \\mu \\mI_p}}}.\n \\end{align}\n Furthermore, as $p, q \\to \\infty$, $|\\mu - \\tfrac{1}{ \\widetilde{v}(\\lambda)}| \\xrightarrow{\\text{a.s.}} 0$, where\n \\begin{align}\n \\widetilde{v}(\\lambda) = \\tfrac{1}{q} \\tr \\bracket{\\big( \\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q\n \\big)^{-1} },\n \\end{align}\n and $|\\lambda_0 + \\lambda_{\\min}^{+}(\\mS^\\ctransp \\mA \\mS)| \\xrightarrow{\\text{a.s.}} 0$. \n\\end{theorem}\n\n\\begin{proof}[Proof sketch]\nWe begin by considering the case that $\\mA$ satisfies $\\limsup \\bignorm[\\rm op]{\\mA^{-1}} < \\infty$. Then we can rewrite the left-hand side of \\cref{eq:thm:sketched-pseudoinverse-A-half} or \\cref{eq:thm:sketched-pseudoinverse}\nsuch that we can apply \\cref{cor:basic-ridge-asympequi-in-r} with $\\mX = \\sqrt{q} \\mS^\\ctransp \\mA^{1\/2}$, $\\lambda = -z$, and $\\mu = -\\zeta$. For any $\\lambda > -\\liminf z_0$,\n\\begin{subequations}\n\\begin{align}\n \\mA^{1\/2} \\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q } \\mS^\\ctransp \\mA^{1\/2} &= \n \\mA^{1\/2} \\mS \\mS^\\ctransp \\mA^{1\/2} \\biginv{\\mA^{1\/2} \\mS \\mS^\\ctransp \\mA^{1\/2} + \\lambda \\mI_p } \\\\\n & = \\mI_p - \\lambda \\biginv{\\mA^{1\/2} \\mS \\mS^\\ctransp \\mA^{1\/2} + \\lambda \\mI_p } \\\\\n & \\simeq \\mI_p - \\mu \\biginv{\\mA + \\mu \\mI_p} \\\\\n & = \\mA^{1\/2} \\biginv{\\mA + \\mu \\mI_p} \\mA^{1\/2}.\n\\end{align}\n\\end{subequations}\nWe can then multiply on the right, or both left and right, by $\\mA^{-1\/2}$ to obtain the results in \\cref{eq:thm:sketched-pseudoinverse-A-half} and \\cref{eq:thm:sketched-pseudoinverse}, respectively, by the product rule of asymptotic equivalences. If $\\mA$ does not have a norm-bounded inverse, we can apply the above result for $\\mA_\\delta \\defeq \\mA + \\delta \\mI_p$ for $\\delta > 0$ and make a uniform convergence argument for interchanging limits of $p$ and $\\delta$ to prove the equivalence in \\cref{eq:thm:sketched-pseudoinverse}. We then multiply by $\\mA^{1\/2}$ and make another uniform convergence argument to extend this equivalence to the case $\\lambda = 0$ to obtain the equivalence in \\cref{eq:thm:sketched-pseudoinverse-A-half}. The details can be found in \\Cref{sec:proof:thm:sketched-pseudoinverse} of the supplementary material.\n\nWe also provide an alternative but incomplete proof strategy that is based on Jacobi's formula rather than on \\cref{cor:basic-ridge-asympequi-in-r} in \\Cref{sec:proofs-main-results-jacobi} of the supplementary material. The idea is to relate the derivatives of $\\log |\\mS^\\ctransp (\\mA + t \\mTheta) \\mS + \\lambda \\mI_q|$ and $\\log |\\mA + t \\mTheta + \\mu \\mI_q|$ with respect to each $t=0$ and $\\lambda$---the former yields the $\\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI} \\mS^\\ctransp$ term we are interested in, and the latter yields the Stieltjes transform of the spectrum of $\\mS^\\ctransp \\mA \\mS$. In this way, we can build an asymptotic equivalence (for trace functionals with general $\\mTheta$) using only a spectral relationship (determined by $\\mTheta = \\mI$). We do not need this proof strategy for \\cref{thm:sketched-pseudoinverse}, but we think it may be valuable for proving \\cref{conj:general-free-sketching} in \\Cref{sec:discussion}.\n\\end{proof}\n\nIn words, the sketched pseudoinverse of $\\mA$ with regularization $\\lambda$ is asymptotically equivalent to the regularized inverse of $\\mA$ with regularization $\\mu$, and the relationship between $\\lambda$ and $\\mu$ asymptotically depends only on $\\mA$, $p$, and $q$. As mentioned in \\Cref{sec:preliminaries}, this implies for example that the elements of the sketched pseudoinverse converge to the elements of the ridge-regularized inverse. We illustrate this in \\cref{fig:empirical-concentration}, where for a diagonal $\\mA$, the off-diagonals of the sketched pseudoinverse quickly converge to zero as $p$ increases, while the diagonals converge to the diagonals of the regularized inverse of $\\mA$.\n\nIn the case where $\\lambda = 0$ and $q < p$, we see that \\cref{eq:sketched-modified-lambda} reduces to $q = \\tr [\\mA \\inv{\\mA + \\mu \\mI_p}]$, which is exactly the same as the result obtained by \\cite{pmlr-v108-mutny20a} for the expected pseudoinverse under the determinantal point process (DPP), with $q$ playing the role of the expected subsample size of the DPP and $\\mu$ being the DPP scale factor. To our knowledge, the regularized pseudoinverse using the DPP has not been further investigated, so we cannot compare our equivalence beyond the $\\lambda = 0$ case. \nIn addition, it also coincides precisely with the optimal choice of $\\alpha = \\tfrac{q}{p}$ for subsampled ordinary least squares ensembles in \\cite{lejeune_javadi_baraniuk_2020} when $\\mu$ is the optimal ridge regularization strength. \n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=5in]{figures\/empirical_concentration.pdf}\n \\caption{Empirical density histograms over 20 trials demonstrating the concentration of the elements of $\\mS \\inv{\\mS^\\transp \\mA \\mS + \\lambda \\mI} \\mS^\\transp$ for real Gaussian $\\mS$ and diagonal $\\mA$ taking values $\\set{0, 1, 2}$ with equal frequency along the diagonal. We choose $\\lambda = 1$ and $q = \\floor{\\alpha p}$ for $\\alpha = 0.8$ over $p \\in \\set{60, 300, 1500}$. As expected by \\cref{thm:sketched-pseudoinverse}, the individual elements of the sketched pseudoinverse converge to those of $\\inv{\\mA + \\mu \\mI}$, where for this problem $\\mu \\approx 1.63$. Therefore, the diagonals concentrate with equal mass around $\\set{1 \/ (a + \\mu) : a \\in \\set{0, 1, 2}}$ (black, dotted), and the off-diagonals concentrate around 0.}\n \\label{fig:empirical-concentration}\n\\end{figure}\n\n\nBelow we provide several remarks on the assumptions and implications\nof \\cref{thm:sketched-pseudoinverse}. It will be useful to interpret the equations in terms of the sketching aspect ratio $\\alpha \\defeq \\tfrac{q}{p}$.\n\n\n\n\\begin{remark}\n [Normalization choice for the sketching matrix]\n \\label{rem:norm-preserving-sketch}\n We remark that the normalization factor $\\sqrt{q}$ in $\\sqrt{q} {\\mathbf{S}}$\n of the sketching matrix is such that\n the norm of the rows of ${\\mathbf{S}}$ is $1$ in expectation.\n This is done so that \n $\\mathbb{E}[ \\| {\\mathbf{S}}^\\ctransp {\\mathbf{x}} \\|_2^2] = \\| {\\mathbf{x}} \\|_2^2$\n as $\\mathbb{E}[{\\mathbf{S}} {\\mathbf{S}}^\\ctransp] = {\\mathbf{I}}_p$.\n One can alternately consider sketching matrices with normalization \n $\\sqrt{p} {\\mathbf{S}}$ such that the columns have norm $1$ in expectation.\n It is easy to write an equivalent version of \\Cref{thm:sketched-pseudoinverse}\n with such a normalization. \n We choose to focus on the former scaling\n because it is more common in practice.\n\\end{remark}\n\n\\begin{remark}\n [On assumptions]\n The assumptions imposed in \\cref{thm:sketched-pseudoinverse}\n are quite mild.\n In particular, the sequences of matrices ${\\mathbf{A}}$ \n being sketched can be random, so long as they are independent of ${\\mathbf{S}}$.\n Furthermore, the spectrum of the sequences of matrices ${\\mathbf{A}}$\n need not converge to a fixed spectrum.\n The aspect ratio $\\alpha$ of the sketching matrices ${\\mathbf{S}}$\n also need not converge to a fixed number.\n The reason this is possible is because\n we are not expressing the sketched resolvent\n in terms of the limiting spectrum of ${\\mathbf{S}}$ and ${\\mathbf{A}}$,\n but rather relating it through ${\\mathbf{A}}$ and a parameter $\\mu$\n that depends on $\\alpha$ and ${\\mathbf{A}}$ \n (and the original regularization level $\\lambda$),\n which allows us to keep our assumptions weak. \n\\end{remark}\n\n\n\\begin{remark}\n [Case of $\\lambda = 0$]\n While the form in \\cref{eq:thm:sketched-pseudoinverse} is the most general, it does not hold for $\\lambda = 0$ if the sketch size is larger than the rank of $\\mA$, since the inverse is unbounded. However, in machine learning settings such as ridgeless regression, we only need to evaluate the regularized pseudoinverse $\\mS (\\mS^\\ctransp \\tfrac{1}{n} \\mX^\\ctransp \\mX \\mS + \\lambda \\mI_p)^{-1} \\mS^\\ctransp \\tfrac{1}{\\sqrt{n}} \\mX$. Thus, we can apply the form in \\cref{eq:thm:sketched-pseudoinverse-A-half} with $\\mA^{1\/2} = (\\tfrac{1}{n} \\mX^\\ctransp \\mX)^{1\/2}$, which is sufficient for any downstream analysis.\n\\end{remark}\n\n\\begin{remark}\n [Alternate form of equivalence representation]\n Expressed in terms of $\\widetilde{v}(\\lambda)$,\n the equivalence \\cref{eq:thm:sketched-pseudoinverse} becomes \n \\begin{equation}\n {\\mathbf{S}} \\big( {\\mathbf{S}}^\\ctransp {\\mathbf{A}} {\\mathbf{S}} + \\lambda {\\mathbf{I}}_q \\big)^{-1} {\\mathbf{S}}^\\ctransp\n \\simeq\n \\widetilde{v}(\\lambda) \\inv{\\widetilde{v}(\\lambda) {\\mathbf{A}} + {\\mathbf{I}}_p},\n \\end{equation}\n and the fixed-point equation \\cref{eq:sketched-modified-lambda} becomes\n \\begin{equation}\n \\lambda \n = \\frac{1}{\\widetilde{v}(\\lambda)}\n - \\tfrac{1}{q} \\tr \\bracket{{\\mathbf{A}} \\inv{\\widetilde{v}(\\lambda) {\\mathbf{A}} + {\\mathbf{I}}_p} }.\n \\end{equation}\n\\end{remark}\n\n\n\\subsection{Second-order equivalence}\n\\label{sec:second-order-sketch-equi}\n\nAlthough the equivalence in \\cref{thm:sketched-pseudoinverse} holds for first order trace functionals, this equivalence does not hold for higher order functionals. To intuitively understand why, it is helpful to reason about the asymptotic equivalence similarly to an equivalence of expectation in classical random variables. That is, we may have two random variables $X, Y$ with $\\expect{X} = \\expect{Y}$, but this does not allow us to make any conclusions about the relationship between $\\expect{X^k}$ and $\\expect{Y^k}$ for $k > 1$. In the same way, our first-order asymptotic equivalence does not directly tell us higher order equivalences.\n\nFortunately, however, because of the resolvent structure of the regularized pseudoinverse, we can cleverly apply the derivative rule of the calculus of asymptotic equivalences to obtain a second order equivalence from the first order equivalence. Such a derivative trick has been employed in several prior works; see, e.g., \\cite{dobriban_wager_2018, karoui_kolsters_2011, ledoit_peche_2011, liu_dobriban_2019}. This approach could in principle be repeated for higher order functionals.\n\n\n\\begin{theorem}\n [Second-order isotropic sketching equivalence]\n \\label{lem:second-order-sketch}\n Consider the setting of \\cref{thm:sketched-pseudoinverse}. If $\\mPsi \\in \\complexset^{p \\times p}$ is a deterministic or random positive semidefinite matrix independent of $\\mS$ with $\\norm[\\rm op]{\\mPsi}$ uniformly bounded in $p$, then if either $\\lambda \\neq 0$ or $\\limsup \\tfrac{q}{p} < \\liminf r(\\mA)$,\n \\begin{align}\n \\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q} \\mS^\\ctransp \\mPsi \\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q} \\mS^\\ctransp\n \\simeq\n \\inv{\\mA + \\mu \\mI_p} (\\mPsi + \\mu' \\mI_p) \\inv{\\mA + \\mu \\mI_p},\n \\end{align}\n where $\\mu$ is as in \\cref{thm:sketched-pseudoinverse}, and\n \\begin{align}\n \\label{eq:mu-prime}\n \\mu' = \\frac{\\frac{1}{q} \\tr \\bracket{\\mu^3 \\inv{\\mA + \\mu \\mI_p} \\mPsi \\inv{\\mA + \\mu \\mI_p} }}{\\lambda + \\frac{1}{q} \\tr \\bracket{\\mu^2 \\mA \\paren{\\mA + \\mu \\mI_p}^{-2}} } \\geq 0.\n \\end{align}\n\\end{theorem}\n\n\\begin{proof}[Proof]\nBy assumption, there exists $M < \\infty$ such that $M > \\limsup \\bignorm[\\rm op]{\\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q}}$ and $M > \\limsup \\bignorm[\\rm op]{\\inv{\\mA + \\mu \\mI_p}}$ almost surely (see proof details for \\cref{thm:sketched-pseudoinverse} in the supplementary material). Define $\\mB_z \\defeq \\mA + z \\mPsi$. Then for all $z \\in D$, where\n\\begin{align}\n D = \\bigset{z \\in \\complexset \\colon \\limsup \\bigparen{|z| M \\norm[\\rm op]{\\mPsi} \\max \\bigset{\\norm[\\rm op]{\\mS}^2, 1}} < \\tfrac{1}{2}},\n\\end{align} we have that $\\max \\bigset{\\limsup \\bignorm[\\rm op]{\\biginv{\\mS^\\ctransp \\mB_z \\mS + \\lambda \\mI_q}}, \\limsup \\bignorm[\\rm op]{\\inv{\\mB_z + \\mu \\mI_p}}}\n\\leq 2 M$. Therefore, we can apply the differentiation rule of asymptotic equivalences for all $z \\in D$:\n\\begin{subequations}\n\\begin{align}\n -\\mS \\biginv{\\mS^\\ctransp \\mB_z \\mS &+ \\lambda \\mI_q} \\mS^\\ctransp \\mPsi \\mS \\biginv{\\mS^\\ctransp \\mB_z \\mS + \\lambda \\mI_q} \\mS^\\ctransp\n = \\tfrac{\\partial}{\\partial z} \\mS \\biginv{\\mS^\\ctransp \\mB_z \\mS + \\lambda \\mI_q} \\mS^\\ctransp \\\\\n &\\simeq \\tfrac{\\partial}{\\partial z} \\biginv{\\mB_z + \\mu(z) \\mI_p} \\\\\n &= - \\biginv{\\mB_z + \\mu(z) \\mI_p} \\paren{\\mPsi + \\tfrac{\\partial}{\\partial z} \\mu(z) \\mI_p} \\biginv{\\mB_z + \\mu(z) \\mI_p}.\n\\end{align}\n\\end{subequations}\nWe let $\\mu'(z) = \\tfrac{\\partial}{\\partial z} \\mu(z)$, and then we can divide \\cref{eq:sketched-modified-lambda} by $\\mu(z)$ and differentiate to obtain\n\\begin{align}\n \\frac{\\lambda \\mu'(z)}{\\mu(z)^2} = \\tfrac{1}{q} \\tr \\bracket{\\mPsi \\inv{\\mB_z + \\mu(z) \\mI_p} - \\mB_z \\inv{\\mB_z + \\mu(z) \\mI_p} \\paren{\\mPsi + \\mu'(z) \\mI_p } \\inv{\\mB_z + \\mu(z) \\mI_p} }.\n\\end{align}\nSolving for $\\mu'(0)$ gives the expression in $\\cref{eq:mu-prime}$. For the non-negativity of $\\mu'$, see \\cref{rem:alt-mu-prime} and its proof.\n\\end{proof}\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=5in]{figures\/empirical_concentration_second.pdf}\n \\caption{\n Empirical density histograms over 20 trials demonstrating the concentration of diagonal elements of $\\mS \\inv{\\mS^\\transp \\mA \\mS + \\lambda \\mI} \\mS^\\transp \\mPsi \\mS \\inv{\\mS^\\transp \\mA \\mS + \\lambda \\mI} \\mS^\\transp$ for $(\\mS, \\mA, \\lambda)$ as in \\cref{fig:empirical-concentration} and $\\mPsi \\in \\set{\\mI_p, \\mA}$. As expected by \\cref{lem:second-order-sketch}, the individual elements of the sketched pseudoinverse converge to those of $\\inv{\\mA + \\mu \\mI}(\\mPsi + \\mu' \\mI) \\inv{\\mA + \\mu \\mI}$ (black, dotted), where $\\mu' \\approx 0.813$ and $0.403$ for $\\mPsi = \\mI_p$ and $\\mA$, respectively.}\n \\label{fig:empirical-concentration-second}\n\\end{figure}\n\n\nThat is, the second-order equivalence is the same as plugging in the first-order equivalence and then adding a non-negative inflation $\\mu' \\inv[2]{\\mA + \\mu \\mI}$. The inflation factor $\\mu'$ depends linearly on the matrix $\\mPsi$, but the inflation is always isotropic, rather than in the direction of $\\mPsi$. It is non-negative in the same way that the variance of an estimator is also non-negative. Examples of quadratic forms where this second-order equivalence can be used include estimation error ($\\mPsi = \\mI$) and prediction error ($\\mPsi = \\mSigma$, the population covariance) in ridge regression problems. We give a demonstration of the concentration in \\cref{fig:empirical-concentration-second}.\nWhile typically $\\mu' > 0$, it can go to 0 in the special case of $\\mu = 0$ and $\\mPsi$ sharing a subspace with $\\mA$, as we discuss in \\cref{rem:mu-prime-to-0}. \n\n\\begin{remark}\n [The case of $\\lambda = 0$]\n Similar to the variant form in \\cref{eq:thm:sketched-pseudoinverse-A-half} of \\cref{thm:sketched-pseudoinverse}, if we consider the slightly different form\n \\begin{align}\n \\mA^{1\/2}\n \\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q} \\mS^\\ctransp \\mPsi &\\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q} \\mS^\\ctransp\n \\mA^{1\/2} \\\\\n &\\simeq\n \\mA^{1\/2}\n \\inv{\\mA + \\mu \\mI_p} (\\mPsi + \\mu' \\mI_p) \\inv{\\mA + \\mu \\mI_p}\n \\mA^{1\/2}\n \\end{align}\n for the second-order resolvent,\n we do not need the $\\lambda \\neq 0$ or $\\limsup \\tfrac{q}{p} < \\liminf r(\\mA)$ restriction as stated in the theorem. Because the proof of this case is entirely analogous to the results in \\cref{thm:sketched-pseudoinverse,lem:second-order-sketch}, we omit the proof.\n\\end{remark}\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\n\n\n\nWe will use the language of asymptotic equivalence of sequences of random matrices\nto state our main results.\nIn this section, we define the notion of asymptotic equivalence,\nreview some of the basic properties that such equivalence satisfies, \nand present an asymptotic equivalence for the ridge resolvent.\nWe then extend that result to handle real-valued resolvents, which\nwill form the building block for our subsequent results.\n\n\nTo begin, consider two sequences ${\\mathbf{A}}_n$ and ${\\mathbf{B}}_n$ of $p(n) \\times q(n)$ matrices, where $p$ and $q$ are increasing in $n$.\nWe will say that ${\\mathbf{A}}_n$ and ${\\mathbf{B}}_n$ are asymptotically equivalent\nif for any sequence of deterministic \nmatrices ${\\bm{\\Theta}}_n$ with trace norm uniformly bounded in $n$,\nwe have $\\tr[{\\bm{\\Theta}}_n ({\\mathbf{A}}_n - {\\mathbf{B}}_n)] \\xrightarrow{\\text{a.s.}} 0$\nas $n \\to \\infty$.\nWe write ${\\mathbf{A}}_n \\simeq {\\mathbf{B}}_n$ to denote this asymptotic equivalence.\nThe notion of \\emph{deterministic} equivalence,\nwhere the right-hand sequence is a sequence of deterministic matrices, \nhas been typically used in random matrix theory to obtain limiting behaviour\nof functionals of random matrices;\nfor example,\nsee \n\\cite{couillet_debbah_silverstein_2011,hachem_loubaton_najim_2006,serdobolskii_2000},\namong others.\nMore recently, the notion of deterministic equivalence \nhas been popularized and developed further \nin \\cite{dobriban_wonder_2020,dobriban_sheng_2021}\\footnote{Note that \n\\cite{dobriban_wonder_2020,dobriban_sheng_2021}\nuse the notation ${\\mathbf{A}}_n \\asymp {\\mathbf{B}}_n$\nto denote deterministic equivalence of sequence ${\\mathbf{A}}_n$ to ${\\mathbf{B}}_n$.\nWe instead use the notation ${\\mathbf{A}}_n \\simeq {\\mathbf{B}}_n$\nto emphasize that this equivalence is asymptotically exact, rather than up to constants.}.\nWe will use a slightly more general notion of asymptotic equivalence\nin this paper, where both sequences of matrices may be random.\n\nThe notion of asymptotic equivalence enjoys some properties\nthat we list next.\nThese are stated in the context of deterministic equivalence\nin \\cite{dobriban_sheng_2021,dobriban_wonder_2020}, but hold more generally for\nasymptotic equivalence.\nFor the statements to follow,\nlet ${\\mathbf{A}}_n$, ${\\mathbf{B}}_n$, ${\\mathbf{C}}_n$, and ${\\mathbf{D}}_n$ be sequences of random or deterministic matrices\n(of appropriate dimensions).\nThen the following properties hold:\n\\begin{enumerate}\n \\item \\textbf{Equivalence.}\n The relation $\\simeq$ is an equivalence relation.\n \\item \\textbf{Sum.}\n If ${\\mathbf{A}}_n \\simeq {\\mathbf{B}}_n$ and ${\\mathbf{C}}_n \\simeq {\\mathbf{D}}_n$,\n then ${\\mathbf{A}}_n + {\\mathbf{C}}_n \\simeq {\\mathbf{B}}_n + {\\mathbf{D}}_n$.\n \\item \\textbf{Product.}\n If ${\\mathbf{A}}_n$ and $\\mB_n$ have operator norm uniformly bounded in $n$ and $\\mA_n \\simeq \\mB_n$, and $\\mC_n$ is independent of $\\mA_n$ and $\\mB_n$ with operator norm bounded in $n$ almost surely, \n then ${\\mathbf{A}}_n {\\mathbf{C}}_n \\simeq {\\mathbf{B}}_n {\\mathbf{C}}_n$.\n \\item \\textbf{Trace.}\n If ${\\mathbf{A}}_n \\simeq {\\mathbf{B}}_n$ for square matrices ${\\mathbf{A}}_n$ and ${\\mathbf{B}}_n$ of dimension $p(n) \\times p(n)$, \n then $\\tfrac{1}{p(n)} \\tr[{\\mathbf{A}}_n] - \\tfrac{1}{p(n)} \\tr[{\\mathbf{B}}_n] \\xrightarrow{\\text{a.s.}} 0$.\n \\item \\textbf{Elements.} If $\\mA_n \\simeq \\mB_n$ for $\\mA_n, \\mB_n$ of dimension $p(n) \\times q(n)$ and $i(n) \\in \\set{1, \\ldots, p(n)}$ and $j(n) \\in \\set{1, \\ldots, q(n)}$, then $[\\mA_n]_{i(n), j(n)} - [\\mB_n]_{i(n), j(n)} \\xrightarrow{\\text{a.s.}} 0$.\n \\item \\textbf{Differentiation.}\n Suppose $f(z, {\\mathbf{A}}_n) \\simeq g(z, {\\mathbf{B}}_n)$\n where the entries of $f$ and $g$ are analytic\n functions in $z \\in D$ and $D$ is an open connected subset of $\\mathbb{C}$.\n Furthermore, suppose for any sequence ${\\bm{\\Theta}}_n$ of deterministic \n matrices with trace norm uniformly bounded in $n$,\n we have that $|\\tr[{\\bm{\\Theta}}_n (f(z, {\\mathbf{A}}_n) - g(z, {\\mathbf{B}}_n))]| \\le M$\n for every $n$ and $z \\in D$ for some constant $M < \\infty$.\n Then we have that $f'(z, {\\mathbf{A}}_n) \\simeq g'(z, {\\mathbf{B}}_n)$\n for every $z \\in D$,\n where the derivatives are taken entry-wise with respect to $z$.\n\\end{enumerate}\n\n\n\n\nThe almost sure convergence in the statements above\nis with respect to the entire randomness in the random variables involved.\nOne can also consider\nthe notion of conditional asymptotic equivalence\nwherein we condition on a sequence of random matrices.\nMore precisely,\nsuppose ${\\mathbf{A}}_n$, ${\\mathbf{B}}_n$ are sequence of random matrices\nthat may depend of another sequence of random matrices ${\\mathbf{Z}}_n$.\nWe call ${\\mathbf{A}}_n$ and ${\\mathbf{B}}_n$ to be asymptotically equivalent\nconditioned on ${\\mathbf{Z}}_n$,\nif for any sequence of deterministic matrices\n${\\bm{\\Theta}}_n$\nwith trace norm uniformly bounded in $n$,\nwe have $\\lim_{n \\to \\infty} \\tr[{\\bm{\\Theta}}_n ({\\mathbf{A}}_n - {\\mathbf{B}}_n)] = 0$ almost surely conditioned on ${\\mathbf{Z}}_n$.\nSimilar properties to those listed above for unconditional asymptotic equivalence\nalso hold for conditional equivalence by considering all the statements\nconditioned on the sequence ${\\mathbf{Z}}_n$.\nIn particular,\nfor the product rule,\nwe require that the sequence ${\\mathbf{C}}_n$ be \\emph{conditionally}\nindependent of ${\\mathbf{A}}_n$ and ${\\mathbf{B}}_n$ given ${\\mathbf{Z}}_n$.\nFinally,\nfor our asymptotic statements,\nwe will work with sequences of matrices,\nindexed by either $n$ or $p$.\nHowever, for notational brevity,\nwe will drop the index from now on whenever it is clear from the context.\n\n\n\n\nEquipped with the notion of asymptotic equivalence,\nbelow we state a result on the asymptotic deterministic\nequivalence for ridge resolvents,\nadapted from Theorem 1 of \\cite{rubio_mestre_2011} and Theorem 3.1 of \\cite{dobriban_sheng_2021},\nthat will form a base for our results.\n\n\\begin{lemma}\n [Basic deterministic equivalent for ridge resolvent, complex-valued regularization]\n \\label{lem:basic-ridge-asympequi}\n Let $\\mZ \\in \\complexset^{n \\times p}$ be a random matrix consisting of i.i.d.\\ random variables that have mean 0, variance 1, and finite \n absolute moment of order\n $8 + \\delta$ for some $\\delta > 0$. Let $\\mSigma \\in \\complexset^{p \\times p}$ be a positive semidefinite matrix with operator norm uniformly bounded in $p$, and let $\\mX = \\mZ \\mSigma^{1\/2}$. \n Then, for $z \\in \\complexset^+$,\n as $n, p \\to \\infty$ such that\n $0 < \\liminf \\tfrac{p}{n} \\le \\limsup \\tfrac{p}{n} < \\infty$,\n we have\n \\begin{equation}\n \\label{eq:basic-ridge-asympequi-in-c}\n \\big( \\tfrac{1}{n} {\\mathbf{X}}^\\ctransp {\\mathbf{X}} - z {\\mathbf{I}}_p \\big)^{-1}\n \\simeq\n \\inv{c(z) {\\bm{\\Sigma}} -z {\\mathbf{I}}_p},\n \\end{equation}\n where $c(z)$ is the unique solution in $\\complexset^-$ to the fixed point equation\n \\begin{equation}\n \\label{eq:basic-ridge-fp-in-c}\n \\frac{1}{c(z)} - 1\n = \\tfrac{1}{n} \\tr \\bracket{{\\bm{\\Sigma}} \\inv{c(z) {\\bm{\\Sigma}} - z {\\mathbf{I}}_p}}.\n \\end{equation}\n Furthermore, \n $\\tfrac{1}{p} \\tr\\bracket{{\\bm{\\Sigma}} (c(z) {\\bm{\\Sigma}} - z {\\mathbf{I}}_p)^{-1}}$\n is a Stieltjes transform of a certain positive measure on $\\mathbb{R}_{\\ge 0}$\n with total mass $\\tfrac{1}{p} \\tr[{\\bm{\\Sigma}}]$.\n\\end{lemma} \nStrictly speaking,\nthe results in \\cite{rubio_mestre_2011} and \\cite{dobriban_sheng_2021}\nrequire that the sequence ${\\bm{\\Sigma}}$ be deterministic.\nHowever, one can take ${\\bm{\\Sigma}}$ to be a random sequence of matrices\nthat are independent of ${\\mathbf{Z}}$;\nsee, for example, \\cite{ledoit_peche_2011}.\nIn this case,\nthe asymptotic equivalence is treated conditionally on ${\\bm{\\Sigma}}$.\n\n\n\\section{Real-valued equivalence}\n\\label{sec:real-valued-equivalence}\n\nFor real-valued negative $z$, corresponding to positive ridge regularization, \nwe remark that one can use \\cref{lem:basic-ridge-asympequi}\nto derive limits of linear and \ncertain non-linear functionals \n(through the calculus rules of asymptotic equivalence)\nof the ridge resolvent \n$(\\tfrac{1}{n} {\\mathbf{X}}^\\ctransp {\\mathbf{X}} - z {\\mathbf{I}}_p)^{-1}$\nby considering $z \\in \\mathbb{C}^{+}$ with $\\Re(z) < 0$\nand letting $\\Im(z) \\to 0$.\nThis follows because\na short calculation (see proof of \\cref{cor:basic-ridge-asympequi-in-r}) \nshows that $\\Im(c(z)) \\to 0$\nas $\\Im(z) \\to 0$ for $z \\in \\mathbb{C}^{+}$ with $\\Re(z) < 0$.\nThus one can recover a real limit\nfrom the right hand side of \\cref{eq:basic-ridge-asympequi-in-c}\nthrough a limiting argument.\nMoreover,\nit is easy to see that\nthe fixed-point equation \\cref{eq:basic-ridge-fp-in-c}\nhas a unique (real) solution $c(z) > 0$\nfor $z \\in \\mathbb{R}_{< 0}$. \n\nHowever, it has recently been pointed out that\nunder certain special data geometry,\nnegative regularization is often beneficial, in real data experiments \\cite{kobak_lomond_sanchez_2020} as well as in theoretical formulations where it can achieve optimal squared prediction risk \\cite{wu_xu_2020}.\nOne can still recover such a case\nby considering $z \\in \\mathbb{C}^{+}$\nwith $\\Re(z) > 0$ over a valid range, and taking the limit as $\\Im(z) \\to 0$.\nHowever, solving the fixed-point equation \\cref{eq:basic-ridge-fp-in-c}\nover reals directly in this case, which is the most efficient way to compute the solution numerically, poses certain subtleties\nas we no longer can guarantee a unique real solution for $c(z)$. \n\nOur next theorem shows how to handle this case.\nWe will make use of this for our results on sketching\nin \\Cref{sec:main_results}, but we believe the result to be of independent interest and worth stating on its own. In addition to enabling the computation of the asymptotic equivalence for non-negative real-valued $z$, it also provides the asymptotic value of $\\lambda_{\\min}^{+}(\\tfrac{1}{n} \\mX^\\ctransp \\mX)$ (given by $z_0$ in the theorem statement) for arbitrary $\\mSigma$, which to our knowledge is the first general characterization of the smallest nonzero eigenvalue of random matrices outside of special cases such as $\\mSigma = \\mI_p$ and some lower bounds (see, e.g., \\cite{bai_silverstein_1998}). We demonstrate the improvement of $z_0$ over these na\\\"ive lower bounds in \\cref{fig:lambda-minnz-bound}.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=5.5in]{figures\/lambda_minnz_bound.pdf}\n \\caption{Plots showing how $z_0$ (solid) from \\cref{eq:basic-ridge-asympequi-in-r:bounds} matches the empirical minimum nonzero eigenvalue (markers) of $\\tfrac{1}{n} \\mX^\\transp \\mX$ when $\\mSigma = \\tfrac{1}{m} \\mY^\\transp \\mY$ for $\\mY \\in \\reals^{m \\times p}$ with i.i.d.\\ $\\normal(0, 1)$ elements, such that the limiting spectrum of $\\mSigma$ follows the $\\mathrm{Marchenko}\\text{--}\\mathrm{Pastur}(\\tfrac{p}{m})$ distribution for $\\tfrac{p}{m} \\in \\set{0.2, 0.9, 5}$. In contrast, the commonly used na\\\"ive bound (dashed)\n $\\liminf \\lambda_{\\min}^{+}(\\tfrac{1}{n} \\mX^\\transp \\mX) \\geq \n \\bigparen{1 - \\sqrt{\\tfrac{p}{m}}}^2 \n \\bigparen{1 - \\sqrt{\\tfrac{p}{n}}}^2 \n \\ind\\set{p < \\max\\set{m, n}}$,\n obtained by multiplying the minimum nonzero eigenvalues of $\\tfrac{1}{n} \\mZ^\\transp \\mZ$ and $\\mSigma$ when at most one of them is singular,\n is quite loose outside of the $m \\gg p$ and $n \\gg p$ cases and fails to capture the correct behavior at all when both are singular ($p > \\max\\set{m, n}$). Empirical values are computed for $p = 500$ for a single trial. }\n \\label{fig:lambda-minnz-bound}\n\\end{figure}\n\n\\begin{theorem}\n[Basic deterministic equivalent for ridge resolvent, real-valued regularization]\n\\label{cor:basic-ridge-asympequi-in-r}\nAssume the setting of \\cref{lem:basic-ridge-asympequi}. \nLet $\\zeta_0, z_0 \\in \\reals$ be the unique solutions, satisfying $\\zeta_0 < \\lambda_{\\min}^{+}({\\bm{\\Sigma}})$,\nto system of equations\n\\begin{align}\n \\label{eq:basic-ridge-asympequi-in-r:bounds}\n 1 = \\tfrac{1}{n} \\tr \\bracket{{\\bm{\\Sigma}}^2 \\paren{{\\bm{\\Sigma}} - \\zeta_0 {\\mathbf{I}}_p}^{-2}}, \\quad \n z_0 = \\zeta_0 \\paren{1 - \\tfrac{1}{n} \\tr \\bracket{{\\bm{\\Sigma}} \\inv{{\\bm{\\Sigma}} - \\zeta_0 {\\mathbf{I}}_p}}}.\n\\end{align}\nThen, for each $z \\in \\reals$ satisfying $z < \\liminf z_0$, as $n, p \\to \\infty$ such that\n$0 < \\liminf \\tfrac{p}{n} \\le \\limsup \\tfrac{p}{n} < \\infty$,\nwe have\n\\begin{align}\n \\label{eq:basic-ridge-asympequi-in-r}\n z \\big( \\tfrac{1}{n} {\\mathbf{X}}^\\ctransp {\\mathbf{X}} - z {\\mathbf{I}}_p \\big)^{-1}\n \\simeq\n \\zeta \\inv{{\\bm{\\Sigma}} - \\zeta {\\mathbf{I}}_p},\n\\end{align}\nwhere $\\zeta \\in \\reals$ is the unique solution in $(-\\infty, \\zeta_0)$ to\nthe fixed-point equation\n\\begin{equation+}\n \\label{eq:basic-ridge-fp-in-r}\n z = \\zeta \\paren{1 - \\tfrac{1}{n} \\tr \\bracket{{\\bm{\\Sigma}} \\inv{{\\bm{\\Sigma}} - \\zeta {\\mathbf{I}}_p}}}.\n\\end{equation+}\nFurthermore, as $n, p \\to \\infty$, $|\\zeta + \\tfrac{1}{v(z)}| \\xrightarrow{\\text{a.s.}} 0$, where \n$v(z)$ is the companion Stieltjes transform of the spectrum of $\\frac{1}{n} \\mX^\\ctransp \\mX$\ngiven by\n\\[\n v(z) = \\tfrac{1}{n} \\tr \\Big[\\big(\\tfrac{1}{n} \\mX \\mX^\\ctransp - z \\mI_n \\big)^{-1}\\Big],\n\\]\nand $|z_0 - \\lambda_{\\min}^{+}(\\frac{1}{n} \\mX^\\ctransp \\mX)| \\xrightarrow{\\text{a.s.}} 0$.\n\\end{theorem}\n\\begin{proof}[Proof sketch]\nTo prove this corollary, we define $\\zeta \\defeq \\tfrac{z}{c(z)}$ to obtain \\cref{eq:basic-ridge-asympequi-in-r} from \\cref{eq:basic-ridge-asympequi-in-c} for $z \\in \\complexset^+$, and also observe that $\\tfrac{1}{\\zeta}$ is the limiting companion Stieltjes transform $v(z)$ of $\\tfrac{1}{n} \\mX \\mX^\\ctransp$ at $z$. This implies that $\\zeta \\in \\complexset^+$ and that the mapping $z \\mapsto \\zeta$ is a holomorphic function on its domain, which includes all real $z < \\liminf \\lambda_{\\min}^{+}(\\tfrac{1}{n} \\mX \\mX^\\ctransp)$. We then identify the analytic continuation of the mapping $z \\mapsto \\zeta$ to the reals, which consists of careful bookkeeping to determine $z_0$, the least positive value of $z$ for which $\\zeta$ does not exist, which must be asymptotically equal to $\\lambda_{\\min}^{+}(\\tfrac{1}{n} \\mX \\mX^\\ctransp)$. \nThe proof details can be found in \\Cref{sec:proof:cor:basic-ridge-asympequi-in-r} \nof the supplementary material.\n\\end{proof}\n\n\\begin{remark}\n [The case of $z = 0$]\n The form of the equivalence \\cref{eq:basic-ridge-asympequi-in-c}\n is slightly different as compared with \\cref{eq:basic-ridge-asympequi-in-r}\n in that the resolvent $(\\tfrac{1}{n} {\\mathbf{X}}^\\ctransp {\\mathbf{X}} - z {\\mathbf{I}}_p)^{-1}$ has a normalizing multiplier of $z$ in the latter case.\n This enables continuity of the left-hand side at $z = 0$, in contrast to\n specializing the equivalence \\cref{eq:basic-ridge-asympequi-in-c}\n to real $z$, where both the left- and right-hand sides may diverge as $z \\nearrow 0$.\n\\end{remark}\n\n\nOur main result in the next section for sketching follows directly from this theorem and shares a very similar form. For this reason, we defer discussion about the interpretation of the solutions to the above equations for our reformulation under the sketching setting; however, analogous interpretations will apply to the above theorem.\n\\section{Properties and examples}\n\\label{sec:properties}\n\nBelow we provide various analytical properties\nof the quantities that appear in \\Cref{thm:sketched-pseudoinverse,lem:second-order-sketch}.\nSee \\Cref{sec:proofs-properties} in the supplementary material for their proofs.\n\n\\subsection{Lower limits}\n\nThe quantities $\\lambda_0$ and $\\mu_0$\nprovide the lower limits of regularization\nin \\Cref{thm:sketched-pseudoinverse}.\nThe following two remarks describe their behaviour\nin terms of $\\alpha$.\n\n\n\\begin{remark}\n [Dependence of $\\mu_0$ and $\\lambda_0$ on $\\alpha$]\n \\label{rem:mu0-lambda0-vs-alpha}\n Writing the first equation in \\cref{eq:mu0-lambda0-fps} as\n \\begin{equation+}\n \\label{eq:mu0-fp-in-alpha}\n \\alpha = \\tfrac{1}{p} \\tr \\bracket{\\mA^2 \\paren{\\mA + \\mu_0 \\mI_p}^{-2}},\n \\end{equation+}\n note that for fixed $\\mA$, $\\mu_0$ only depends on $\\alpha$.\n Furthermore, the equation indeed admits a unique solution for $\\mu_0$ for a given $\\alpha$.\n This can be seen by noting that the function\n $f: \\mu_0 \\mapsto \\tfrac{1}{p} \\tr[{\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}}_p)^{-2}]$\n is monotonically decreasing in $\\mu_0$,\n and \n \\[\n \\tfrac{1}{p} \\lim_{\\mu_0 \\searrow - \\lambda_{\\min}^{+}(\\mA)} \\tr[{\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}}_p)^{-2}] = \\infty,\n \\quad\n \\text{and}\n \\quad\n \\tfrac{1}{p}\\lim_{\\mu_0 \\to \\infty} \\tr[{\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}}_p)^{-2}] = 0.\n \\]\n In addition,\n because $\\mu_0(\\alpha) = f^{-1}(\\alpha)$,\n $\\mu_0$\n is monotonically decreasing in $\\alpha$,\n and $\\lim_{\\alpha \\searrow 0} \\mu_0(\\alpha) = \\infty$\n and $\\lim_{\\alpha \\to \\infty} \\mu_0(\\alpha) = - \\lambda_{\\min}^{+}({\\mathbf{A}})$.\n \n Given $\\mu_0$,\n the second equation in \\cref{eq:mu0-lambda0-fps} then\n provides $\\lambda_0$ as\n \\begin{equation+}\n \\label{eq:lambda0-mu0-in-alpha}\n \\lambda_0 \n = \\mu_0 \\paren{1 - \\tfrac{1}{\\alpha} \\tfrac{1}{p} \\tr \\bracket{\\mA \\inv{\\mA + \\mu_0 {\\mathbf{I}}}}}.\n \\end{equation+}\n For $\\alpha \\in (0, r({\\mathbf{A}}))$,\n $\\lambda_0 : \\alpha \\mapsto \\lambda_0(\\alpha)$\n is monotonically increasing,\n and $\\lim_{\\alpha \\searrow 0} \\lambda_0(\\alpha) = - \\infty$\n and $\\lim_{\\alpha \\to r({\\mathbf{A}})} \\lambda_0(\\alpha) = 0$.\n When $\\alpha = r({\\mathbf{A}})$, $\\mu_0 = 0$,\n and consequently $\\lambda_0 = 0$.\n Finally, for $\\alpha \\in (r({\\mathbf{A}}), \\infty)$,\n $\\lambda_0 : \\alpha \\mapsto \\lambda_0(\\alpha)$\n is monotonically decreasing in $\\alpha$,\n and $\\lim_{\\alpha \\to \\infty} \\lambda_0(\\alpha) = -\\lambda_{\\min}^{+}({\\mathbf{A}})$.\n This follows from a short limiting calculation.\n\\end{remark}\n\n\\begin{remark}\n [Joint sign patterns of $\\mu_0$ and $\\lambda_0$]\n \\label{rem:mu0-lambda0-signs}\n Observe from \\cref{eq:mu0-lambda0-fps}\n the sign pattern summarized in \\cref{tab:mu0-lambda0-signs}.\n \\begin{table}[h!]\n \\label{tab:mu0-lambda0-signs}\n \\centering\n \\caption{Sign patterns of $\\lambda_0$ and $\\mu_0$.}\n \\begin{tabular}{c | c | c | c}\n $\\alpha$ vs. $r({\\mathbf{A}})$ \n & $\\mu_0$ \n & $\\alpha$ vs. $\\tfrac{1}{p} \\tr[{\\mathbf{A}} ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}})^{-1}]$\n & $\\lambda_0$ \\\\\n \\hline\n $\\alpha > r({\\mathbf{A}})$ \n & $< 0$\n & $\\alpha = \\tfrac{1}{p} \\tr[{\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}})^{-2}]\n > \\tfrac{1}{p} \\tr[{\\mathbf{A}} ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}})^{-1}]$ \n & $< 0$ \\\\\n $\\alpha = r({\\mathbf{A}})$ \n & 0\n & $\\alpha \n = \\lim_{x \\searrow 0}\n \\tfrac{1}{p} \\tr[{\\mathbf{A}}^2 ({\\mathbf{A}} + x {\\mathbf{I}})^{-2}]\n = \\lim_{x \\searrow 0}\n \\tfrac{1}{p} \\tr[{\\mathbf{A}} ({\\mathbf{A}} + x {\\mathbf{I}})^{-1}]$\n & 0 \\\\\n $\\alpha < r({\\mathbf{A}})$ \n & $> 0$ \n & $\\alpha = \\tfrac{1}{p} \\tr[{\\mathbf{A}}^2 ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}})^{-2}]\n < \\tfrac{1}{p} \\tr[{\\mathbf{A}} ({\\mathbf{A}} + \\mu_0 {\\mathbf{I}})^{-1}]$ \n & $< 0$\n \\end{tabular}\n \\end{table}\n\\end{remark}\n\n\n\\subsection{First-order equivalence}\n\nIn general,\nthe exact $\\mu$ depends on $\\lambda$, $\\alpha$, and ${\\mathbf{A}}$\nvia the fixed-point equation \\cref{eq:sketched-modified-lambda}.\nHowever, we can infer several properties of the behaviour\nof $\\mu$ as a function of $\\lambda$ and $\\alpha$\nas summarized below.\n\n\n\\begin{proposition}\n [Monotonicities of $\\mu$ in $\\lambda$ and $\\alpha$]\n \\label{prop:monotonicities-lambda-alpha}\n For a fixed $\\alpha \\ge 0$, \n the map $\\lambda \\mapsto \\mu(\\lambda)$,\n where $\\mu(\\lambda)$ is as defined in \\cref{eq:sketched-modified-lambda}\n is monotonically increasing in $\\lambda$\n over $(\\lambda_0, \\infty)$,\n and $\\lim_{\\lambda \\searrow \\lambda_0} \\mu(\\lambda) = \\mu_0$, \n while $\\lim_{\\lambda \\to \\infty} \\mu(\\lambda) = \\infty$.\n For a fixed $\\lambda \\ge 0$,\n the map $\\alpha \\mapsto \\mu(\\alpha)$\n where $\\mu(\\alpha)$ is as defined in \\cref{eq:sketched-modified-lambda}\n is monotonically decreasing in $\\alpha$ over $(0, \\infty)$;\n when $\\lambda < 0$,\n the map $\\alpha \\to \\mu(\\alpha)$ is monotonically decreasing\n over $(0, r({\\mathbf{A}}))$ \n and monotonically increasing over $(r({\\mathbf{A}}), \\infty)$.\n Furthermore,\n for any $\\lambda \\in (\\lambda_0, \\infty)$,\n $\\lim_{\\alpha \\searrow 0} \\mu(\\alpha) = \\infty$,\n and $\\lim_{\\alpha \\to \\infty} \\mu(\\alpha) = \\lambda$.\n\\end{proposition}\n\n\\begin{remark}\n [Joint signs of $\\lambda$ and $\\mu$]\n \\label{rem:joint-signs-lambda-mu}\n When $\\lambda \\ge 0$, \n for any $\\alpha > 0$, \n we have $\\mu \\ge 0$,\n where $\\mu$ is the unique solution to \n \\cref{eq:sketched-modified-lambda} in $(\\mu_0, \\infty)$.\n When $\\lambda < 0$,\n for $\\alpha \\le r({\\mathbf{A}})$,\n we have $\\mu \\ge 0$,\n while for $\\alpha > r({\\mathbf{A}})$,\n we have \n $\\mathrm{sign}(\\mu) \n = \\mathrm{sign}(\\lambda)$.\n\\end{remark}\n\n\n\\begin{proposition}\n [Concavity, bounds, and asymptotic behaviour of $\\mu$ in $\\lambda$]\n \\label{rem:concavity-mu-in-lambda}\n The function $\\lambda \\mapsto \\mu(\\lambda)$,\n where $\\mu(\\lambda)$ is solution to \\cref{eq:sketched-modified-lambda}\n is a concave function over $(\\lambda_0, \\infty)$.\n Furthermore, for any $\\alpha \\in (0, \\infty)$,\n $\\mu(\\lambda) \\le \\lambda + \\tfrac{1}{q} \\tr[\\mA]$ for all $\\lambda \\in (\\lambda_0, \\infty)$; and when $\\alpha \\le r({\\mathbf{A}})$,\n $\\mu(\\lambda) \\geq \\lambda$ for all $\\lambda \\in (\\lambda_0, \\infty)$,\n otherwise $\\mu(\\lambda) \\geq \\lambda$\n for $\\lambda \\geq 0$.\n Additionally,\n $\\lim_{\\lambda \\to \\infty} |\\mu(\\lambda) - (\\lambda + \\tfrac{1}{q} \\tr[\\mA])| = 0$. \n\\end{proposition}\n\n\\subsection{Second-order equivalence}\n\nWe also have a few remarks about the second-order equivalence.\n\n\\begin{remark}\n \\label{rem:alt-mu-prime}\n We have the following alternative form for $\\mu'$:\n \\begin{align}\n \\mu' = \\tfrac{1}{q} \\tr \\bracket{\\mu^2 \\inv{\\mA + \\mu \\mI_p} \\mPsi \\inv{\\mA + \\mu \\mI_p}} \\frac{\\partial \\mu}{\\partial \\lambda}.\n \\end{align}\n Note that the term $\\frac{\\partial{\\mu}}{\\partial \\lambda}$ does not depend in any way on $\\mPsi$, and that the remaining term is well-controlled for any $\\mu > \\mu_0$. Therefore, $\\mu'$ will only diverge when $\\frac{\\partial{\\mu}}{\\partial \\lambda}$ diverges, which occurs as $\\lambda \\to \\lambda_0$. This is clearly visible in \\cref{fig:mu-equiv} (top) as $\\lambda$ approaches $\\lambda_0$, where the slope of the curve tends to infinity. Additionally, because $\\mu$ is increasing in $\\lambda$, this decomposition shows that $\\mu' \\geq 0$.\n\\end{remark}\n\n\\begin{remark}[Vanishing $\\mu'$]\n\\label{rem:mu-prime-to-0}\nIf $\\mathrm{Ker}(\\mA) \\subseteq \\mathrm{Ker}(\\mPsi)$, then as $\\mu \\to 0$, $\\mu' \\to 0$. The best intuition for this is in the case $\\mPsi = \\mA$. Because we can only have $\\mu = 0$ for $\\alpha > r(\\mA)$ and $\\lambda = 0$, we have $\\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q} \\mS^\\ctransp \\mA \\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q} \\mS^\\ctransp \\big|_{\\lambda = 0} = \\mS \\biginv{\\mS^\\ctransp \\mA \\mS + \\lambda \\mI_q} \\mS^\\ctransp \\big|_{\\lambda = 0}$, and the second-order equivalence reduces to the first-order equivalence with no inflation factor. This remarkable property means that i.i.d.\\ sketching leads to extremely accurate estimates with no spectral distortion, but only in low-rank settings with little regularization. \n\\end{remark}\n\n\n\n\\subsection{Illustrative examples}\n\nIn order to better understand \\cref{thm:sketched-pseudoinverse,lem:second-order-sketch}, \nwe consider a few examples with special choices of the matrix ${\\mathbf{A}}$. When the spectrum of $\\mA$ converges to a particular distribution of eigenvalues, $\\mu$ will converge to a value that is deterministic given $\\mA$. %\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/mu_equiv.pdf}\n \\caption{Plots of $\\mu$ as a function of $\\lambda$ and $\\alpha$ for rank-deficient isotropic (left) and Marchenko--Pastur (middle) spectra, normalized so that $\\frac{1}{p} \\tr[\\mA] = r = 1\/2$. \n The values of $\\lambda$ and $\\alpha$ in each location of the plot are indicated by the colormap (right), shared between the two views of each plot. \n As we sweep $\\alpha$, we also plot $(\\alpha, \\lambda_0, \\mu_0)$ (black, dotted). \n We also plot the lines $\\mu = 0$, $\\lambda = 0$, and $\\alpha = r$ (gray, dashed).\n The scaling of the $\\mu$ and $\\lambda$ axes are linear, and the scaling of the $\\alpha$ axis is proportional to $1\/\\alpha$. \n In this way we can clearly capture the general $\\mu \\approx \\lambda + \\tfrac{1}{p} \\tr[A] \/ \\alpha$ relationship for $\\lambda > 0$, as well as the limiting behavior of $\\mu = \\lambda$ for large $\\alpha$.\n The most significant difference between the two distributions is that for the isotropic distribution, $\\lambda_{\\min}^{+}(\\mA) = 1$, while for the Marchenko--Pastur case, $\\lambda_{\\min}^{+}(\\mA) = (\\sqrt{2} - 1)^2\/2 \\approx 0.0859$, limiting the achievable negative values of $\\mu$ when $\\lambda < 0$ and $\\alpha > r$.\n }\n \\label{fig:mu-equiv}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=5in]{figures\/mu_prime.pdf}\n \\caption{Plot of $\\mu'$ as a function of $\\mu$ and $\\alpha$ for the rank-deficient isotropic spectrum with $r = 1\/2$ for $\\mPsi \\in \\set{\\mI_p, \\mA}$. In both cases, as $\\mu \\to \\mu_0$ (dashed), $\\mu' \\to \\infty$. Otherwise, $\\mu'$ is not too large. For $\\mPsi = \\mI_p$, $\\mu'$ decays slowly in $\\alpha$ and $\\mu$. However, for $\\mPsi = \\mA$, there is a regime for $\\alpha > r$ around $\\mu = 0$ for which $\\mu'$ tends to zero. Thus, the unregularized pseudo-inverse preserves $\\mA$ remarkably well on its range when the sketch size is greater than the rank of the matrix, but outside of the range of $\\mA$, it has non-negligible error.}\n \\label{fig:mu-prime}\n\\end{figure}\n\n\n\\subsubsection{Isotropic rank-deficient matrix}\n\nFor the first example, let $0 < r \\le 1$ be a real number.\nWe then consider $\\mA = \\begin{bsmallmatrix}\\mI_{\\floor{r p}} & \\vzero \\\\ \\vzero & \\vzero \\end{bsmallmatrix}$ such that $r(\\mA) \\to r$ as $p \\to \\infty$. We have chosen the standard basis representation of this matrix, but the following results also hold for any $\\mA$ that is isotropic on a subspace, regardless of basis. Such an $\\mA$ includes settings such as $\\mA = \\mX^\\transp \\mX$ where $\\mX \\in \\reals^{n \\times p}$ is an orthogonal design matrix with orthonormal rows. In this case,\n\\begin{align}\n \\mu = \\frac{\\lambda + \\tfrac{r}{\\alpha} - 1 + \\sqrt{(\\lambda + \\frac{r}{\\alpha} - 1)^2 + 4 \\lambda}}{2}.\n\\end{align}\nFurthermore, we have simple forms for $\\mu_0$ and $\\lambda_0$:\n\\begin{align}\n \\mu_0 = \\sqrt{\\tfrac{r}{\\alpha}} - 1, \\quad\n \\lambda_0 = - \\paren{1 - \\sqrt{\\tfrac{r}{\\alpha}}}^2.\n\\end{align}\nThe expression for $\\lambda_0$ can also be obtained directly from the minimum nonzero eigenvalue of the Marchenko--Pastur distribution with aspect ratio $\\tfrac{\\alpha}{r}$ and variance scaling $\\tfrac{r}{\\alpha}$, which describes $\\mS^\\ctransp \\mA \\mS$. In the case $\\lambda = 0$, we have a very simple expression for $\\mu$:\n\\begin{align}\n \\mu = \\begin{cases}\n \\frac{r}{\\alpha} - 1 & \\text{if } \\alpha < r, \\\\\n 0 & \\text{otherwise}.\n \\end{cases}\n\\end{align}\nWe can also obtain the limiting behavior of $\\mu$ for large $\\lambda$ or small $\\alpha$:\n\\begin{align}\n \\label{eq:limiting-mu}\n \\lim_{\\lambda + \\tfrac{r}{\\alpha} \\to \\infty} \\frac{\\mu}{\\lambda + \\tfrac{r}{\\alpha}} = 1.\n\\end{align}\nIn \\Cref{fig:mu-equiv} (left), we plot $\\mu$ as a function of both $\\lambda$ and $\\alpha$.\nWe see that even for modest values of $\\lambda > 0$ or $\\alpha < r$, the relationship $\\mu \\sim \\lambda + \\tfrac{r}{\\alpha}$ holds quite accurately. We see a clear transition point at $\\alpha = r$ where $\\lambda_0 = 0$, and on either side of which $\\lambda_0$ decreases. Other properties from the previous sections, such as monotonicity, concavity in $\\lambda$, and sign patterns are clearly visible in this plot as well.\nWe also plot $\\mu'$ as a function of $\\mu$ and $\\alpha$ in \\Cref{fig:mu-prime}, where we see that the inflation vanishes for $\\mPsi = \\mA$ only if $\\alpha > r$ and $\\mu = 0$. It is non-negligible otherwise, and tends to infinity as $\\mu$ tends to $\\mu_0$ for each $\\alpha$.\n\n\\subsubsection{Marchenko--Pastur spectrum}\n\nWe also consider the case when ${\\mathbf{A}}$ is a random matrix\nof the form ${\\mathbf{A}} = \\tfrac{1}{n} {\\mathbf{Z}}^\\top {\\mathbf{Z}}$,\nwhere ${\\mathbf{Z}} \\in \\mathbb{R}^{n \\times p}$ contains i.i.d.\\ entries\nof mean $0$, variance $1$, and bounded moments of order $4 + \\delta$\nfor some $\\delta > 0$.\nThis case is of interest for real data settings\nwhere ${\\mathbf{A}}$ will be a sample covariance matrix.\nIn this case, the spectrum of ${\\mathbf{A}}$ can be computed explicitly and is given by the Marchenko--Pastur law.\nComputing $\\mu$ explicitly in this case is possible,\nbut cumbersome. We instead provide numerical illustrations\non the behaviour of $\\mu$ as a function of $\\alpha$ and $\\lambda$.\n\nFrom \\cref{fig:mu-equiv} (middle), we can see that the behavior of $\\mu$ for the Marchenko--Pastur spectrum is not substantially different from the rank-deficient isotropic spectrum. The only regime that differs significantly is when $\\alpha > r(\\mA)$ and $\\lambda < 0$, where $\\lambda_0$ is much closer to $0$ than in the isotropic case, and so there is no equivalence for more negative values of $\\lambda$.\n\nIt is also worth noting that when $\\alpha < r({\\mathbf{A}}) < 1$,\nthe na\\\"ive bound on the smallest\nregularization $\\lambda$ permissible is $0$\n(as explained in the caption of \\Cref{fig:lambda-minnz-bound}).\nHowever, from \\Cref{fig:mu-equiv}\nwe observe that the equivalence in \\cref{thm:sketched-pseudoinverse} holds even for quite negative $\\lambda$ (blue region), contrary to this na\\\"ive bound.\nIn fact, the true bound is almost the same as the rank-deficient isotropic case, $\\lambda_0 = - \\paren{1 - \\sqrt{\\tfrac{r}{\\alpha}}}^2$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzlfq b/data_all_eng_slimpj/shuffled/split2/finalzlfq new file mode 100644 index 0000000000000000000000000000000000000000..72e273d970d0a73ac9a250ac3e4a14a6dd102afe --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzlfq @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nVery few symmetries in nature are manifestly realized. Why do I \nthink then that the breaking of CP invariance is very special -- \nmore subtle, more fundamental and more profound than parity \nviolation? \n\\begin{itemize}\n\\item \nParity violation tells us that nature makes a difference between \n\"left\" and \"right\" -- but not which is which! For the \nstatement that neutrinos emerging from pion decays are \nleft- rather than right-handed implies the use of \npositive instead of negative pions. \"Left\" and \n\"right\" is thus defined in terms of \"positive\" \nand \"negative\", respectively. This is like saying that \nyour left thumb is on your right hand -- certainly \ncorrect, yet circular and thus not overly useful. \n\nOn the other hand CP violation manifesting itself through \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{{\\rm BR}(K_L \\rightarrow l^+ \\nu \\pi ^-)}\n{{\\rm BR}(K_L \\rightarrow l^- \\bar \\nu \\pi ^+)} \\simeq 1.006J\\neq 1 \n\\eeq \nallows us to define \"positive\" and \"negative\" in \nterms of \n{\\em observation} rather than {\\em convention}, and \nsubsequently likewise for \"left\" and \"right\". \n\\item \nThe limitation on CP invariance in the $K^0 - \\bar K^0$ \nmass matrix \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm Im} M_{12} \\simeq 1.1 \\cdot 10^{-8} \\; \\; {\\rm eV} \\; \\; \n\\hat = \\; \\; \\frac{{\\rm Im} M_{12}}{m_K} \\simeq 2.2 \\cdot \n10^{-17} \n\\eeq\nrepresents the most subtle symmetry violation \nactually observed to date. \n\n\\item \nCP violation constitutes one of the three essential ingredients in any \nattempt to understand the observed baryon number of the universe \nas a dynamically generated quantity rather than an initial \ncondition \\cite{DOLGOV2}. \n\n\\item \nDue to CPT invariance -- on which I will not cast any doubt during \nthese lectures -- CP breaking implies a violation of \ntime reversal invariance \n\\footnote{{\\em Operationally} one defines time reversal as the \nreversal \nof {\\em motion}: $\\vec p \\rightarrow - \\vec p$, $\\vec j \\rightarrow - \\vec j$ for \nmomenta $\\vec p$ and angular momenta $\\vec j$.}. That nature \nmakes an \nintrinsic distinction between past and future on the \n{\\em microscopic} level that cannot be explained by statistical \nconsiderations is an utterly amazing observation. \n\n\\item \nThe fact that time reversal represents a very peculiar operation can \nbe expressed also in a less emotional way, namely through \n{\\em Kramers' Degeneracy} \n\\cite{KRAMERS}. The time reversal operator $\\bf T$ \nhas to be {\\em anti}-unitary; ${\\bf T}^2$ then has eigenvalues \n$\\pm 1$. Consider the sector of the Hilbert space with \n${\\bf T}^2 = -1$ and assume the dynamics to conserve ${\\bf T}$; \ni.e., the Hamilton operator $\\bf H$ and ${\\bf T}$ commute. It is \neasily shown that if $|E\\rangle$ is an eigenvector of $\\bf H$, so is \n${\\bf T}|E\\rangle$ -- with the {\\em same} eigenvalue. Yet \n$|E\\rangle$ and ${\\bf T}|E\\rangle$ are -- that is the main \nsubstance of this theorem -- orthogonal to each other! Each \nenergy eigenstate in the Hilbert sector with ${\\bf T}^2 = -1$ \nis therefore at least doubly degenerate. This degeneracy is realized \nin nature through {\\em fermionic spin} degrees. Yet it is \nquite remarkable that the time reversal operator $\\bf T$ already \nanticipates this option -- and the qualitative difference \nbetween fermions and bosons -- through ${\\bf T}^2 = \\pm 1$ -- \n{\\em without} any explicit reference to spin! \n\n\\end{itemize}\n\n\\subsection{General Description of Particle-Antiparticle Oscillations}\nA symmetry $\\bf S$ can be manifestly realized in two different \nways: \n\\begin{itemize}\n\\item \nThere exists a pair of degenerate states that transform into each \nother under $\\bf S$. \n\\item \nWhen there is an {\\em un}paired state it has to be an eigenstate of \n$\\bf S$. \n\\end{itemize}\nThe observation of $K_L$ decaying into a $2\\pi$ state -- which \nis CP even -- and a CP odd $3\\pi$ combination therefore \nestablishes CP violation only because $K_L$ and $K_S$ are \n{\\em not} mass degenerate. \n\nIn general, decay rates can exhibit CP violation in three different \nmanners, \nnamely through \n\\begin{itemize}\n\\item \nthe {\\em existence} of a reaction, like \n$K_L \\rightarrow \\pi \\pi$, \n\\item \na {\\em difference} in CP conjugate rates, like \n$K_L \\rightarrow l^- \\bar \\nu \\pi ^+$ vs. \n$K_L \\rightarrow l^+ \\nu \\pi ^-$, \n\\item \na decay rate evolution that is {\\em not a purely \nexponential} function of the proper time of decay; i.e., \nif one finds for a CP {\\em eigenstate} $f$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{d}{dt} e^{\\Gamma t}{\\rm rate} \n(K_{neutral}(t) \\rightarrow f) \\neq 0 \n\\eeq \nfor all (real) values of $\\Gamma$, then CP symmetry must be \nbroken. This is easily proven: \nif CP invariance holds, the decaying state \nmust be a CP eigenstate like the final state $f$; yet in that case \nthe decay rate evolution must be purely exponential -- unless \nCP is violated. Q.E.D. \n\n\\end{itemize} \nThe whole formalism of particle-antiparticle oscillations is \nactually a straightforward application of basic quantum mechanics. \nI will describe it in terms of strange mesons; the generalization to \nany other flavour or quantum number (like beauty or charm) is \nobvious. In the absence of weak forces one has two mass degenerate \nand stable mesons $K^0$ and $\\bar K^0$ carrying definite \nstrangeness $+1$ and $-1$, respectively, since the strong and \nelectromagnetic \nforces conserve this quantum number. The addition of the weak \nforces changes the picture qualitatively: strangeness is no longer \nconserved, kaons become unstable and the new mass eigenstates \n-- being linear superpositions of $K^0$ and $\\bar K^0$ -- no longer \ncarry definite strangeness. The violation of the quantum number \nstrangeness has lifted the degeneracy: we have two physical \nstates $K_L$ and $K_S$ with different masses and lifetimes: \n$\\Delta m_K = m_L - m_K \\neq 0 \\neq \\Delta \\tau = \n\\tau _L - \\tau _S$. \n\nIf CP is conserved in the $\\Delta S=2$ transitions the mass \neigenstates $K_1$ and $K_2$ have to be CP eigenstates as pointed out \nabove: $|K_1\\rangle = |K_+\\rangle$, $|K_2\\rangle = |K_-\\rangle$, \nwhere ${\\bf CP}|K_{\\pm}\\rangle \\equiv \\pm |K_{\\pm}\\rangle$. \nUsing the phase convention \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|\\bar K^0 \\rangle \\equiv - {\\bf C} |K^0\\rangle \n\\eeq \nthe time evolution of a state that starts out as a $K^0$ is given by \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|K^0(t)\\rangle = \n\\frac{1}{\\sqrt{2}} e^{im_1t} e^{-\\frac{\\Gamma _1}{2}t} \n\\left( |K_+\\rangle + e^{i\\Delta m_Kt} e^{-\\frac{\\Delta \\Gamma}{2}t}\n|K_-\\rangle \\right) \n\\eeq \nThe intensity of an initially pure $K^0$ beam traveling in vacuum will then exhibit the \nfollowing time profile: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nI_{K^0}(t) = |\\langle K^0|K^0(t)\\rangle |^2 = \n\\frac{1}{4} e^{-\\Gamma _1 t } \n\\left( 1 + e^{\\Delta \\Gamma _Kt} + 2e^{\\frac{\\Delta \\Gamma _K}{2} t} \n{\\rm cos}\\Delta m_K t\\right) \n\\eeq \nThe orthogonal state $|\\bar K^0 (t)\\rangle$ that was absent initially \nin this beam gets regenerated {\\em spontaneously}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nI_{\\bar K^0}(t) = |\\langle \\bar K^0|K^0(t)\\rangle |^2 = \n\\frac{1}{4} e^{-\\Gamma _1 t } \n\\left( 1 + e^{\\Delta \\Gamma _K t} - 2e^{\\frac{\\Delta \\Gamma _K}{2} t} \n{\\rm cos}\\Delta m_K t\\right) \n\\eeq \nThe oscillation rate expressed through $\\Delta m_K$ and \n$\\Delta \\Gamma _K$ is naturally calibrated by the average \ndecay rate $\\bar \\Gamma _K \\equiv \n\\frac{1}{2}( \\Gamma _1 + \\Gamma _2)$: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nx_K \\equiv \\frac{\\Delta m_K}{\\bar \\Gamma _K} \\simeq 0.95 \n\\; \\; \\; , \\; \\; \\; \ny_K \\equiv \\frac{\\Delta \\Gamma _K}{2\\bar \\Gamma _K} \\simeq 1 \n\\eeq \nTwo comments are in order at this point: \n\\begin{itemize}\n\\item \nIn any such binary quantum system there will be two lifetimes. \nThe fact that they differ so spectacularly for neutral kaons \n-- $\\tau (K_L) \\sim 600 \\cdot \\tau (K_S)$ -- is \ndue to a kinematical accident: the only available nonleptonic \nchannel for the CP odd kaon is the 3 pion channel, for which \nit has barely enough mass. \n\\item \n$\\Delta m_K \\simeq 3.7 \\cdot 10^{-6}$ eV is often related \nto the kaon mass: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\Delta m_K}{m_K} \\simeq 7 \\cdot 10^{-15} \n\\label{STRIKING} \n\\eeq \nwhich is obviously a very striking number. Yet \nEq.(\\ref{STRIKING}) somewhat overstates the point. \nThe kaon mass has nothing really to do with the \n$K_L-K_S$ mass difference \n\\footnote{It would not be much more absurd to relate \n$\\Delta m_K$ to the mass of an elephant!} and actually is \nmeasured relative to $\\Gamma _K$. There is however \none exotic application where it makes sense to state \nthe ratio $\\Delta m_K\/m_K$, and that is in the context \nof antigravity where one assumes matter and antimatter \nto couple to gravity with the opposite sign. The gravitational \npotential $\\Phi$ \nwould then produce a {\\em relative} phase between $K^0$ and \n$\\bar K^0$ of 2 $m_K\\Phi t$. In the earth's potential this would \nlead to a gravitational oscillation time of \n$10^{-15}$ sec, which is much shorter than the lifetimes or \nthe weak oscillation time; $K^0 - \\bar K^0$ oscillations could \nthen not be observed \\cite{GOOD}. \nThere are some loopholes in this argument -- \nyet I consider it still intriguing or at least entertaining. \n\n\n\\end{itemize}\n\n \n\\section{CP Phenomenology in $K_L$ Decays}\n\\subsection{General Formalism}\n\nOscillations become more complex once CP symmetry is broken in \n$\\Delta S=2$ transitions, as seen from solving the (free) \nSchr\\\" odinger equation \n\\begin{equation}} \\def\\eeq{\\end{equation} \ni\\frac{d}{dt} \\left( \n\\begin{array}{ll}\nK^0 \\\\\n\\bar K^0\n\\end{array} \n\\right) = \\left( \n\\begin{array}{ll}\nM_{11} - \\frac{i}{2} \\Gamma _{11} & \nM_{12} - \\frac{i}{2} \\Gamma _{12} \\\\ \nM^*_{12} - \\frac{i}{2} \\Gamma ^*_{12} & \nM_{22} - \\frac{i}{2} \\Gamma _{22} \n\\end{array}\n\\right) \n\\left( \n\\begin{array}{ll}\nK^0 \\\\\n\\bar K^0\n\\end{array} \n\\right) \n\\label{SCHROED} \n\\eeq\nCPT invariance imposes \n\\begin{equation}} \\def\\eeq{\\end{equation} \nM_{11}= M_{22} \\; \\; , \\; \\; \\Gamma _{11} = \\Gamma _{22} \\; . \n\\label{CPTMASS}\n\\eeq \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n{\\em Homework Problem \\#1}: \n\\end{center}\nWhich physical situation is \ndescribed by an equation analogous to Eq.(\\ref{SCHROED}) \nwhere however the two diagonal matrix elements differ \n{\\em without} violating CPT? \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $\n\\end{center} \nThe mass eigenstates obtained through diagonalising this matrix \nare given by (for details see \\cite{LEE,BOOK}) \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|K_S\\rangle \\equiv |K_1\\rangle = \n\\frac{1}{\\sqrt{|p|^2 + |q|^2}} \\left( p |K^0 \\rangle + \nq |\\bar K^0\\rangle \\right) \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|K_L\\rangle \\equiv |K_2\\rangle = \n\\frac{1}{\\sqrt{|p|^2 + |q|^2}} \\left( p |K^0 \\rangle - \nq |\\bar K^0\\rangle \\right) \n\\eeq \nwith \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{q}{p} = \\sqrt{\\frac{M_{12}^* - \\frac{i}{2} \\Gamma _{12}^*}\n{M_{12} - \\frac{i}{2} \\Gamma _{12}}}\n\\eeq \nand eigenvalues \n\\begin{equation}} \\def\\eeq{\\end{equation} \nM_S - \\frac{i}{2}\\Gamma _S = M_{11} - \\frac{i}{2} \\Gamma _{11} \n- \\sqrt{\\left( M_{12} - \\frac{i}{2} \\Gamma _{12}\\right) \n\\left( M^*_{12} - \\frac{i}{2} \\Gamma ^*_{12}\\right)} \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \nM_L - \\frac{i}{2}\\Gamma _L = M_{11} - \\frac{i}{2} \\Gamma _{11} \n+ \\sqrt{\\left( M_{12} - \\frac{i}{2} \\Gamma _{12}\\right) \n\\left( M^*_{12} - \\frac{i}{2} \\Gamma ^*_{12}\\right)} \n\\eeq \nThese states are conveniently expressed in terms of the \nCP eigenstates $|K_{\\pm}\\rangle$: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|K_S \\rangle = \\frac{1+q\/p}{\\sqrt{2(1+ |q\/p|^2)}} \n\\left( |K_+\\rangle + \\bar \\epsilon |K_-\\rangle\\right) \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|K_L \\rangle = \\frac{1+q\/p}{\\sqrt{2(1+ |q\/p|^2)}} \n\\left( |K_-\\rangle + \\bar \\epsilon |K_+\\rangle\\right) \n\\eeq \nwhere the parameter \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\bar \\epsilon \\equiv \\frac{1-q\/p}{1+q\/p} \n\\eeq \nreflects the CP impurity in the state vector. \n\nA few comments -- some technical, some not -- \nmight elucidate the situation: \n\\begin{itemize}\n\\item \nIf there is no relative phase between $M_{12}$ and \n$\\Gamma _{12}$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm arg} \\frac{M_{12}}{\\Gamma _{12}} =0 \n\\eeq \nthen $q\/p =1$ and the state vectors conserve CP: \n$\\bar \\epsilon =0$. \n\\item \nYet for our later \ndiscussion one should take note that \n$q\/p$ -- and therefore also $\\bar \\epsilon$ -- \n{\\em by itself cannot} be an observable. For a change in the \nphase {\\em convention} adopted for defining $\\bar K^0$ does \nnot leave it invariant: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|\\bar K^0 \\rangle \\rightarrow e^{i\\xi}|\\bar K^0 \\rangle \\; \n\\Longrightarrow \\; \n(M_{12}, \\Gamma _{12}) \\rightarrow e^{i\\xi} \n(M_{12}, \\Gamma _{12}) \\; \n\\Longrightarrow \\; \n\\frac{q}{p} \\rightarrow e^{-i\\xi} \\frac{q}{p} \\; !\n\\eeq\nOn the other hand $|q\/p|$ is independant of the phase \nconvention and its deviation from unity is one measure \nof CP violation. \n\\item \nOn very general grounds -- without recourse to any model -- \none can infer that CP violation in the neutral kaon system \nhas to be small. CP invariance implies the two mass eigenstates \n$K_L$ and $K_S$ to be orthogonal -- as can be read off \nexplicitely from the general expression \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\langle K_L |K_S \\rangle = \\frac{1 -|q\/p|^2} {1 +|q\/p|^2}\n\\eeq\nThe Bell-Steinberger relation allows to place a bound on \nthis scalar product from inclusive decay rates \n\\cite{LEE,BOOK}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\langle K_L |K_S \\rangle \\leq \\sqrt{2} \n\\sum _f \\sqrt{\\frac{\\Gamma _L^f\\Gamma _S^f}{\\Gamma _S^2} } \n\\leq \\sqrt{2} \\sqrt{\\frac{\\Gamma _L}{\\Gamma _S}} \n\\simeq 0.06 \n\\label{BELL} \n\\eeq \nThere is no input from any CP measurement. What is essential, \nthough, is the huge lifetime ratio. \n\\item \nThere are actually two processes underlying the transition \n$K_L \\rightarrow 2\\pi$: $\\Delta S=2$ forces generate the mass eigenstates \n$K_L$ and $K_S$ whereas $\\Delta S=1$ dynamics drive the decays \n$K \\rightarrow 2\\pi$. Thus CP violation can enter in two a priori independant \nways, namely through the $\\Delta S=2$ and the $\\Delta S=1$ \nsector. This distinction can be made explicit in terms of the \ntransition amplitudes: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\eta _{+-} \\equiv \n\\frac{A(K_L \\rightarrow \\pi ^+ \\pi ^-)}{A(K_S \\rightarrow \\pi ^+ \\pi ^-)} \\equiv \n\\epsilon _K+ \\epsilon ^{\\prime} \\; , \\; \n\\eta _{00} \\equiv \n\\frac{A(K_L \\rightarrow \\pi ^0 \\pi ^0)}{A(K_S \\rightarrow \\pi ^0 \\pi ^0)} \\equiv \n\\epsilon _K- 2\\epsilon ^{\\prime} \n\\eeq\nThe quantity $\\epsilon _K$ describes the CP violation common to the \n$K_L$ decays; it thus characterizes the decaying {\\em state} and is \nreferred to as {\\em CP violation in the mass matrix} or \n{\\em superweak CP violation}; $\\epsilon ^{\\prime}$ on the other \nhand differentiates between different channels and thus \ncharacterizes {\\em decay} dynamics; it is called {\\em direct \nCP violation}. \n\\item \n{\\em Maximal} parity and\/or charge conjugation violation \ncan be defined by saying there is no right-handed neutrino \nand\/or left-handed antineutrino, respectively. Yet \n{\\em maximal} CP violation {\\em cannot} be defined in an analogous \nway: for the existence of the right-handed antineutrino which is the \nCP conjugate to the left-handed neutrino is already required by \nCPT invariance. \n\n\\end{itemize} \n\n\\subsection{Data}\nThe data on CP violation in neutral kaon decays are as follows: \n\\begin{enumerate} \n\\item \n{\\em Existence} of $K_L \\rightarrow \\pi \\pi$:\n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{l}\n{\\rm BR}(K_L \\rightarrow \\pi ^+ \\pi ^-) = (2.067 \\pm 0.035) \\cdot 10^{-3} \\\\ \n{\\rm BR}(K_L \\rightarrow \\pi ^0 \\pi ^0) = (0.936 \\pm 0.020) \\cdot 10^{-3} \\\\ \n\\end{array}\n\\label{ETADATA}\n\\eeq \n\\item \nSearch for {\\em direct} CP violation: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\epsilon ^{\\prime}}{\\epsilon _K} \\simeq \n{\\rm Re} \\frac{\\epsilon ^{\\prime}}{\\epsilon _K} = \n\\left\\{ \n\\begin{array}{ll}\n(2.3 \\pm 0.65) \\cdot 10^{-3} & NA\\; 31 \\\\\n(1.5 \\pm 0.8) \\cdot 10^{-3} & PDG \\; '96\\; average \\\\\n(0.74 \\pm 0.52 \\pm 0.29) \\cdot 10^{-3} & E\\; 731 \\\\\n\\end{array} \n\\right. \n\\label{DIRECTCPDATA} \n\\eeq \n\\item \nRate {\\em difference} in semileptonic decays: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\delta _l\\equiv \n\\frac{\\Gamma (K_L \\rightarrow l^+ \\nu \\pi ^-) - \n\\Gamma (K_L \\rightarrow l^- \\bar \\nu \\pi ^+)}\n{\\Gamma (K_L \\rightarrow l^+ \\nu \\pi ^-) + \n\\Gamma (K_L \\rightarrow l^- \\bar \\nu \\pi ^+)} = \n(3.27 \\pm 0.12)\\cdot 10^{-3} \\; , \n\\label{SLDIFFDATA} \n\\eeq\nwhere an average over electrons and muons has been taken. \n\\item \n{\\em T violation}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\Gamma (K^0 \\Rightarrow \\bar K^0) - \n\\Gamma (\\bar K^0 \\Rightarrow K^0)} \n{\\Gamma (K^0 \\Rightarrow \\bar K^0) + \n\\Gamma (\\bar K^0 \\Rightarrow K^0)} = \n(6.3 \\pm 2.1 \\pm 1.8) \\cdot 10^{-3} \\; \\; \\; \nCPLEAR \n\\label{CPLEAR}\n\\eeq \nfrom a third of their data set \\cite{CPLEARBLOCH}. \nIt would be premature to claim this asymmetry has been \nestablished; yet it represents an intriguingly direct test of time \nreversal violation and is sometimes referred to as the Kabir \ntest . It requires tracking the flavour identity of \nthe {\\em decaying} meson as a $K^0$ or $\\bar K^0$ through its \nsemileptonic decays -- $\\bar K^0 \\rightarrow l^- \\bar \\nu \\pi ^+$ vs. \n$K^0 \\rightarrow l^+ \\nu \\pi ^-$ -- and also of the {\\em initially \nproduced} kaon. The latter is achieved through correlations \nimposed by associated production. The CPLEAR collaboration \nstudied low energy proton-antiproton annihilation \n\\begin{equation}} \\def\\eeq{\\end{equation} \np \\bar p \\rightarrow K^+ \\bar K^0 \\pi ^- \\; \\; vs. \\; \\; \np \\bar p \\rightarrow K^- K^0 \\pi ^+ \\; ; \n\\eeq \nthe charged kaon reveals whether a $K^0$ or a $\\bar K^0$ was \nproduced in association with it. In the future the CLOE collaboration \nwill study T violation in $K^0 \\bar K^0$ production at DA$\\Phi$NE: \n\\begin{equation}} \\def\\eeq{\\end{equation} \ne^+ e^- \\rightarrow \\phi (1020) \\rightarrow K^0 \\bar K^0 \n\\eeq \n\\end{enumerate}\n\n\\subsection{Phenomenological Interpretation}\n\\subsubsection{Semileptonic Transitions}\nCPT symmetry imposes constraints well beyond the equality of \nlifetimes for particles and antiparticles: certain {\\em sub}classes of \ndecay rates have to be equal as well. For example one finds \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Gamma (\\bar K^0 \\rightarrow l^- \\bar \\nu \\pi ^+) = \n\\Gamma (K^0 \\rightarrow l^+ \\nu \\pi ^-) \n\\eeq \nThe rate asymmetry in semileptonic decays listed in \nEq.(\\ref{SLDIFFDATA}) thus reflects pure superweak CP violation: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\delta _l = \\frac{ 1 - |q\/p|^2}{1+|q\/p|^2} \n\\eeq\nFrom the measured value of $\\delta _l$ one then obtains \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left| \\frac{q}{p}\\right| = 1 + (3.27 \\pm 0.12) \\cdot 10^{-3} \n\\eeq \nSince one has for the $K^0 - \\bar K^0$ system specifically \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left| \\frac{q}{p}\\right| \\simeq \n1 + \\frac{1}{2} {\\rm arg}\\frac{M_{12}}{\\Gamma _{12}} \n\\eeq \none can express this kind of CP violation through a phase: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Phi (\\Delta S=2) \\equiv {\\rm arg}\\frac{M_{12}}{\\Gamma _{12}} = \n(6.54 \\pm 0.24)\\cdot 10^{-3} \n\\label{PHI2}\n\\eeq\nThe result of the Kabir test, Eq.(\\ref{CPLEAR}), yields: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Phi (\\Delta S=2) = (6.3 \\pm 2.1 \\pm 1.8)\\cdot 10^{-3}\\; , \n\\eeq \nwhich is of course consistent with Eq.(\\ref{PHI2}). \n\nUsing the measured value of $\\Delta m_K\/\\Delta \\Gamma _K$ one \ninfers \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{M_{12}}{\\Gamma _{12}} = - (0.4773 \\pm 0.0023)\n\\left[ 1 - i (6.54 \\pm 0.24)\\cdot 10^{-3})\\right] \n\\eeq \n\\subsubsection{Nonleptonic Transitions}\nFrom Eq.(\\ref{ETADATA}) one deduces \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{l} \n|\\eta _{+-}| = (2.275 \\pm 0.019)\\cdot 10^{-3} \\\\ \n|\\eta _{00}| = (2.285 \\pm 0.019)\\cdot 10^{-3} \\\\ \n\\end{array}\n\\eeq \nAs mentioned before the ratios $\\eta _{+-,00}$ are sensitive also \nto direct CP violation generated by a phase between the \ndecay amplitudes $A_{0,2}$ for $K_L\\rightarrow (\\pi \\pi )_I$, where the \nsubscript $I$ denotes the isospin of the $2\\pi$ system: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Phi (\\Delta S=1) \\equiv {\\rm arg}\\frac{A_2}{A_0} \n\\eeq\nOne finds \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\eta _{+-} \\simeq \\frac{i \\tilde x}{2\\tilde x+i}\n\\left[ \\Phi (\\Delta S=2) + 2 \\omega \\Phi (\\Delta S=1) \\right] \\; , \n\\eeq \nwith \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\tilde x \\equiv \\frac{\\Delta m_K}{\\Delta \\Gamma _K} = \n\\frac{\\Delta m_K}{\\Gamma (K_S)} \n=\\frac{1}{2} x_K \\simeq 0.477 \\; \\; , \\; \\; \n\\omega \\equiv \\left| \\frac{A_2}{ A_0}\\right| \\simeq 0.05\n\\eeq\nwhere the second quantity represents the observed\nenhancement of $A_0$ for which a name -- \"$\\Delta I=1\/2$ rule\" -- \nyet no quantitative dynamical explanation has been found. \nEquivalently one can write \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\epsilon ^{\\prime}}{\\epsilon _K} \\simeq 2 \\omega \n\\frac{\\Phi (\\Delta S=1)}{\\Phi (\\Delta S=2)}\n\\eeq \nThe data on $K_L \\rightarrow \\pi \\pi$ can thus be expressed as \nfollows \\cite{WINSTEIN} \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{l} \n\\Phi (\\Delta S=2) = (6.58 \\pm 0.26) \\cdot 10^{-3} \\\\ \n\\Phi (\\Delta S=1) = (0.99 \\pm 0.53) \\cdot 10^{-3} \n\\end{array}\n\\eeq \n\\subsubsection{Resume}\nThe experimental results can be summarized as follows: \n\\begin{itemize}\n\\item \nThe decays of neutral kaons exhibit unequivocally CP violation \nof the superweak variety, which is expressed through the \nangle $\\Phi (\\Delta S=2)$. The findings from \nsemileptonic and nonleptonic transitions concur to an impressive \ndegree. \n\\item \nDirect CP violation still has not been established. \n\\item \nA theorist might be forgiven for mentioning that the evolution of the \nmeasurements over the last twenty odd \nyears has not followed the straight line this brief summary \nmight suggest to the uninitiated reader. \n\\end{itemize} \n\n\\section{Theoretical Implementation of CP Violation}\n\n\\subsection{Some Historical Remarks}\nTheorists can be forgiven if they felt quite pleased with the state \nof their craft in 1964:\n\\begin{itemize}\n\\item \nThe concept of (quark) families had emerged, at least in a \nrudimentary form. \n\\item \nMaximal parity and charge conjugation violations had been found in \nweak charged current interactions, yet CP invariance apparently \nheld. Theoretical pronouncements were made ex cathedra why this \nhad to be so!\n\\item \n{\\em Pre}dictions of the existence of two kinds of neutral \nkaons with different lifetimes and masses had been \nconfirmed by experiment \\cite{PAIS}. \n\\end{itemize}\nThat same year the reaction $K_L \\rightarrow \\pi ^+ \\pi ^-$ was \ndiscovered \\cite{FITCH}! Two things should be noted here. \nThe Fitch-Cronin experiment had predecessors: rather than \nbeing an isolated effort it was the culmination of a whole \nresearch program. Secondly there was at least one theoretical \nvoice, namely that of Okun \\cite{OKUN}, who in 1962\/63 had listed a \ndedicated search for $K_L \\rightarrow \\pi \\pi$ as one of the most important \nunfinished tasks. Nevertheless for the vast majority of the community the \nFitch-Cronin observation came as a shock and caused considerable \nconsternation among theorists. Yet -- to their credit -- these data \nand their consequence, namely that CP invariance was broken, \nwere soon accepted as facts. This was phrased -- though \n{\\em not explained} -- in terms of the Superweak Model \n\\cite{WOLFSW} later that same year. \n\nIn 1970 the renormalizability of the $SU(2)_L\\times U(1)$ \nelectroweak gauge theory was proven. I find it quite amazing \nthat it was still not realized that the physics known at that time \ncould not produce CP violation. As long as one had to struggle \nwith infinities in the theoretical description one could be forgiven \nfor not worrying unduly about a tiny quantity like \nBR$(K_L \\rightarrow \\pi ^+ \\pi ^-) \\simeq 2.3 \\cdot 10^{-3}$. Yet no such \nexcuse existed any longer once a renormalizable theory had been \ndeveloped! The existence of the Superweak Model somewhat \nmuddled the situation in this respect: for it provides merely \na classification of the dynamics underlying CP violation rather \nthan a dynamical description itself. \n\nThe paper by Kobayashi and Maskawa \\cite{KM}, \nwritten in 1972 and published in 1973, was the first \n\\begin{itemize}\n\\item \nto state clearly that the \n$SU(2)_L\\times U(1)$ gauge theory even with two complete \nfamilies \\footnote{Remember this was still before the $J\/\\psi$ \ndiscovery!} is necessarily CP-invariant and \n\\item \nto list the possible extensions that could generate CP \nviolation; among them -- as one option -- was the three (or more) \nfamily scenario now commonly referred to as the KM ansatz. They \nalso discussed the impact of right-handed currents and of a \nnon-minimal Higgs sector. \n\\end{itemize}\n\n\\subsection{The Minimal Model: The KM Ansatz}\nOnce a theory reaches a certain degree of complexity, many potential \nsources of CP violation emerge. Popular examples of such a scenario \nare provided by models implementing supersymmetry or its \nlocal version, supergravity; hereafter both are referred to as \nSUSY. In my lectures I will however focus on the minimal \ntheory that can support CP violation, namely the Standard Model \nwith three families. All of its dynamical elements have been \nobserved -- except for the Higgs boson, of course. \n\n\\subsubsection{Weak Phases like the Scarlet Pimpernel}\n\nWeak interactions at low energies are described by four-fermion \ninteractions. The most general expression for spin-one \ncouplings are \n$$ \n{\\cal L}_{V\/A} = \\left( \\bar \\psi _1 \\gamma _{\\mu}\n(a + b \\gamma _5)\\psi _2\\right) \n\\left( \\bar \\psi _3 \\gamma _{\\mu}\n(c + d \\gamma _5)\\psi _4\\right) + \n$$\n\\begin{equation}} \\def\\eeq{\\end{equation} \n+ \\left( \\bar \\psi _2 \\gamma _{\\mu}\n(a^* + b^* \\gamma _5)\\psi _1\\right) \n\\left( \\bar \\psi _4 \\gamma _{\\mu}\n(c^* + d^* \\gamma _5)\\psi _3\\right)\n\\eeq \nUnder CP these terms transform as follows: \n$$ \n{\\cal L}_{V\/A} \\stackrel{CP}{\\Longrightarrow} \nCP {\\cal L}_{V\/A} (CP)^{\\dagger} = \n\\left( \\bar \\psi _2 \\gamma _{\\mu}\n(a + b \\gamma _5)\\psi _1\\right) \n\\left( \\bar \\psi _4 \\gamma _{\\mu}\n(c + d \\gamma _5)\\psi _3\\right) + \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n+ \\left( \\bar \\psi _1 \\gamma _{\\mu}\n(a^* + b^* \\gamma _5)\\psi _2\\right) \n\\left( \\bar \\psi _3 \\gamma _{\\mu}\n(c^* + d^* \\gamma _5)\\psi _4\\right)\n\\eeq \nIf $a,b,c,d$ are real numbers, one obviously has \n${\\cal L}_{V\/A}= CP {\\cal L}_{V\/A} (CP)^{\\dagger} $ and CP \nis conserved. Yet CP is {\\em not necessarily} broken if these \nparameters are complex, as we will explain specifically \nfor the Standard Model. \n\n{\\em Weak Universality} arises naturally whenever the weak \ncharged current interactions are described through a \n{\\em single} non-abelian gauge group -- $SU(2)_L$ in the case \nunder study. For the single {\\em self}-coupling of the gauge bosons \ndetermines also their couplings to the fermions; \none finds for the quark couplings to the charged $W$ bosons: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal L}_{CC} = \ng \\bar U_L^{(0)}\\gamma _{\\mu}D_L^{(0)} W^{\\mu} + \n\\bar U_R^{(0)} {\\bf M}_U U_L^{(0)} + \n\\bar D_R^{(0)} {\\bf M}_D D_L^{(0)} + h.c. \n\\eeq \nwhere $U$ and $D$ denote the up- and down-type quarks, \nrespectively: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nU= (u,c,t) \\; \\; \\; , \\; \\; \\; \\; D=(d,s,b) \n\\eeq \nand ${\\bf M_U}$ and ${\\bf M_D}$ \ntheir 3$\\times$3 mass matrices. In general \nthose will not be diagonal; to find the physical states, one has to \ndiagonalize these matrices: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\bf M}^{diag}_U = {\\bf K}^U_R{\\bf M}_U ({\\bf K}^U_L)^{\\dagger} \\; \\; , \\; \\; \n{\\bf M}^{diag}_D = {\\bf K}^D_R{\\bf M}_D ({\\bf K}^D_L)^{\\dagger}\n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \nU_{L,R} = {\\bf K}^U_{L,R} U^{(0)}_{L,R} \\; \\; , \\; \\; \nD_{L,R} = {\\bf K}^D_{L,R} D^{(0)}_{L,R}\n\\eeq \nwith ${\\bf K}_{L,R}^{U,D}$ representing four unitary 3$\\times$3 matrices. \nThe coupling of these physical fermions to $W$ bosons is then given \nby \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal L}_{CC} = \ng \\bar U_L({\\bf K}_L^U)^{\\dagger}{\\bf K}_L^D\\gamma _{\\mu}D W^{\\mu} + \n\\bar U_R {\\bf M}^{diag}_U U_L + \n\\bar D_R {\\bf M}^{diag}_D D_L + h.c. \n\\eeq \nand the combination $({\\bf K}_L^U)^{\\dagger}{\\bf K}_L^D \n\\equiv {\\bf V}_{CKM}$ \nrepresents the KM matrix, which obviously has to be unitary \nlike $K^U$ and $K^D$. Unless the \nup- and down-type mass matrices are aligned in flavour \nspace (in which case they would be diagonalized by the \nsame operators ${\\bf K}_{L,R}$) one has ${\\bf V}_{CKM} \\neq 1$. \n\nIn the neutral current sector one has \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal L}_{NC} = g^{\\prime} \n\\bar U_L^{(0)} \\gamma _{\\mu}U_L^{(0)}Z_{\\mu} = \ng^{\\prime} \n\\bar U_L \\gamma _{\\mu}U_LZ_{\\mu}\n\\eeq \nand likewise for $U_R$ and $D_{L,R}$; i.e. {\\em no} \nflavour changing neutral currents are generated, let alone \nnew phases. CP violation thus has to be embedded into the \ncharged current sector. \n\nIf ${\\bf V}_{CKM}$ is real (and thus orthogonal), CP symmetry is \nconserved in the weak interactions. Yet the occurrance of \ncomplex matrix elements does not {\\em automatically} signal \nCP violation. This can be seen through a straightforward \n(in hindsight at least) algebraic argument. A unitary \n$N\\times N$ matrix contains $N^2$ independant real \nparameters; $2N-1$ of those can be eliminated through \nre-phasing of the $N$ up-type and $N$ down-type fermion \nfields (changing all fermions by the {\\em same} phase obviously \ndoes not affect ${\\bf V}_{CKM}$). Hence there are $(N-1)^2$ \nreal physical parameters in such an $N \\times N$ matrix. \nFor $N=2$, i.e. two families, one recovers a familiar result, \nnamely there is just one mixing angle, the Cabibbo \nangle. For $N=3$ there are four real physical parameters, \nnamely three (Euler) angles -- and one phase. It is the latter \nthat provides a gateway for CP violation. For $N=4$ Pandora's \nbox opens up: there would be 6 angles and 3 phases. \n\nPDG suggests a \"canonical\" parametrization for the $3\\times 3$ CKM \nmatrix: \n$$ \n{\\bf V}_{CKM} = \n\\left( \n\\begin{array}{ccc} \nV(ud) & V(us) & V(ub) \\\\\nV(cd) & V(cs) & V(cb) \\\\\nV(td) & V(ts) & V(tb) \n\\end{array} \n\\right) \n$$\n\\begin{equation}} \\def\\eeq{\\end{equation} \n= \\left( \n\\begin{array}{ccc} \nc_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i \\delta _{13}} \\\\\n- s_{12}c_{23} - c_{12}s_{23}s_{13}e^{i \\delta _{13}} &\nc_{12}c_{23} - s_{12}s_{23}s_{13}e^{i \\delta _{13}} & \nc_{13}s_{23} \\\\\ns_{12}s_{23} - c_{12}c_{23}s_{13}e^{i \\delta _{13}} &\n- c_{12}s_{23} - s_{12}c_{23}s_{13}e^{i \\delta _{13}} &\nc_{13}c_{23} \n\\end{array}\n\\right) \n\\label{PDGKM} \n\\eeq\nwhere \n\\begin{equation}} \\def\\eeq{\\end{equation} \nc_{ij} \\equiv {\\rm cos} \\theta _{ij} \\; \\; , \\; \\; \ns_{ij} \\equiv {\\rm sin} \\theta _{ij}\n\\eeq \nwith $i,j = 1,2,3$ being generation labels. \n\nThis is a completely general, yet not unique parametrisation: a \ndifferent set of \nEuler angles could be chosen; the phases can be shifted around \namong the matrix elements \nby using a different phase convention. In that sense one can refer to \nthe KM phase as the Scarlet Pimpernel: \"Sometimes here, sometimes \nthere, sometimes everywhere!\" \n\nUsing just the observed hierarchy \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|V(ub)| \\ll |V(cb)| \\ll |V(us)| , |V(cd)| \\ll 1\n\\label{HIER}\n\\eeq \none can, as first realized by Wolfenstein, expand \n${\\bf V}_{CKM}$ in powers of the Cabibbo angle $\\theta _C$: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\bf V}_{CKM} = \n\\left( \n\\begin{array}{ccc} \n1 - \\frac{1}{2} \\lambda ^2 & \\lambda & \nA \\lambda ^3 (\\rho - i \\eta + \\frac{i}{2} \\eta \\lambda ^2) \\\\\n- \\lambda & 1 - \\frac{1}{2} \\lambda ^2 - i \\eta A^2 \\lambda ^4 & \nA\\lambda ^2 (1 + i\\eta \\lambda ^2 ) \\\\ \nA \\lambda ^3 (1 - \\rho - i \\eta ) \\\\\n& - A\\lambda ^2 & 1 \n\\end{array}\n\\right) \n+ {\\cal O}(\\lambda ^6) \n\\label{WOLFKM}\n\\eeq \nwhere \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\lambda \\equiv {\\rm sin} \\theta _C\n\\eeq\nFor such an expansion in powers of $\\lambda$ to be self-consistent, \none has to require that $|A|$, $|\\rho |$ and $|\\eta |$ are of order \nunity. Numerically we obtain \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\lambda = 0.221 \\pm 0.002 \n\\label{KMNUm1} \n\\eeq \nfrom $|V(us)|$, \n\\begin{equation}} \\def\\eeq{\\end{equation} \nA = 0.81 \\pm 0.06 \n\\label{KMNUM2}\n\\eeq \nfrom $|V(cb)| \\simeq 0.040 \\pm 0.002|_{exp} \\pm 0.002|_{theor}$ \nand \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\sqrt{\\rho ^2 + \\eta ^2} \\sim 0.38 \\pm 0.11 \n\\label{KMNUM3} \n\\eeq\nfrom $|V(ub)| \\sim (3.2 \\pm 0.8)\\cdot 10^{-3}$. \n\nWe see that the CKM matrix is a very special unitary matrix: \nit is almost diagonal, it is almost symmetric and the matrix \nelements get smaller the more one moves away from the \ndiagonal. \nNature most certainly has encoded a profound message in \nthis peculiar pattern. Alas -- we have not succeeded yet in \ndeciphering it! I will return to this point at the end of my \nlectures. \n\n\\subsubsection{Unitarity Triangles}\n\nThe qualitative difference between a two and a three family scenario \ncan be seen also in a less abstract way. \nConsider $\\bar K^0 \\rightarrow \\pi ^+ \\pi ^-$; it can proceed through \na tree-level process \n$[s\\bar d] \\rightarrow [d \\bar u][u\\bar d]$ , in which case its \nweak couplings are \ngiven by $V(us)V^*(ud)$. Or it can oscillate first to $K^0$ \nbefore decaying; i.e., on the quark level it is the \ntransition $[s\\bar d] \\rightarrow [d\\bar s] \\rightarrow [d \\bar u][u\\bar d]$ \ncontrolled by \n$\\left( V(cs)\\right) ^2 \\left( V^*(cd)\\right) ^2 V^*(us)V(ud)$. \nAt first sight it would seem that those two combinations of \nweak parameters are not only different, but should also exhibit a \nrelative phase. Yet the latter is not so -- if there are two families \nonly! In that case the four quantities $V(ud)$, $V(us)$, $V(cd)$ \nand $V(cs)$ have to form a unitary $2\\times 2$ which leads to the \nconstraint \n\\begin{equation}} \\def\\eeq{\\end{equation} \nV(ud)V^*(us) + V(cd)V^*(cs) = 0 \n\\label{UNIT2FAM}\n\\eeq \nUsing Eq.(\\ref{UNIT2FAM}) twice one gets \n$$ \n\\left( V(cs) V^*(cd)\\right) ^2 V^*(us)V(ud) = \n- |V(cd)V(cs)|^2 V(cs)V^*(cd) = \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n= |V(cd)V(cs)|^2 V^*(ud) V(us) \\; ; \n\\eeq \n i.e., the two combinations $V^*(ud)V(us)$ and \n$\\left( V(cs)\\right) ^2 \\left( V^*(cd)\\right) ^2 V^*(us)V(ud)$ are \nactually parallel to each other with {\\em no} \nrelative phase. A penguin \noperator with a charm quark as the internal fermion \nline generates another contribution to \n$K_L \\rightarrow \\pi ^-\\pi ^+$, this one controlled by $V(cs)V^*(cd)$. Yet \nthe unitarity condition Eq.(\\ref{UNIT2FAM}) forces this contribution \nto be antiparallel to $V^*(ud)V(us)$; i.e., again no relative phase. \n\nThe situation changes fundamentally for three families: the weak \nparameters $V(ij)$ now form a $3\\times 3$ matrix and the condition \nstated in Eq.(\\ref{UNIT2FAM}) gets extended: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nV(ud)V^*(us) + V(cd)V^*(cs) + V(td)V^*(ts) = 0 \n\\label{UNIT3FAM}\n\\eeq \n{\\em This is a triangle relation in the complex plane.} \nThere emerge now \nrelative phases between the weak parameters and the \nloop diagrams with internal charm and top quarks can generate \nCP asymmetries. \n\nUnitarity imposes altogether nine algebraic conditions on the \nmatrix elements of ${\\bf V}_{CKM}$, of which six are triangle relations \nanalogous to Eq.(\\ref{UNIT3FAM}). \nThere are several nice features about this representation in terms of \ntriangles; I list four now and others later: \n\\begin{enumerate}\n\\item \nThe {\\em shape} of each triangle is independant of the phase \nconvention adopted for the quark fields. Consider for example \nEq.(\\ref{UNIT3FAM}): changing the phase of any of the \nup-type quarks will not affect the triangle at all. Under \n$|s\\rangle \\rightarrow |s\\rangle e^{i \\phi _s}$ the whole triangle will \nrotate around the left end of its base line by an angle \n$\\phi _s$ -- yet the shape of the triangle -- in contrast \nto its orientation in the complex plane -- remains the same! \nThe angles inside the triangles are thus observables; \nchoosing an orientation for the triangles is then a matter \nof convenience. \n\\item \nIt is easily shown that all six KM triangles possess \nthe same area. \nMultiplying Eq.(\\ref{UNIT3FAM}) by the phase \nfactor $V^*(ud)V(us)\/|V(ud)V(us)|$, which does not change the \narea, yields \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|V(ud)V(us)| + \\frac{V^*(ud)V(us)V(cd)V^*(cs)}{|V(ud)V(us)|} + \n\\frac{V^*(ud)V(us)V(td)V^*(ts)}{|V(ud)V(us)|} = 0 \n\\eeq \n$$ \n{\\rm area (triangle \\; of \\; Eq.(\\ref{UNIT3FAM})}) = \n\\frac{1}{2} |{\\rm Im}V(ud)V(cs)V^*(us)V^*(cd)| = \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n= \\frac{1}{2} |{\\rm Im}V(ud)V(ts)V^*(us)V^*(td)|\n\\eeq\nMultiplying Eq.(\\ref{UNIT3FAM}) instead by the phase \nfactors $V^*(cd)V(cs)\/|V(cd)V(cs)|$ or $V^*(td)V(ts)\/|V(td)V(ts)|$ \none sees that the area of this triangle can be expressed in other ways \nstill. Among them is \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm area (triangle \\; of \\; Eq.(\\ref{UNIT3FAM})}) = \n\\frac{1}{2} |{\\rm Im}V(cd)V(ts)V^*(cs)V^*(td)| \n\\eeq \nDue to the unitarity relation \n\\begin{equation}} \\def\\eeq{\\end{equation} \nV^*(cd)V(td) + V^*(cb)V(tb) = - V^*(cs)V(ts) \n\\label{UNIT3FAM2}\n\\eeq\none has \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm area ( triangle \\; of \\; Eq.(\\ref{UNIT3FAM})}) = \n\\frac{1}{2} |{\\rm Im}V(cd)V(tb)V^*(cb)V^*(td)| \n\\eeq \n-- yet this is exactly the area of the triangle defined by \nEq.(\\ref{UNIT3FAM2})! \nThis is the re-incarnation of the original \nobservation that there is a {\\em single irreducible} \nweak phase for three families. \n\\item \nIn general one has for the area of these triangles \n$$ \nA_{CPV}(\\rm every \\; triangle) = \\frac{1}{2} J \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \nJ = {\\rm Im}V^*(km) V(lm)V(kn)V^*(ln) = \n{\\rm Im}V^*(mk) V(ml)V(nk)V^*(nl) \n\\label{KMAREA}\n\\eeq\nirrespective of the indices $k,l,m,n$; $J$ is obviously re-phasing \ninvariant. \n\\item \nIf there is a representation of $V_{CKM}$ where \nall phases were confined to a $2\\times 2$ \nsub-matrix exactly rather than approximately, then one can \nrotate all these phases away; i.e., CP is conserved in such a \nscenario! Consider again the triangle described by \nEq.(\\ref{UNIT3FAM}): it can always be rotated such that its \nbaseline -- $V(ud)V^*(us)$ -- \nis real. Then Im$V(td)V^*(ts)$ = - Im$V(cd)V^*(cs)$ holds. \nIf, for example, there were no phases in the third row and column, \none would have Im$(V(td)V^*(ts)) =0$ and therefore \nIm$V(cd)V^*(cs) =0$ as well; i.e., $V(ud)V^*(us)$ and \n$V(cd)V^*(cs)$ were real relative to each other; \ntherefore $J=0$, i.e. all six triangles had zero area meaning \nthere are no relative weak phases! \n\\end{enumerate} \n\n\\subsection{Evaluating $\\epsilon _K$ and $\\epsilon ^{\\prime}$} \nIn calculating observables in a given theory -- in the case under \nstudy $\\epsilon _K$ and $\\epsilon ^{\\prime}$ \nwithin the KM Ansatz -- one is faced with the `Dichotomy of the Two \nWorlds', namely \n\\begin{itemize}\n\\item \none world of {\\em short}-distance physics where even the strong \ninteractions can be treated {\\em perturbatively} in terms of \nquarks and gluons and in which theorists like to work, and \n\\item \nthe other world of {\\em long}-distance physics where one has \nto deal with hadrons the behaviour of which is controlled \nby {\\em non}-perturbative dynamics and where, by the way, \neveryone, including theorists, lives. \n\\end{itemize}\nAccordingly the calculational task is divided into two \nparts, namely first determing the relevant \ntransition operators in the short-distance world and then \nevaluating their matrix elements in the hadronic world. \n\n\\subsubsection{$\\Delta S=2$ Transitions}\nSince the {\\em elementary} interactions in the \nStandard Model can change strangeness at most by one unit, \nthe $\\Delta S=2$ amplitude driving $K^0 - \\bar K^0$ \noscillations is obtained by iterating the \nbasic $\\Delta S=1$ coupling: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal L}_{eff} (\\Delta S=2) = {\\cal L}(\\Delta S=1) \\otimes \n{\\cal L}(\\Delta S=1) \n\\eeq \nThere are actually two ways in which the $\\Delta S=1$ transition \ncan be iterated: \n\n{\\bf (A)} \nThe resulting $\\Delta S=2$ transition is described by \na {\\em local} operator. The celebrated box diagram makes \nthis connection quite transparent. The contributions that do \n{\\em not} depend on the mass of the internal quarks cancel against \neach other due to the GIM mechanism. Integrating over the internal \nfields, namely the $W$ bosons and the top and charm quarks \n\\footnote{The up quarks act merely as a subtraction term here.} \nthen yields a convergent result: \n$$ \n{\\cal L}_{eff}^{box}(\\Delta S=2, \\mu ) = \n\\left( \\frac{G_F}{4\\pi }\\right) ^2 \\cdot \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\cdot \\left[ \\xi _c^2 E(x_c) \\eta _{cc} + \n\\xi _t^2 E(x_t) \\eta _{tt} + \n2\\xi _c \\xi _t E(x_c, x_t) \\eta _{ct}\n \\right] \\cdot [\\alpha _S(\\mu ^2)]^{-6\/27} \n\\left( \\bar s \\gamma _{\\mu}(1- \\gamma _5) d\\right) ^2 \n\\label{LAGDELTAS2}\n\\eeq \nwith $\\xi _i$ denoting combinations of KM parameters \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\xi _i = V(is)V^*(id) \\; , \\; \\; i=c,t \\; ; \n\\eeq \n$E(x_i)$ and $E(x_c,x_t)$ reflect the box loops with equal and \ndifferent internal quarks, respectively \\cite{INAMI}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nE(x_i) = x_i \n\\left( \n\\frac{1}{4} + \\frac{9}{4(1- x_i)} - \\frac{3}{2(1- x_i)^2} \n\\right) \n- \\frac{3}{2} \\left( \\frac{x_i}{1-x_i}\\right) ^3 \n{\\rm log} x_i \n\\eeq \n$$ \nE(x_c,x_t) = x_c x_t \n\\left[ \\left( \n\\frac{1}{4} + \\frac{3}{2(1- x_t)} - \\frac{3}{4(1- x_t)^2} \\right) \n\\frac{{\\rm log} x_t}{x_t - x_c} + (x_c \\leftrightarrow x_t) - \n\\right. \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left. - \\frac{3}{4} \\frac{1}{(1-x_c)(1- x_t)} \\right] \n\\eeq\n\\begin{equation}} \\def\\eeq{\\end{equation} \nx_i = \\frac{m_i^2}{M_W^2} \n\\eeq \nand $\\eta _{ij}$ containing the QCD radiative corrections from \nevolving the effective Lagrangian from $M_W$ down to \nthe internal quark mass. The factor $[\\alpha _S(\\mu ^2)]^{-6\/27}$ \nreflects the fact that a scale \n$\\mu$ must be introduced at which the four-quark operator \n$\\left( \\bar s \\gamma _{\\mu}(1- \\gamma _5) d\\right) ^2 $ is \ndefined. This dependance on the auxiliary variable \n$\\mu$ drops out when one takes the matrix element of this \noperator (at least when one does it correctly). \nIncluding next-to-leading log \ncorrections one finds (for $m_t \\simeq 180$ GeV) \\cite{BURAS}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\eta _{cc} \\simeq 1.38 \\pm 0.20 \\; , \\; \\; \n\\eta _{tt} \\simeq 0.57 \\pm 0.01 \\; , \\; \\; \n\\eta _{cc} \\simeq 0.47 \\pm 0.04 \n\\eeq \n\n{\\bf (B)} \nHowever there is also a {\\em non}-local $\\Delta S=2$ \noperator generated from the iteration of ${\\cal L}(\\Delta S=1)$. \nIt presumably provides a major contribution to $\\Delta m_K$. \nYet for $\\epsilon _K$ it is not sizeable within the KM ansatz \n\\footnote{This can be inferred from the observation that \n$|\\epsilon ^{\\prime}\/\\epsilon _K|\\ll 0.05$} and will be \nignored here. \n\nEven for a local four-fermion operator it is non-trivial to \nevaluate an on-shell matrix element \nbetween hadron states since that is \nclearly controlled by non-perturbative dynamics. Usually one \nparametrizes this matrix element as follows: \n$$ \n\\matel{\\bar K^0}{(\\bar s \\gamma _{\\mu}(1-\\gamma _5)d) \n(\\bar s \\gamma _{\\mu}(1-\\gamma _5)d)}{K^0} = \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n= \\frac{4}{3} B_K \n\\matel{\\bar K^0}{(\\bar s \\gamma _{\\mu}(1-\\gamma _5)d)}{0} \n\\matel{0}{(\\bar s \\gamma _{\\mu}(1-\\gamma _5)d)}{K^0} = \n\\frac{4}{3} B_K f_K^2m_K\n\\label{BAGFACT} \n\\eeq \nThe factor $B_K$ is -- for historical reasons of no consequence now -- \noften called the bag factor; $B_K = 1$ is referred to as \n{\\em vacuum saturation} or {\\em factorization ansatz} since it \ncorresponds to a situation where inserting the vacuum intermediate \nstate into Eq.(\\ref{BAGFACT}) reproduces the full result \nafter all colour contractions of the quark lines have been included. \nSeveral theoretical techniques have been employed to estimate the \nsize of $B_K$; their findings are listed in \nTable \\ref{TABLEBAG}. \n\\begin{table}\n\\begin{tabular} {|l|l|}\n\\hline \nMethod & $B_K$ \\\\ \n\\hline \n\\hline \nLarge $N_C$ Expansion & $\\frac{3}{4}$\\\\\n\\hline \nLarge $N_C$ Chiral Pert. with loop correction & $0.66 \\pm 0.1$\\\\\n\\hline \nLattice QCD & $0.84 \\pm 0.2$ \\\\\n\\hline \n\\end{tabular}\n\\centering\n\\caption{Values of $B_K$ from various theoretical techniques} \n\\label{TABLEBAG} \n\\end{table} \nThese results, which are all consistent with each other and with \nseveral phenomenological studies as well, can be summarized as \nfollows: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nB_K \\simeq 0.8 \\pm 0.2 \n\\label{BAG} \n\\eeq \nSince the size of this matrix element is determined \nby the strong interactions, one indeed expects $B_K \\sim 1$. \n\nWe have assembled all the ingredients now for calculating \n$\\epsilon _K$. The starting point is given by \n\\footnote{The exact expression is \n$|\\epsilon _K| = \\frac{1}{\\sqrt{2}}\n\\left| \\frac{{\\rm Im}M_{12}}{\\Delta m_K} - \\xi _0\\right| $ \nwhere $\\xi _0$ denotes the phase of the \n$K^0 \\rightarrow (\\pi \\pi )_{I =0}$ isospin zero amplitude; its \ncontribution is \nnumerically irrelevant.}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|\\epsilon _K| \n\\simeq \n\\frac{1}{\\sqrt{2}}\n\\left| \\frac{{\\rm Im}M_{12}}{\\Delta m_K} \\right| \n\\eeq \nThe CP-odd part Im$M_{12}$ is obtained from \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm Im} M_{12} = \n{\\rm Im}\\matel{K^0}{{\\cal L}_{eff}(\\Delta S=2)}{\\bar K^0} \n\\eeq \nwhereas for $\\Delta m_K$ one inserts the experimental value, \nsince the long-distance contributions to $\\Delta m_K$ are not under \ntheoretical control. One then finds \n$$ \n|\\epsilon _K|_{KM} \\simeq |\\epsilon _K|_{KM}^{box} \\simeq \n$$ \n$$ \n\\simeq \\frac{G_F^2}{6 \\sqrt{2} \\pi ^2} \n\\frac{M_W^2m_K f_K^2 B_K}{\\Delta m_K} \n\\left[ {\\rm Im}\\xi _c^2 E(x_c) \\eta _{cc} + \n {\\rm Im}\\xi _t^2 E(x_t) \\eta _{tt} + \n2{\\rm Im}(\\xi _c\\xi _t) E(x_c,x_t) \\eta _{ct} \\right] \n$$ \n$$ \n\\simeq 1.9 \\cdot 10^4 B_K \\left[ {\\rm Im}\\xi _c^2 E(x_c) \\eta _{cc} + \n {\\rm Im}\\xi _t^2 E(x_t) \\eta _{tt} + \n2{\\rm Im}(\\xi _c\\xi _t) E(x_c,x_t) \\eta _{ct} \\right] \n\\simeq \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\simeq \n7.8 \\cdot 10^{-3} \\eta B_K (1.3 - \\rho ) \n\\label{EPSKM} \n\\eeq \nwhere I have used the numerical values for the KM parameters \nlisted above and \n$x_t \\simeq 5$ corresponding to $m_t = 180$ GeV. \n\nTo reproduce the observed value of $|\\epsilon _K|$ one needs \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\eta \\simeq \\frac{0.3}{B_K} \\frac{1}{1.3 - \\rho } \n\\label{EPSREP} \n\\eeq\nFor a given $B_K$ one thus obtains another $\\rho - \\eta $ \nconstraint. Since $B_K$ is not precisely known \n\\footnote{Some might argue that this is an understatement.} \none has a fairly broad band in the $\\rho - \\eta $ plane \nrather than a line. Yet I find it quite remarkable and very \nnon-trivial that Eq.(\\ref{EPSREP}) {\\em can} be \nsatisfied since \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{0.3}{B_K} \\sim 0.3 \\div 0.5 \n\\eeq \nwithout stretching any of the parameters or bounds, in particular \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\sqrt{\\rho ^2 + \\eta ^2} \\sim 0.38 \\pm 0.11 \\; . \n\\eeq \n While this does of course not amount to a {\\em pre}diction, \none should keep in mind for proper perspective \nthat in the 1970's and early \n1980's values like $|V(cb)| \\sim 0.04$ and \n$|V(ub)| \\sim 0.004$ would have seemed quite unnatural; claiming \nthat the top quark mass had to be 180 GeV would have been \noutright preposterous even in the 1980's! Consider a \nscenario with $|V(cb)| \\simeq 0.04$ and $|V(ub)| \\simeq 0.003$, \nyet $m_t \\simeq 40$ GeV; in the mid 80's this would have appeared \nto be quite natural (and there had even been claims that top quarks \nwith a mass of $40\\pm 10$ GeV had been discovered). In that \ncase one would need \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\eta \\sim \\frac{0.75}{B_K} \n\\label{EPS40} \n\\eeq \nto reproduce $|\\epsilon _K|$. Such a large value for $\\eta$ would hardly \nbe compatible with what we know about $|V(ub)|$ \n\\footnote{For some time it was thought that $B_K \\simeq 0.3 \\div \n0.5$ was the best estimate. This would make satisfying \nEq.(\\ref{EPS40}) completely out of the question!}. \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n{\\em Homework Problem \\# 2}: \n\\end{center}\nEq.(\\ref{EPSKM}) suggests that \na non-vanishing value for $\\epsilon _K$ is generated from the \nbox diagram with internal charm quarks only -- \nIm$\\xi _c^2\\; E(x_c) = - \\eta A^2 \\lambda ^6 E(x_c) \\neq 0$ -- \n{\\em without} top quarks. How does this match up with the \nstatement that the intervention of three families \nis needed for a CP asymmetry to arise? \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $\n\\end{center} \n\n\\subsubsection{$\\Delta S=1$ Decays}\nAt first one might think that no {\\em direct} CP asymmetry \ncan arise in $K \\rightarrow \\pi \\pi$ decays since it requires the interplay \nof three quark families. Yet upon further reflection one realizes \nthat a one-loop diagram produces the so-called Penguin \noperator which changes isopin by half a unit only, \nis {\\em local} and contains a CP {\\em odd} component since it involves virtual charm \nand top quarks. With direct CP violation thus being \nof order $\\hbar$, i.e. a pure quantum \neffect, one suspects already at this point that it will be reduced \nin strength. \n\nThe quantity $\\epsilon ^{\\prime}$ is suppressed relative to \n$\\epsilon _K$ due to two other reasons: \n\\begin{itemize}\n\\item \nThe GIM factors are actually quite different for $\\epsilon _K$ and \n$\\epsilon ^{\\prime}$: in the former case they are of the type \n$(m_t^2 - m_c^2)\/M_W^2$, in the latter log$(m_t^2\/m_c^2)$. Both \nof these expressions vanish for $m_t=m_c$, yet for the realistic \ncase $m_t \\gg m_c$ they behave very differently: $\\epsilon _K$ \nis much more enhanced by the large top mass than \n$\\epsilon ^{\\prime}$. This means of course that \n$|\\epsilon ^{\\prime}\/\\epsilon _K|$ is a rather steeply decreasing \nfunction of $m_t$. \n\\item \nThere are actually two classes of Penguin operators contributing \nto $\\epsilon ^{\\prime}$, namely strong as well as electroweak \nPenguins. The latter become relevant \nsince they are more enhanced than the former for very heavy top \nmasses due to the coupling of the longitudinal virtual $Z$ boson \n(the re-incarnation of one of the original Higgs fields) to \nthe internal top line. Yet electroweak and strong Penguins contribute \nwith the opposite sign! \n\\end{itemize}\nCPT invariance together with the measured $\\pi \\pi$ phase shifts \ntells us that the two complex quantities $\\epsilon ^{\\prime}$ and \n$\\epsilon _K$ are almost completely real to each other; i.e., \ntheir ratio is practically real: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\epsilon ^{\\prime}}{\\epsilon _K} \\simeq 2 \\omega \n\\frac{\\Phi (\\Delta S=1)}{\\Phi (\\Delta S=2)}\n\\label{EPSPOVEREPSTH} \n\\eeq \nwhere, as defined before, \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\omega \\equiv \\frac{|A_2|}{|A_0|} \\simeq 0.05 \\; \\; , \\; \\; \n\\Phi (\\Delta S=2) \\equiv {\\rm arg}\\frac{M_{12}}{\\Gamma _{12}} \n\\; \\; , \\Phi (\\Delta S=1) \\equiv {\\rm arg} \\frac{A_2}{A_0}\n\\eeq\nEq.(\\ref{EPSPOVEREPSTH}) makes two points obvious:\n\\begin{itemize}\n\\item \nDirect CP violation -- $\\epsilon ^{\\prime} \\neq 0$ -- \nrequires a relative phase between the isospin 0 and 2 amplitudes; \ni.e., $K \\rightarrow (\\pi \\pi )_0$ and $K \\rightarrow (\\pi \\pi )_2$ have to exhibit \ndifferent CP properties. \n\\item \nThe observable ratio $\\epsilon ^{\\prime} \/ \\epsilon _K$ is \n{\\em artifically reduced} by the \nenhancement of the $\\Delta I =1\/2$ amplitude, as \nexpressed through $\\omega$. \n\\end{itemize} \nSeveral $\\Delta S=1$ transition operators contribute to \n$\\epsilon ^{\\prime}$ and their renormalization has to be treated \nquite carefully. Two recent detailed analyses yield \n\\cite{BURASPRIME,CIUCHINIPRIME}\n\\begin{equation}} \\def\\eeq{\\end{equation} \n-2.1 \\cdot 10^{-4} \\leq \\frac{\\epsilon ^{\\prime}}{\\epsilon _K} \\leq \n13.3 \\cdot 10^{-4} \n\\label{BURAS1} \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\epsilon ^{\\prime}}{\\epsilon _K} = \n(4.6 \\pm 3.0 \\pm 0.4)\\cdot 10^{-4} \n\\label{CIUCHINI} \n\\eeq \nThese results are quite consistent with each other and show \n\\begin{itemize}\n\\item \nthat the KM ansatz leads to a prediction typically in the range \n{\\em below} $10^{-3}$, \n\\item \nthat the value could happen to be zero or \neven slightly negative and \n\\item \nthat large theoretical uncertainties persist due to cancellations among \nvarious contributions. \n\\end{itemize} \nThis last (unfortunate) point can be illustrated also by comparing \nthese predictions with older ones made before top quarks were \ndiscovered and their mass measured; those old predictions \n\\cite{FRANZINI} are \nvery similar to Eqs.(\\ref{BURAS1},\\ref{CIUCHINI}), once the now \nknown value of $m_t$ has been inserted. \n\nTwo new experiments running now -- NA 48 at CERN and KTEV at \nFNAL -- and one expected to start up soon -- CLOE at DA$\\Phi $NE -- \nexpect to measure $\\epsilon ^{\\prime}\/\\epsilon _K$ with a \nsensitivity of $\\simeq \\pm 2\\cdot 10^{-4}$. Concerning their future \nresults one can distinguish four scenarios: \n\\begin{enumerate}\n\\item \nThe `best' scenario: \n$\\epsilon ^{\\prime}\/\\epsilon _K \\geq 2 \\cdot 10^{-3}$. One would \nthen \nhave established unequivocally direct CP violation of a strength that \nvery probably reflects the intervention of new physics beyond the \nKM ansatz. \n\\item \nThe `tantalizing' scenario: \n$1\\cdot 10^{-3}\\leq \\epsilon ^{\\prime}\/\\epsilon _K \n\\leq 2 \\cdot 10^{-3}$. It would be tempting to interprete this \ndiscovery of direct CP violation as a sign for new physics -- yet \none could not be sure! \n\\item \nThe `conservative' scenario: \n$\\epsilon ^{\\prime}\/\\epsilon _K \\simeq \n{\\rm few}\\cdot10^{-4} > 0$. This strength of direct CP violation could \neasily be accommodated \nwithin the KM ansatz -- yet no further constraint would \nmaterialize. \n\\item \nThe `frustrating' scenario: \n$\\epsilon ^{\\prime}\/\\epsilon _K \\simeq 0$ within errors! \nNo substantial conclusion could be drawn then concerning the \npresence or absence of direct CP violation, and the allowed KM \nparameter space would hardly shrink. \n\\end{enumerate}\n\n\n\\section{`Exotica'}\nIn this section I will discuss important possible manifestations \nof CP and\/or T violation that are exotic only in the sense that \nthey are unobservably small with the KM ansatz. \n\n\\subsection{$K_{3\\mu}$ Decays}\nIn the reaction \n\\begin{equation}} \\def\\eeq{\\end{equation} \nK^+ \\rightarrow \\mu ^+ \\nu \\pi ^0\n\\eeq \none can search for a transverse polarisation of the emerging \nmuons: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nP_{\\perp}^{K^+}(\\mu ) \\equiv \n\\langle \\vec s (\\mu) \\cdot \n(\\vec p (\\mu ) \\times \\vec p (\\pi ^0))\\rangle \n\\eeq\nwhere $\\vec s$ and $\\vec p$ denote spin and momentum, \nrespectively. \nThe quantity $P_{\\perp}(\\mu )$ constitutes a {\\em T-odd} \ncorrelation: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left. \n\\begin{array}{l}\n\\vec p \\Rightarrow - \\vec p \\\\ \n\\vec s \\stackrel{T}{\\Rightarrow} - \\vec s\n\\end{array} \n\\right\\} \\leadsto \nP_{\\perp}(\\mu ) \\stackrel{T}{\\Rightarrow} - P_{\\perp}(\\mu ) \n\\eeq \nOnce a {\\em non-}vanishing value has been observed for a \nparity-odd correlation one has unequivocally found a manifestation \nof parity violation. From $P_{\\perp}^{K^+}(\\mu ) \\neq 0$ one can \ndeduce that T is violated -- yet the argument is more subtle as can \nbe learnt from the following homework problem. \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n{\\em Homework Problem \\#3}: \n\\end{center}\nConsider \n\\begin{equation}} \\def\\eeq{\\end{equation} \nK_L \\rightarrow \\mu ^+ \\nu \\pi ^- \n\\eeq \nDoes $P_{\\perp}^{K_L}(\\mu ) \\equiv \n\\langle \\vec s(\\mu ) \\cdot (\\vec p(\\mu ) \\times \\vec p(\\pi ^-)) \n\\rangle \\neq 0$ necessarily imply that T invariance does not hold in \nthis reaction?\n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n\\end{center}\nData on $P_{\\perp}^{K^+}(\\mu )$ are still consistent with zero \n\\cite{SCHMIDT}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nP_{\\perp}^{K^+}(\\mu ) = ( -1.85 \\pm 3.60) \\cdot 10^{-3} \\; ; \n\\label{YALE}\n\\eeq \nyet being published in 1981 they are ancient by the standards of our \ndisciplin. \n\nOn general grounds one infers that \n\\begin{equation}} \\def\\eeq{\\end{equation} \nP_{\\perp}^{K^+}(\\mu ) \\propto {\\rm Im} \\frac{f_-^*}{f_+} \n\\label{POLTH} \n\\eeq \nholds where $f_-\\, [f_+]$ denotes the chirality changing \n[conserving] decay amplitude. Since $f_-$ practically vanishes \nwithin the Standard Model, one obtains a fortiori \n$P_{\\perp}^{K^+}(\\mu )|_{KM} \\simeq 0$. \n\nYet in the presence of charged Higgs fields one has \n$f_- \\neq 0$. CPT implies that \n$P_{\\perp}^{K^+}(\\mu ) \\neq 0$ represents CP violation as \nwell, and actually one of the {\\em direct} variety. A rather \nmodel independant guestimate on how large such an effect \ncould be is obtained from the present bound on \n$\\epsilon ^{\\prime}\/\\epsilon _K$: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nP_{\\perp}^{K^+}(\\mu ) \\leq 20 \\cdot \n(\\epsilon ^{\\prime}\/\\epsilon _K) \\cdot \\epsilon _K \\leq \n10^{-4} \n\\eeq \nwhere the factor 20 allows for the `accidental' reduction of \n$\\epsilon ^{\\prime}\/\\epsilon _K$ by the $\\Delta I=1\/2$ rule: \n$\\omega \\simeq 1\/20$. This bound is a factor of 100 larger \nthan what one could obtain within KM. It could actually be bigger \nstill since there is a loophole in this generic \nargument: Higgs couplings to leptons could be strongly \nenhanced through a large ratio of vacuum expectation values $v_1$ \nrelative to $v_3$, where $v_1$ controls the couplings to \nup-type quarks and $v_3$ to leptons. \nThen \n\\begin{equation}} \\def\\eeq{\\end{equation} \nP_{\\perp}^{K^+}(\\mu )|_{Higgs} \\leq {\\cal O}(10^{-3}) \n\\eeq \nbecomes conceivable with the Higgs fields as heavy as \n80 - 200 GeV \n\\cite{GARISTO}. Such Higgs exchanges would be quite insignificant \nfor $K_L \\rightarrow \\pi \\pi $! \n\nSince $K_{\\mu 3}$ studies provide such a unique \nwindow onto Higgs dynamics, I find it mandatory to probe for \n$P_{\\perp}(\\mu ) \\neq 0$ in a most determined way. \nIt is gratifying to note that an on-going KEK experiment will be \nsensitive to $P_{\\perp}(\\mu )$ down to the $10^{-3}$ level -- \nyet I strongly feel one should not stop there, but push \nfurther down to the $10^{-4}$ level. \n\\subsection{Electric Dipole Moments}\nConsider a system -- such as an elementary particle or \nan atom -- in a weak external electric field $\\vec E$. The \nenergy shift of this system due to the electric field \ncan then be expressed through an expansion in powers \nof $\\vec E$ \\cite{BERN}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Delta E = \\vec d \\cdot \\vec E + d_{ij}E_i E_j + \n{\\cal O}(|\\vec E|^3) \n\\label{ESHIFTDIP}\n\\eeq \nwhere summation over the indices $i,j$ is understood. The \ncoefficient $\\vec d$ of the term linear in $\\vec E$ is called \nelectric dipole moment or sometimes permanent \nelectric dipole moment (hereafter referred to as EDM) whereas that \nof the quadratic \nterm is often named an {\\em induced} dipole moment. \n\nFor an elementary object one has \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\vec d = d \\vec j \n\\eeq \nwhere $\\vec j$ denotes its total angular momentum since that \nis the only available vector. Under time reversal one finds \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{l}\n\\vec j \\; \\stackrel{T}{\\Rightarrow} \\; - \\vec j \\\\ \n\\vec E \\; \\stackrel{T}{\\Rightarrow} \\vec E \\; . \n\\end{array} \n\\eeq \nTherefore \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm T \\; invariance} \\leadsto d = 0 \\; ; \n\\eeq \ni.e., such an electric dipole moment has to vanish, unless T is \nviolated (and likewise for parity). \n\nThe EDM is at times confused with an induced electric dipole \nmoment objects can possess due to their internal structure. To \nillustrate that consider an atom with two {\\em nearly degenerate} \nstates of opposite parity: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\bf P}|\\pm \\rangle = \\pm |\\pm \\rangle \\; , \\; \n{\\bf H}|\\pm \\rangle = E_{\\pm} |\\pm \\rangle \\; , \\; E_+ < E_- \\; , \\; \n\\frac{E_- - E_+}{E_+} \\ll 1 \n\\eeq \nPlaced in a constant external electric field $\\vec E$ the states \n$|\\pm \\rangle $ will mix to produce new energy eigenstates; \nthose can be found by diagonalising the matrix of the \nHamilton operator: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nH = \n\\left( \n\\begin{array}{ll}\nE_+ & \\Delta \\\\ \n\\Delta & E_- \n\\end{array} \n\\right) \n\\eeq \nwhere $\\Delta = \\vec d_{ind} \\cdot \\vec E$ with \n$\\vec d_{ind}$ being the transition matrix element between \nthe $|+\\rangle$ and $|-\\rangle$ states induced by the electric \nfield. The two new energy eigenvalues are \n\\begin{equation}} \\def\\eeq{\\end{equation} \nE_{1,2} = \\frac{1}{2} (E_+ + E_-) \\pm \n\\sqrt{\\frac{1}{4} (E_+ - E_-)^2 + \\Delta ^2} \n\\eeq \nFor $E_+ \\simeq E_-$ one has \n\\begin{equation}} \\def\\eeq{\\end{equation} \nE_{1,2} \\simeq \\frac{1}{2} (E_+ + E_-) \\pm |\\Delta | \\; ; \n\\eeq \ni.e., the energy shift appears to be linear in $\\vec E$: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Delta E = E_2 - E_1 = 2 |\\vec d_{ind} \\cdot \\vec E| \n\\label{FAKELINEAR} \n\\eeq \nYet with $\\vec E$ being sufficiently small one arrives at \n$4(\\vec d_{ind}\\cdot \\vec E)^2 \\ll (E_+ - E_-)^2$ and therefore \n\\begin{equation}} \\def\\eeq{\\end{equation} \nE_1 \\simeq E_- + \\frac{(\\vec d_{ind}\\cdot \\vec E)^2}{E_- - E_+} \\; , \\; \nE_2 \\simeq E_+ - \\frac{(\\vec d_{ind}\\cdot \\vec E)^2}{E_- - E_+} \\; ; \n\\eeq\ni.e., the induced energy shift is {\\em quadratic} in $\\vec E$ \nrather than \nlinear and therefore does {\\em not} imply T violation! The distinction \nbetween an EDM and an induced electric dipole moment is somewhat \nsubtle -- yet it can be established in an unequivocal way by \nprobing for a linear Stark effect with weak electric fields. A more \ncareful look at Eq.(\\ref{FAKELINEAR}) already indicates that. \nFor the energy shift stated there does not change under \n$\\vec E \\Rightarrow - \\vec E$ as it should for an EDM which also \nviolates parity! \n\nThe data for neutrons read: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nd_n = \\left\\{ \n\\begin{array}{l} \n(-3 \\pm 5)\\cdot 10^{-26} \\; ecm \\; \\; \\; \\; \\; {\\rm ILL} \\\\ \n(2.6 \\pm 4 \\pm 1.6)\\cdot 10^{-26} \\; ecm \\; \\; \\; \\; \\; {\\rm LNPI} \n\\end{array}\n\\right. \n\\label{EDMNEUT}\n\\eeq \nThese numbers and the experiments leading to them are very \nimpressive: \n\\begin{itemize} \n\\item \nOne uses neutrons emanating from a reactor and \nsubsequently cooled down \nto a temperature of order $10^{-7}$ eV. This is comparable to the \nkinetic energy a neutron gains when dropping 1 m in the \nearth's gravitational field. \n\\item \nExtrapolating the ratio between the neutron's radius \n-- $r_N \\sim 10^{-13}$ cm -- with its EDM of no more than \n$10^{-25}$ ecm to the earth's case, one would say that it \ncorresponds to a situation where one has searched for a displacement \nin the earth's mass distribution of order $10^{-12} \\cdot r_{earth} \n\\sim 10^{-3} {\\rm cm} = 10$ microns! \n\\end{itemize} \n\nA truly dramatic increase in sensitivity for the \n{\\em electron's} EDM has \nbeen achieved over the last few years: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nd_e = (-0.3 \\pm 0.8) \\cdot 10^{-26} \\; \\; e \\, cm \n\\label{EDMEL}\n\\eeq \nThis quantity is searched for through measuring electric dipole \nmoments of {\\em atoms}. At first this would seem to be a \nlosing proposition theoretically: for according to Schiff's \ntheorem an atom when placed inside an external electric \nfield gets deformed in such a way that the electron's EDM is \ncompletely shielded; i.e., $d_{atom} = 0$. This theorem \nholds true in the nonrelativistic limit, yet is vitiated by relativistic \neffects. Not surprisingly the latter are particularly large for \nheavy atoms; one would then expect the electron's \nEDM to be only partially shielded: $d_{atom} = S\\cdot d_e$ with \n$S < 1$. Yet amazingly -- and highly welcome of \ncourse -- the electron's EDM can actually get magnified by two to three \norders of magnitude in the atom's electric dipole moment; for \nCaesium one has \\cite{BERN} \n\\begin{equation}} \\def\\eeq{\\end{equation} \nd_{Cs} \\simeq 100 \\cdot d_e \n\\eeq\nThis enhancement factor is the theoretical reason behind the \ngreatly improved sensitivity for $d_e$ as expressed through \nEq.(\\ref{EDMEL}); the other one is experimental, namely the great \nstrides made by laser technology applied to atomic physics. \n\nThe quality of the number in Eq.(\\ref{EDMEL}) can be illustrated \nthrough a comparison with the electron's magnetic moment. \nThe electromagnetic form factor $\\Gamma _{\\mu}(q)$ \nof a particle like the electron evaluated at momentum \ntransfer $q$ contains two tensor terms: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nd_{atom} = \n\\frac{1}{2m_e}\\sigma _{\\mu \\nu} q^{\\nu}\\left[ i F_2(q^2) + \nF_3(q^2) \\gamma _5 \\right] + ... \n\\eeq \nIn the nonrelativistic limit one finds for the EDM: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nd_e = - \\frac{1}{2m_e} F_3(0) \n\\eeq \nOn the other hand one has \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{1}{2}(g-2) = \\frac{1}{e} F_2(0) \n\\eeq \nThe {\\em precision} with which $g-2$ is known for the \nelectron -- $\\delta [(g-2)\/2] \\simeq 10^{-11}$ -- \n(and which represents one of the great success stories of \nfield theory) corresponds to an {\\em uncertainty} in the electron's \n{\\em magnetic} moment \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\delta \\left[ \\frac{1}{2m_e} F_2(0)\\right] \\simeq \n2\\cdot 10^{-22} \\; \\; e\\, cm\n\\eeq \nthat is several orders of magnitude larger than the bound on its \nEDM! \n\nSince the EDM is, as already indicated above, described by a \ndimension-five operator in the Lagrangian \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal L}_{EDM} = - \\frac{i}{2} d \n\\bar \\psi \\sigma _{\\mu \\nu}\\gamma _5 \\psi F^{\\mu \\nu} \n\\eeq\nwith $F^{\\mu \\nu}$ denoting the electromagnetic field strength \ntensor, one can calculate $d$ within a given theory of CP violation \nas a finite quantity. Within the KM ansatz one finds that the \nneutron's EDM is zero for all practical purposes \n\\footnote{I ignore here the Strong CP Problem, which is \ndiscussed in the next section.}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left. d_N\\right| _{KM} < 10^{-30} \\; \\; e\\, cm \n\\eeq\nand likewise for $d_e$. Yet again that is due to very specific features \nof the KM mechanism and the chirality structure of the Standard \nModel. In alternative models -- where CP violation enters \nthrough {\\em right}-handed currents or a non-minimal \nHiggs sector (with or without involving SUSY) -- one finds \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left. d_N\\right| _{New\\; Physics} \\sim \n10^{-27} - 10^{-28} \\; \\; e\\, cm \n\\eeq \nas reasonable benchmark figures. \n\n\\section{The Strong CP Problem}\n\\subsection{The Problem}\n\nIt is often listed among the attractive features of QCD that it `naturally' conserves \nbaryon number, flavour, parity and CP. Actually the last two points \nare not quite true, \nwhich had been overlooked for \nsome time \nalthough it can be seen in different ways \n\\cite{PECCEI}. Consider \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal L}_{eff} = \\sum _q \\bar q \\left( i \\not{D} - m_q\\right) q \n- \\frac{1}{4} G \\cdot G + \n\\frac{\\theta g_S^2}{32 \\pi ^2} \nG\\cdot \\tilde G \n\\label{QCDCP} \n\\eeq \nwhere $D_{\\mu}$, \n$G$ and \n$\\tilde G$ denote the covariant derivative, the \ngluon field strength tensor and its dual, respectively:\n\\begin{equation}} \\def\\eeq{\\end{equation} \nD_{\\mu} = \\partial _{\\mu} + i g_S A^i_{\\mu}t^i \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \nG_{\\mu \\nu} \\equiv G_{\\mu \\nu}^i t^i \\; \\; , \\; \\; \nG_{\\mu \\nu} ^i = \\partial _{\\mu} A^i_{\\nu} - \n\\partial _{\\nu} A^i_{\\mu} + g_S if_{ijk}A^j_{\\mu}A^k_{\\nu} \\; \\; \n, \\; \\; \\tilde G_{\\mu \\nu} \\equiv \\frac{i}{2} \n\\epsilon _{\\mu \\nu \\alpha \\beta} G_{\\alpha \\beta} \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \nG\\cdot G \\equiv G_{\\mu \\nu} G_{\\mu \\nu} \\; \\; , \n\\; \\; \nG\\cdot \\tilde G \\equiv G_{\\mu \\nu} \\tilde G_{\\mu \\nu} \n\\eeq\nIn adding the operator $G \\cdot \\tilde G$ \nto the usual QCD Lagrangian we have followed a general \ntenet of quantum field theory: any Lorentz scalar gauge invariant \noperator of dimension four has to be included in the Lagrangian \nunless there is a specific reason -- in particular a symmetry \nrequirement -- that enforces its absence. For otherwise \nradiative corrections will resurrect such an operator with \na (logarithmically) divergent coefficient! \n\nSuch an operator exists also in an \nabelian gauge theory like QED where the field strength tensor \ntakes on a simpler form: $F_{\\mu \\nu} = \n\\partial _{\\mu} A_{\\nu} - \\partial _{\\nu} A_{\\mu}$. One \nthen finds \n\\begin{equation}} \\def\\eeq{\\end{equation} \nF_{\\mu \\nu} \\tilde F_{\\mu \\nu} = \n\\partial _{\\mu} K^{QED}_{\\mu} \\; , \n\\; \\; K^{QED}_{\\mu} = 2 \\epsilon _{\\mu \\alpha \\beta \\gamma } \nA_{\\alpha} \\partial _{\\beta} A_{\\gamma} \\; ; \n\\eeq \ni.e., this extra term can be reduced to a total \nderivative which is usually dropped without further ado \nas physically irrelevant. \n\nFor nonabelian gauge theories one obtains \n\\begin{equation}} \\def\\eeq{\\end{equation} \nG_{\\mu \\nu}\\tilde G_{\\mu \\nu} = \\partial _{\\mu} K_{\\mu} \\; , \n\\; \\; K_{\\mu} = 2 \\epsilon _{\\mu \\alpha \\beta \\gamma } \n\\left( A_{\\alpha} \\partial _{\\beta} A_{\\gamma} + \n\\frac{2}{3} i g_S A_{\\alpha}A_{\\beta} A_{\\gamma} \n\\right) \\; . \n\\label{KTOPO} \n\\eeq \nThe extra term is still a total derivative and our first reaction \nwould be to just drop it for that very reason. \nAlas this time we would be wrong in doing so! \nLet us recapitulate the usual argument. If a term in the \nLagrangian can be expressed as a total divergence \nlike $\\partial _{\\mu}K_{\\mu}$ than its contribution to the \n{\\em action} which determines the dynamics can be \nexpressed as the integral of the current $K$ over a surface \nat infinity. Yet with physical observables having to \nvanish rapidly at infinity to yield finite values for energy etc., \nsuch integrals are expected to yield zero. The field strength indeed \ngoes to zero at infinity -- but not necessarily the gauge potentials \n$A_{\\mu}$! The field configuration at large \nspace-time distances has to approach that of a \nground state for which $G_{\\mu \\nu} = 0$ holds. Yet the \nlatter property \ndoes not suffice to define the ground state {\\em uniquely}: \nit \nstill allows ground states to differ by pure gauge configurations \nwhich obviously satisfy $G_{\\mu \\nu} = 0$. \nThis is also true for abelian gauge \ntheories, yet remains without dynamical significance. The structure \nof nonabelian gauge theories on the other hand is much more \ncomplex and they possess an infinity of {\\em inequivalent} \nstates defined by $G_{\\mu \\nu}=0$ \n\\cite{REBBI}. Their differences can be expressed \nthrough topological characteristics of their gauge field \nconfigurations. To be more precise: these states can be characterised \nby an integer, the so-called {\\em winding number}; accordingly they \nare denoted by $|n \\rangle$. They are {\\em not} gauge \ninvariant. Not surprisingly \nthen transitions between states \n$|n_1\\rangle$ and $|n_2\\rangle $ with $n_1 \\neq n_2$ \ncan take place. The net change $\\Delta n$ in winding number between \n$t=-\\infty$ and $t=\\infty$ is described by their $K$ charge, \nthe space integral of the zeroth component of the current \n$K_{\\mu}$ defined in Eq.(\\ref{KTOPO}). \nA gauge invariant state is constructed as a \nlinear superposition of the states $|n \\rangle$ labeled by a real \nparameter $\\theta$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|\\theta \\rangle = \\sum _n e^{-i n\\theta }|n\\rangle \\; . \n\\eeq \nOne easily shows that for a \ngauge invariant operator $O_{g.inv.}$ \n$\\matel{\\theta}{O_{g.inv.}}{\\theta ^{\\prime}}=0$ \nholds if $\\theta \\neq \\theta ^{\\prime}$. \nWe thus see that the state space of QCD consists of \n{\\em disjoint} sectors built up from ground states $|\\theta \\rangle $. \n\nFor vacuum-to-vacuum transitions one then finds \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\langle \\theta _+|\\theta _-\\rangle = \n\\sum _{n,m} e^{i\\theta (m - n)}\\langle m_+|n_-\\rangle = \n\\sum _{\\Delta n}e^{i \\Delta n \\theta}\n\\sum _n \\langle (n+\\Delta n)_+| n_- \\rangle \\; , \n\\; \\; \\Delta n = m - n \\; , \n\\eeq \nwhich can be reformulated in the path integral formalism \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\langle \\theta _+|\\theta _-\\rangle = \n\\sum _{\\Delta n} \\sum _{\\rm fields} \ne^{i \\int d^4x {\\cal L}_{eff}} \n\\delta (\\Delta n - \\frac{g_S^2}{16 \\pi ^2} \\int d^4x G \\cdot \n\\tilde G) \\; ; \n\\eeq \ni.e., the $G \\cdot \\tilde G$ term in Eq.(\\ref{QCDCP}) acts as a \nLagrangian multiplier implementing the change in winding \nnumber $\\Delta n$. \n\nThis is easily generalized to any transition amplitude, and \nthe situation can be summarized as follows: \n\\begin{itemize}\n\\item \nThere is an infinity set of {\\em inequivalent} groundstates in QCD \nlabeled by a real parameter $\\theta$. \n\\item \nThe dependance of observables on $\\theta$ can be determined by \nemploying the {\\em effective} Lagrangian of \nEq.(\\ref{QCDCP}). \n\n\\end{itemize} \n\nThe problem with this additional term in the Lagrangian is \nthat $G\\cdot \\tilde G$ -- in contrast to $G \\cdot G$ -- \nviolates both parity and time reversal invariance! \nThis is best seen by expressing $G_{\\mu \\nu}$ and its dual \nthrough the colour electric and colour magnetic fields $\\vec E$ \nand $\\vec B$, respectively: \n\\begin{eqnarray} \nG\\cdot G \\propto |\\vec E|^2 +|\\vec B|^2 \n&\\stackrel{{\\bf P}, {\\bf T}}{\\Longrightarrow}& \n|\\vec E|^2 +|\\vec B|^2 \\\\ \nG \\cdot \\tilde G \\propto 2\\vec E \\cdot \\vec B \n&\\stackrel{{\\bf P}, {\\bf T}}{\\Longrightarrow}& \n- 2\\vec E \\cdot \\vec B\n\\end{eqnarray} \nsince \n\\begin{eqnarray} \n\\vec E \\stackrel{{\\bf P}}{\\Longrightarrow} - \\vec E \n\\; \\; \\; &,& \\; \\; \\; \n\\vec B \\stackrel{{\\bf P}}{\\Longrightarrow} \\vec B \\\\ \n\\vec E \\stackrel{{\\bf T}}{\\Longrightarrow} \\vec E \n\\; \\; \\; &,& \\; \\; \\; \n\\vec B \\stackrel{{\\bf T}}{\\Longrightarrow} - \\vec B \\; ; \n\\end{eqnarray} \ni.e., for $\\theta \\neq 0$ neither parity nor time reversal invariance \nare fully conserved by QCD. This is the {\\em Strong CP Problem}. \n\nThe problem which resides in gluodynamics spreads into \nthe quark sector through the `chiral' anomaly \n\\cite{ABJANOM}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\partial _{\\mu} J_{\\mu}^5 = \n\\partial _{\\mu} \\sum _q \\bar q_L \\gamma _{\\mu}q_L = \n\\frac{g_S^2}{32\\pi ^2} G \\cdot \\tilde G \\neq 0 \\; ; \n\\label{TRIANOM} \n\\eeq \ni.e., the axial current of massless quarks, which is conserved \n{\\em classically}, ceases to be so on the quantum level \n\\footnote{This is why it is called an anomaly.}. This \nchiral anomaly is also called the `triangle' anomaly because it \nis produced by a diagram with a triangular fermion loop. \n\nThere are two further aspects to the anomaly expressed in \nEq.(\\ref{TRIANOM}): \n\\begin{itemize}\n\\item \nThe anomaly actually solves one long standing puzzle of \n{\\em strong} dynamics, the `U(1) Problem': \nIn the limit of massless \n$u$ and $d$ quarks QCD would appear to have a {\\em global} \n$U(2)_L \\times U(2)_R$ invariance. While the vectorial part \n$U(2)_{L+R}$ is a manifest symmetry, the axial part \n$U(2)_{L-R}$ is spontaneously realized leading to the emergence \nof four Goldstone bosons. In the presence of quark masses those \nbosons acquire a mass as well. The pions readily play the part, \nbut the $\\eta$ meson does not \n\\footnote{An analogous discussion can be given with $s$ quarks included. The spontaneous breaking of the global \n$U(3)_{L-R}$ symmetry leads to the existence of nine \nGoldstone bosons, yet the $\\eta ^{\\prime}$ meson is far \ntoo heavy for this role.}! Yet from the anomaly one infers that \ndue to quantum corrections the axial $U(1)_{L-R}$ was never there \nin the first place even for massless quarks: therefore only \nthree Goldstone bosons are predicted, the pions! \n\\item \nOn the other hand the anomaly aggravates the \nStrong CP Problem \nwhen electroweak dynamics are included. For the quarks \nacquire their masses from the Higgs mechanism driving \nthe phase transition \n\\begin{equation}} \\def\\eeq{\\end{equation} \nSU(2)_L \\times U(1) \\leadsto U(1)_{QED} \n\\eeq \nThe resulting quark mass ${\\cal M}_{quark}$ matrix cannot be expected to be diagonal and Hermitian ab initio; \nit will have to be diagonalized through \nchiral rotations of the quark fields: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal M}_{quark}^{diag} = \nU_R^{\\dagger} {\\cal M}_{quark} U_L\n\\eeq \nExactly because of the axial anomaly this induces an additional \nterm in the Lagrangian of the Standard Model: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal L}_{SM,eff} = {\\cal L}_{QCD} + {\\cal L}_{SU(2)_L \\times U(1)} \n+ \\frac{\\bar \\theta g_S^2}{32 \\pi ^2} G \\cdot \\tilde G \n\\eeq \nwhere \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\bar \\theta = \\theta _{QCD} + \\Delta \\theta _{EW}\\; , \n\\; \\; \\Delta \\theta _{EW} = {\\rm arg \\; det} U_R^{\\dagger} U_L \n\\eeq \nSince the electroweak sector has to contain sources of \nCP violation other than $G \\cdot \\tilde G$, the second term \nin $\\bar \\theta$ has no a priori reason to vanish. \n\\end{itemize}\n\n\n\\subsection{The Neutron Electric Dipole Moment}\nSince the gluonic operator $G \\cdot \\tilde G$ does not change \nflavour one suspects right away that its most noticeable impact \nwould be to generate an electric dipole moment (EDM) \nfor neutrons. \nThis is indeed the case, yet making this connection more concrete \nrequires a more sophisticated argument. In the context of the Strong CP Problem one views the neutron \nEDM -- $d_N$ -- as due to the photon coupling to a virtual \nproton or pion in a fluctuation of the neutron: \n$ \nn \\; \\Longrightarrow \\; p^* \\pi ^* \\; \\Longrightarrow \\; n\n$. \nOf the two effective pion nucleon couplings in this one-loop process \none is produced by ordinary strong forces and conserves \nP, T and CP; \nthe other one is induced by $G\\cdot \\tilde G$. The first \nestimate was obtained by Baluni \n\\cite{BALUNI} in a nice paper using bag model \ncomputations of the transition amplitudes between \nthe neutron and its excitations: \n$d_N \\simeq 2.7 \\cdot 10^{-16} \\; \\bar \\theta \\; e cm$. \nIn \\cite{CREWTHER} \nchiral perturbation theory instead was employed: \n$d_N \\simeq 5.2 \\cdot 10^{-16} \\; \\bar \\theta \\; e cm$. \nMore recent estimates yield values in roughly the same range: \n$d_N \\simeq (4\\cdot 10^{-17} \\div 2\\cdot 10^{-15}) \\bar \\theta \n\\; e\\; cm$ \\cite{PECCEI}. Hence \n\\begin{equation}} \\def\\eeq{\\end{equation} \nd_N \\sim {\\cal O}(10^{-16} \\bar \\theta ) \\; \\; e \\; cm \n\\eeq \nand \none infers \n\\begin{equation}} \\def\\eeq{\\end{equation} \nd_N \\leq 1.1 \\cdot 10^{-25} \\; \\; e \\; cm \\; \\; \\; 95\\% \\; C.L. \n\\; \\; \\; \\leadsto \\; \\; \\; \n\\bar \\theta < 10^{-9 \\pm 1} \n\\label{THETABOUND}\n\\eeq \nAlthough $\\theta _{QCD}$ is a QCD parameter it might not necessarily \nbe of order unity; nevertheless its truly tiny size begs for \nan explanation. The only kind of explanation that is usually \naccepted as `natural' in our community is one based on \nsymmetry. Such an explanation has been put forward based \non a so-called Peccei-Quinn symmetry. Yet before we start speculating too wildly, we want \nto see whether there are no more mundane explanations. \n\n\\subsection{Are There Escape Hatches?}\nOne could argue that the Strong CP Problem is fictitious \nusing one of two lines of reasoning: \n\\begin{itemize}\n\\item \nBeing the coefficient of a dimension-four operator \n$\\bar \\theta$ \ncan in general \\footnote{Exceptions will be mentioned below.} \nbe renormalized to any value, including zero. \nThis is technically correct; however $\\theta \\leq {\\cal O}(10^{-9})$ \nis viewed as highly `unnatural': \n\\begin{itemize} \n\\item \nA priori there is no reason why $\\theta _{QCD}$ and \n$\\Delta \\theta _{EW}$ should practically vanish. \n\\item \nEven if $\\theta _{QCD} = 0 = \\Delta \\theta _{EW}$ were set \n{\\em by fiat} quantum corrections to $\\Delta \\theta _{EW}$ are \ntypically much larger than $10^{-9}$ and ultimately actually \ninfinite. \n\\item \nTo expect that $\\theta _{QCD}$ and $\\Delta \\theta _{EW}$ \ncancel as to render $\\bar \\theta$ sufficiently tiny would \nrequire fine tuning of a kind which would have to strike even \na skeptic as unnatural. For $\\theta _{QCD}$ reflects dynamics \nof the strong sector and $\\Delta \\theta _{EW}$ that of the \nelectroweak sector. \n\\item \nIn models where CP symmetry is realized in a spontaneous \nfashion one has $\\theta _{tree} = 0$ and \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\bar \\theta = \\delta \\theta _{ren} \n\\eeq \nturns out to be a finite and calculable quantity that has no \napparent reason to be smaller than, say, $10^{-4}$. \n\\end{itemize}\n\\item \nA more respectable way out is provided by the \nfollowing observation: if one of the quark masses \nvanishes \nthe resulting chiral invariance would remove any \n$\\bar \\theta$ dependance of observables by rotating it \n-- through the anomaly -- into the quark mass matrix with its zero \neigenvalue. However most authors argue quite forcefully that \nneither the up quark nor a fortiori the down quark mass can \nvanish \\cite{PECCEI}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nm_d (1\\; \\,\\mbox{GeV}) > m_u (1\\; {\\,\\mbox{GeV}}) \\simeq 5\\; \\,\\mbox{MeV} \n\\eeq \nwhere the notation shows that one has to use the running mass \nevaluated at a scale of 1 GeV. \n\\end{itemize} \n\n\n\\subsection{Peccei-Quinn Symmetry}\nAs just argued $\\bar \\theta \\leq {\\cal O}(10^{-9})$ \ncould hardly come about accidentally; an organizing principle had \nto arrange various contributions and corrections in such a way as to \nrender the required cancellations. There is the general \nphilosophy that such a principle has to come from an underlying \nsymmetry. We have already sketched such an approach: a global chiral invariance allows to rotate the \ndependance on $\\bar \\theta$ away; we failed however in our \nendeavour because this symmetry is broken by $m_q \\neq 0$. Is \nit possible to invoke some other variant of chiral symmetry for \nthis purpose even if it is spontaneously broken? \nOne \nparticularly intriguing ansatz is to re-interprete a physical \nquantity that is conventionally taken to be a constant as a \ndynamical degree of freedom that relaxes itself to a certain \n(desired) value in response to forces acting upon it. One early \nexample is provided by the original Kaluza-Klein theory \n\\cite{KALUZA} \ninvoking a six-dimensional `space'-time manifold: two compactify \ndynamically and thus lead to the quantization of electric and \nmagnetic charge. \n\nSomething similar has been suggested by \nPeccei and Quinn \\cite{PQ}. They augmented the Standard Model \nby a global $U(1)_{PQ}$ symmetry -- now referred to as \nthe Peccei-Quinn symmetry -- that is axial. The spontaneous breaking of this symmetry gives rise to a Goldstone boson -- named the axion -- \nwith zero mass on the Lagrangian level. Goldstone couplings \nto other fields usually have to be derivative, i.e. involve \n$\\partial _{\\mu}a(x)$, but not $a(x)$ directly. Since \n$U(1)_{PQ}$ is axial it exhibits a triangle anomaly again; \nthis is implemented in the effective Lagrangian by having a \nterm linear in the axion field coupled to $G\\cdot \\tilde G$: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal L}_{eff} = {\\cal L}_{SM} + \\frac{g_S^2\\bar \\theta}{32\\pi ^2} \nG \\cdot \\tilde G + \\frac{g_S^2}{32\\pi ^2}\\frac{\\xi}{\\Lambda _{PQ}} \naG \\cdot \\tilde G - \\frac{1}{2}\\partial _{\\mu} a \\partial _{\\mu} a \n+ {\\cal L}_{int}(\\partial _{\\mu} a,\\psi ) \n\\label{LAGAXION}\n\\eeq\nThe size of the parameters $\\Lambda _{PQ}$ and \n$\\xi$ and the form of \n${\\cal L}_{int}(\\partial _{\\mu} a,\\psi )$ describing the \n(purely derivative) coupling of the axion field to other fields \n$\\psi$ \ndepend on how the Peccei-Quinn symmetry is specifically \nrealized. \n\nThe term $a G \\cdot \\tilde G$ represents an {\\em explicit} \nbreaking of the $U(1)_{PQ}$ symmetry. This gives \nrise to an axion mass. Yet the primary role of \n$a G \\cdot \\tilde G$ is to make the $ G \\cdot \\tilde G$ term \ndisappear from the effective Lagrangian. $U(1)_{PQ}$ \ninvariance being realized {\\em spontaneously} means \nthat $a(x)$ acquires a vacuum expectation value (=VEV) \n$\\langle a\\rangle$; the physical axion excitations \nare then described by the shifted field \n$ \na_{phys}(x) = a(x) - \\langle a \\rangle \n$ \nand one rewrites Eq.(\\ref{LAGAXION}) as follows: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal L}_{eff} = {\\cal L}_{SM} + \\frac{g_S^2}{32\\pi ^2} \n\\bar \\theta G \\cdot \\tilde G - \n\\frac{1}{2}\\partial _{\\mu} a_{phys} \\partial _{\\mu} a_{phys} \n+\\frac{g_S^2}{32\\pi ^2}\\frac{\\xi}{\\Lambda _{PQ}} \na_{phys} G \\cdot \\tilde G \n+ {\\cal L}_{int}(\\partial _{\\mu} a_{phys},\\psi ) \n\\eeq \nwhere now \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\bar \\theta = \\theta _{QCD} + \n{\\rm arg\\, det} U_R^{\\dagger} U_L - \n\\frac{\\langle a\\rangle }{f_a} \\; , \n\\; \\; f_a = \\frac{\\Lambda _{PQ}}{\\xi}\\; ; \n\\eeq \ni.e., the size of the \ncoefficient of the $G \\cdot \\tilde G$ operator is \ndetermined by the VEV of the axion field. \n\nThe term $a G \\cdot \\tilde G$ generates an effective potential \nfor the axion field; its minimum defines the ground state: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\langle \\frac{\\partial V_{eff}}{\\partial a}\\rangle \\equiv \n- \\frac{\\xi }{\\Lambda _{PQ}} \\frac{g_S^2}{32 \\pi ^2} \n\\langle G \\cdot \\tilde G \\rangle = 0 \n\\eeq \nThat means that the term $a G \\cdot \\tilde G$ -- the reincarnation \nof the anomaly -- singles out one of the previously degenerate \n$\\bar \\theta$ states as the true ground state. This is not \nsurprising since $a G \\cdot \\tilde G$ is {\\em not} invariant \nunder $U(1)_{PQ}$. It is a pleasant surprise, though, that \nfor this lowest energy state $G\\cdot \\tilde G$ settles into a \n{\\em vanishing} expectation value thus banning \nthe Strong CP Problem {\\em dynamically}. \n\n\\subsection{The Dawn of Axions -- and Their Dusk?}\nRather than ending here the story contains another twist or \ntwo. The breaking of $U(1)_{PQ}$ gives rise to a Nambu-Goldstone \nboson. Actually the axion is, as already mentioned, a \npseudo-Nambu-Goldstone \nboson; for it acquires a mass due to the anomaly: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nm_a^2 \\sim \\frac{G \\cdot \\tilde G}{\\Lambda _{PQ}^2} \n\\sim {\\cal O}\\left( \\frac{\\Lambda _{QCD}^4}{\\Lambda _{PQ}^2} \n\\right) \n\\eeq \nSince one expects on general grounds \n$\\Lambda _{PQ} >> \\Lambda _{QCD}$ one is dealing with a \nvery light boson. The question is how light would the \naxion be. \n\nThe electroweak scale $v_{EW} = \n\\left( \\sqrt{2}G_F\\right) ^{-\\frac{1}{2}} \\simeq 250$ GeV \nprovides the discriminator for two scenarios: \n\\begin{itemize} \n\\item \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Lambda _{PQ} \\sim v_{EW} \n\\eeq \nIn that case axions can or even should be seen in accelerator based \nexperiment. Such scenarios are referred to as {\\em visible} axions. \n\\item \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Lambda _{PQ} \\gg v_{EW} \n\\eeq \nSuch axions could not be \nfound in accelerator based experiments; therefore they are \ncalled {\\em invisible} scenarios. Yet that does not mean that \nthey necessarily escape detection! For they could be of \ngreat significance for the formation of stars, whole galaxies \nand even the universe. \n\n\\end{itemize} \n\n\\subsubsection{Visible Axions}\nThe simplest scenario involves two $SU(2)_L$ doublet Higgs fields \nthat possess opposite hypercharge \n\\footnote{In the Standard Model the Higgs doublet and its \ncharge conjugate fill this role.}. \nThey also carry a $U(1)$ \ncharge {\\em in addition} to the hypercharge; this second (and \nglobal) $U(1)$ is identified with the PQ symmetry, and the axion is \nits pseudo-Nambu-Goldstone boson. \nThe anomaly induces a mass for the axion: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nm_a \\simeq \\frac{m_{\\pi}F_{\\pi}}{v}N_{fam} \n\\left( x + \\frac{1}{x} \\right) \n\\frac{\\sqrt{m_um_d}}{(m_u + m_d)} \\simeq \n25 \\; N_{fam} \\left( x + \\frac{1}{x} \\right) \\; \\; {\\rm KeV} \n\\eeq \nwhere $N_{fam}$ denotes the number of families. \nSuch an axion is almost certainly much lighter than the \npion. Depending on the axion's mass two cases have to be \ndecided:\n\\begin{itemize}\n\\item \nIf $m_a > 2 m_e$, the axion decays very rapidly into electrons and \npositrons: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\tau (a \\rightarrow e^+ e^-) \\simeq 4 \\cdot 10^{-9}\n\\left( \\frac{1\\; {\\rm MeV}}{m_a} \\right) \n\\frac{x^2 \\; {\\rm or} \\; 1\/x^2}{\\sqrt{1- \\frac{4m_e^2}{m_a^2}}} \\; \\; {\\rm sec} \n\\eeq \n\\item \nIf on the other hand $m_a < 2 m_e$ then the axion decays \nfairly slowly into two photons: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\tau (a \\rightarrow \\gamma \\gamma ) \\sim \n{\\cal O}\\left( \\frac{100 \\; {\\rm KeV}}{m_a} \n\\right) \\; \\; {\\rm sec} \n\\eeq \n\\end{itemize} \nI have presented here a very rough sketch of scenarios \nwith visible axions since we can confidently declare that they \nhave been \nruled out experimentally. They have been looked for in beam \ndump experiments -- without success. Yet the more telling \nblows have come from searches in rare decays:\n\\begin{itemize}\n\\item \nFor long-lived axions -- $m_a < 2 m_e$ -- one expects a \ndominating contribution to $K^+ \\rightarrow \\pi ^+ + \\; nothing$ from \n\\begin{equation}} \\def\\eeq{\\end{equation} \nK^+ \\rightarrow \\pi ^+ + a \n\\label{KTOAXION}\n\\eeq \nwith the axion decayig well outside the detector. For the two-body \nkinematics of Eq.(\\ref{KTOAXION}) one has very tight bounds from \npublished data \\cite{ADLER}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nBR(K^+ \\rightarrow \\pi ^+ X^0) < 5.2 \\cdot 10^{-10} \\; \\; \\; 90\\%\\; C.L.\n\\eeq\nfor $X^0$ being a practically massless and noninteracting particle. \nTheoretically one would expect:\n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left. BR(K^+ \\rightarrow \\pi ^+ a) \\right|_{theor} \\sim \n3\\cdot 10^{-5} \\cdot \\left( x + 1\/x \\right) ^{-2} \n\\label{KTOAXTH}\n\\eeq\nAlthough Eq.(\\ref{KTOAXTH}) does not represent a precise prediction \nthe discrepancy between expectation and observation is \nconclusive. \n\\item \nOne arrives at the same conclusion that \n{\\em long-lived visible} axions \ndo not exist from the absence of {\\em quarkonia} decay \ninto them: neither $J\/\\psi \\rightarrow a\\gamma$ nor \n$\\Upsilon \\rightarrow a \\gamma$ has been seen. \n\\item \nThe analysis is a bit more involved for \n{\\em short-lived} axions -- $m_a > 2 m_e$. Yet again \ntheir absence has been established through a combination \nof experiments. Unsuccessful searches for \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\pi ^+ \\rightarrow a e^+ \\nu \n\\eeq \nfigure prominently in this endeavour. Likewise the \n{\\em absence} of axion driven \n{\\em nuclear de-excitation} has been \nestablished on a level that appears to be conclusive \n\\cite{NUCLEAR}. \n\\subsubsection{Invisible Axions}\nInvisible axion scenarios involve a complex \nscalar field $\\sigma$ that \n(i) is an $SU(2)_L$ {\\em singlet}, \n(ii) \ncarries a PQ charge and \n(iii) \npossesses a huge VEV $\\sim \\Lambda _{PQ} >> v_{ew}$. \n\\end{itemize}\nThe reasons underlying those requirements should be obvious; \nthey can be realized in two distinct (sub-)scenarios: \n\\begin{enumerate}\n\\item \nOnly presumably very heavy new quarks carry a PQ charge. This \nsituation is referred to as {\\em KSVZ} axion \\cite{KSVZAX}. The minimal \nversion can do with a single $SU(2)_L$ Higgs doublet. \n\\item \nAlso the known quarks and leptons carry a PQ charge. Two \n$SU(2)_L$ Higgs doublets are then required in addition to \n$\\sigma$. The fermions do not couple directly to \n$\\sigma$, yet become sensitive to PQ breaking through \nthe Higgs potential. This is referred to as {\\em DFSZ} axion \n\\cite{DFSZAX}. \n\\end{enumerate} \nFrom current algebra one infers for the axion mass in either \ncase \n\\begin{equation}} \\def\\eeq{\\end{equation} \nm_a \\simeq 0.6 \\; {\\rm eV} \\cdot \\frac{10^7 \\; {\\rm GeV}}{f_a} \n\\eeq \nThe most relevant coupling of such axions is to two photons \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal L}( a \\rightarrow \\gamma \\gamma ) = \n- \\tilde g_{a\\gamma \\gamma} \\frac{\\alpha}{\\pi} \n\\frac{a(x)}{f_a} \\vec E \\cdot \\vec B \\; , \n\\eeq \nwhere $\\tilde g_{a\\gamma \\gamma}$ is a model dependant \ncoefficient of order unity. \n\nAxions with such tiny masses have lifetimes easily in excess \nof the age of the universe. Also their couplings to other \nfields are so minute that they would not betray their \npresence -- hence their name {\\em invisible} axions -- \nunder ordinary circumstances! \nYet in astrophysics and cosmology more favourable \nextra-ordinary conditions can arise. \n\nThrough their couplings to electrons axions would provide \na cooling mechanism to {\\em stellar evolution}. Not \nsurprisingly their greatest impact occurs for the lifetimes of \nred giants and the supernovae like SN 1987a. The actual \nbounds depend on the model -- whether it is a \nKSVZ or DFSZ axion -- but relatively mildly only. \nAltogether astrophysics tells us that {\\em if} \naxions exist one has \n\\begin{equation}} \\def\\eeq{\\end{equation} \nm_a < 3 \\cdot 10^{-3} \\; \\; {\\rm eV} \n\\eeq \n\n{\\em Cosmology} on the other hand provides us with a \n{\\em lower} bound through a very intriguing line of \nreasoning. At temperatures $T$ above $\\Lambda _{QCD}$ \nthe axion is massless and all values of \n$\\langle a(x)\\rangle$ are equally likely. For \n$T \\sim 1$ GeV the anomaly induced potential \nturns on driving $\\langle a(x)\\rangle$ \nto a value as to yield $\\bar \\theta =0$ at the new potential \nminimum. The energy stored previously as {\\em latent heat} \nis then released into axions oscillating around its new VEV. \nPrecisely because of the invisible axion's couplings are so \nimmensely suppressed the energy cannot be dissipated into \nother degrees of freedom. We are then dealing with a fluid of \naxions. Their typical momenta is the inverse of their \ncorrelation length which in turn cannot exceed their horizon; \none finds \n\\begin{equation}} \\def\\eeq{\\end{equation} \np_a \\sim \\left( 10^{-6}\\; {\\rm sec}\\right) ^{-1} \n\\sim 10^{-9}\\; {\\rm eV} \n\\eeq \nat $T \\simeq 1$ GeV; i.e., the axions despite their minute \nmass form a very {\\em cold} fluid and actually represent \na candidate \nfor cold dark matter. Their contribution to the density of the \nuniverse relative to its critical value is \\cite{SIKIVIE}\n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Omega _a = \\left( \\frac{0.6 \\cdot 10^{-5}\\; {\\rm eV}}\n{m_a}\\right) ^{\\frac{7}{6}}\\cdot \n\\left( \\frac{200\\; {\\rm MeV}}\n{\\Lambda _{QCD}}\\right) ^{\\frac{3}{4}}\\cdot \n\\left( \\frac{75 \\; {\\rm km\/sec\\, \\cdot Mpc}}\n{H_0}\\right) ^2 \\; ; \n\\eeq \n$H_0$ is the present Hubble expansion rate. For axions not to overclose the universe one thus has to require: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nm_a \\geq 10^{-6} \\; {\\rm eV} \n\\eeq\nor \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Lambda _{PQ} \\leq 10^{12} \\; {\\rm GeV} \n\\eeq \nThis means also that we might be existing in a bath \nof cold axions still making up a significant fraction of the \nmatter of the universe. \n\nIngenious suggestions have been made to search for such \ncosmic background axions. The main handle one has on them \nis their coupling to two photons. They can be detected \nby stimulating the conversion \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm axion} \\; \\; \\stackrel{\\vec B}\n{\\longrightarrow} \\; \\; {\\rm photon} \n\\eeq \nin a strong magnetic field $\\vec B$: \nthe second photon \nwhich is virtual in this process effects the interaction \nwith the inhomogeneous magnetic field in the cavity. The \navailable microwave technology allows an impressive \nexperimental sensitivity. No signal has been found yet, but the search continues and soon should reach a level where one has a \ngood chance to see a signal \\cite{LLNL}. \n\n\\subsection{The Pundits' Judgement}\nThe story of the Strong CP Problem is a particularly intriguing one. \nWe -- like most though not all of our community -- find the \ntheoretical arguments persuasive that there is a problem that \nhas to be resolved. The inquiry has been based on an impressive \narsenal of theoretical reasoning and has inspired fascinating \nexperimental undertakings. \n\nLike many modern novels the problem -- if its is indeed \none -- has not found any resolution. On he other hand it has \nthe potential to lead the charge towards a new paradigm \nin high energy physics. \n\n\n\\section{Summary on the CP Phenomenology with Light Degrees of \nFreedom}\nTo summarize our discussion up to this point: \n\\begin{itemize} \n\\item \nThe following data represent the most sensitive probes: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm BR}(K_L \\rightarrow \\pi ^+ \\pi ^-) = 2.3 \\cdot 10^{-3} \\neq 0 \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{{\\rm BR}(K_L \\rightarrow l^+ \\nu \\pi ^-)}\n{{\\rm BR}(K_L \\rightarrow l^- \\nu \\pi ^+)} \\simeq 1.006 \\neq 1 \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm Re} \\frac{\\epsilon ^{\\prime}}{\\epsilon _K} = \n\\left\\{ \n\\begin{array}{l} \n(2.3 \\pm 0.7) \\cdot 10^{-3} \\; \\; NA\\, 31 \\\\ \n(0.6 \\pm 0.58 \\pm 0.32 \\pm 0.18) \\cdot 10^{-3} \\; \\; \nE\\, 731 \n\\end{array} \n\\right. \n\\eeq\n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm Pol}_{\\perp}^{K^+}(\\mu ) = (-1.85\\pm 3.60) \\cdot 10^{-3} \n\\eeq\n\\begin{equation}} \\def\\eeq{\\end{equation} \nd_N < 12 \\cdot 10^{-26} \\; \\; e\\, cm \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \nd_{Tl} = (1.6 \\pm 5.0) \\cdot 10^{-24} \\; \\; e\\, cm \n\\stackrel{theor.}{\\leadsto} \nd_e = (-2.7 \\pm 8.3) \\cdot 10^{-27} \\; \\; e\\, cm\n\\eeq \n\\item \nAn impressive amount of experimental ingenuity, acumen and \ncommitment \nwent into producing this list. We know that CP violation \nunequivocally exists in nature; it can be \ncharacterized by a {\\em single} non-vanishing quantity: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm Im} M_{12} \\simeq 1.1 \\cdot 10^{-8} \\; eV \\neq 0 \n\\eeq \n\\item \nThe `Superweak Model' states that there just happens to \nexist a $\\Delta S=2$ interaction that is fundamental or effective -- \nwhatever the case may be -- generating Im$M_{12} =$ \nIm$M_{12}|_{exp}$ while $\\epsilon ^{\\prime} =0$. It \nprovides merely \na {\\em classification} for possible dynamical implementations \nrather than such a dynamical implementation itself. \n\n\\item \nThe KM ansatz allows us to incorporate CP violation into the \nStandard Model. Yet it does not regale us with an understanding. \nInstead it relates the origins of CP violation to central \nmysteries of the Standard Model: Why are there families? \nWhy are there three of those? What is underlying \nthe observed pattern in the fermion masses? \n\\item \nStill the KM ansatz succeeds in {\\em accommodating} the data in an \nunforced way: $\\epsilon _K$ emerges to be naturally \nsmall, $\\epsilon ^{\\prime}$ naturally tiny (once the huge \ntop mass is built in), the EDM's for neutrons [electrons] \nnaturally (tiny)$^2$ [(tiny)$^3$] etc. \n\n\\end {itemize} \n\n\n\n\\section{CP Violation in Beauty Decays -- The KM Perspective}\nThe KM predictions for strange decays and electric dipole \nmoments given above will be subjected to sensitive tests in the \nforeseeable future. Yet there is one question that most naturally \nwill come up in this context: \"Where else to look?\" \nI will show below that on very general grounds one has to \nconclude that the decays of beauty hadrons provide by far the \noptimal lab. Yet first I want to make some historical remarks. \n\n\\subsection{The Emerging Beauty of $B$ Hadrons}\n\\subsubsection{Lederman's Paradise Lost -- and Regained!}\nIn 1970 Lederman's group studying the Drell-Yan process \n\\begin{equation}} \\def\\eeq{\\end{equation} \np p \\rightarrow \\mu ^+ \\mu ^- X\n\\eeq \nat Brookhaven observed a shoulder in the di-muon mass \ndistribution around 3 GeV. 1974 saw the `Octobre Revolution' when \nTing et al. and Richter et al. found a narrow resonance -- the $J\/\\psi$ \n-- with a mass of 3.1 GeV at Brookhaven and SLAC, respectively, and \nannounced it. In 1976\/77 Lederman's group working at \nFermilab saw \na structure -- later referred to as the Oops-Leon -- around 6 GeV, \nwhich then disappeared. In 1977 Lederman et al. discovered \nthree resonances in the mass range of 9.5 - 10.3 GeV, the \n$\\Upsilon$, $\\Upsilon ^{\\prime}$ and $\\Upsilon ^{\\prime \\prime}$! \nThat shows that persistence can pay off -- at least sometimes and for \nsome people. \n\n\\subsubsection{Longevity of Beauty}\nThe lifetime of weakly decaying beauty quarks can be related \nto the muon lifetime \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\tau (b) \\sim \\tau (\\mu ) \\left( \\frac{m_{\\mu}}{m_b} \\right) ^5 \n\\frac{1}{9} \\frac{1}{|V(cb)|^2} \\sim \n3 \\cdot 10^{-14} \\left| \\frac{{\\rm sin}\\theta _C}{V(cb)}\\right| ^2 \n\\; {\\rm sec} \n\\eeq \nfor a $b$ quark mass of around 5 GeV; the factor 1\/9 reflects \nthe fact that the virtual $W ^-$ boson in $b$ quark decays can \nmaterialize as a $d \\bar u$ or $s \\bar c$ in three colours each and \nas three lepton pairs. I have ignored phase space corrections here. \nSince the $b$ quark has to decay outside its own family one would \nexpect $|V(cb)| \\sim {\\cal O}({\\rm sin}\\theta _C) = \n|V(us)|$. Yet starting in 1982 data showed a considerably longer \nlifetime \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\tau ({\\rm beauty}) \\sim 10^{-12} \\; {\\rm sec} \n\\eeq \nimplying \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|V(cb)| \\sim {\\cal O}({\\rm sin}^2\\theta _C) \\sim 0.05 \n\\eeq \nThe technology to resolve decay vertices for objects of such \nlifetimes happened to have just been developed -- for charm \nstudies! \n\n\\subsubsection{The Changing Identity of Neutral $B$ Mesons}\nSpeedy $B_d - \\bar B_d$ oscillations were discovered by ARGUS in \n1986: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nx_d \\equiv \\frac{\\Delta m(B_d)}{\\Gamma (B_d)} \\simeq \n{\\cal O}(1)\n\\eeq \nThese oscillations can then be tracked like the decays. This \nobservation was also the first evidence that top quarks had to be \nheavier than orginally thought, namely $m_t \\geq M_W$. \n\n\\subsubsection{Beauty Goes to Charm (almost always)}\nIt was soon found that $b$ quarks exhibit a strong preference \nto decay into charm rather than up quarks \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left| \\frac{V(ub)}{V(cb)} \\right| ^2 \\ll 1 \n\\eeq \nestablishing thus the hierarchy \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|V(ub)|^2 \\ll |V(cb)|^2 \\ll |V(us)|^2 \\ll 1\n\\eeq \n\n\\subsubsection{Resume}\nWe will soon see how all these observations form crucial inputs \nto the general message that big CP asymmetries should emerge in \n$B$ decays and that they (together with interesting rare decays) \nare within reach of experiments. It is for this reason that I strongly \nfeel that the only appropriate name for this quantum number is \n{\\em beauty}! A name like bottom would not do it justice. \n\n\\subsection{The KM Paradigm of Huge CP Asymmetries}\n\\subsubsection{Large Weak Phases!}\nThe Wolfenstein representation expresses the CKM matrix as an \nexpansion: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nV_{CKM} = \n\\left( \n\\begin{array}{ccc} \n1 & {\\cal O}(\\lambda ) & {\\cal O}(\\lambda ^3) \\\\ \n{\\cal O}(\\lambda ) & 1 & {\\cal O}(\\lambda ^2) \\\\ \n{\\cal O}(\\lambda ^3) & {\\cal O}(\\lambda ^2) & 1 \n\\end {array} \n\\right) \n\\; \\; \\; , \\; \\; \\; \\lambda = {\\rm sin}\\theta _C \n\\eeq \nThe crucial element in making this expansion meaningful is the \n`long' lifetime of beauty hadrons of around 1 psec. That number \nhad to change by an order of magnitude -- which is out of the \nquestion -- to invalidate the conclusions given below for the \nsize of the weak phases. \n\nThe unitarity condition yields 6 triangle relations: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(ud)V(us) + &V^*(cd)V(cs) + &V^*(td) V(ts) = \n\\delta _{ds}= 0 \\\\\n{\\cal O}(\\lambda ) & {\\cal O}(\\lambda ) & {\\cal O}(\\lambda ^5) \n\\end{array} \n\\label{TRI1} \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(ud)V(cd) + &V^*(us)V(cs) + &V^*(ub) V(cb) = \n\\delta _{uc}= 0 \\\\\n{\\cal O}(\\lambda ) & {\\cal O}(\\lambda ) & {\\cal O}(\\lambda ^5) \n\\end{array} \n\\label{TRI2} \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(us)V(ub) + &V^*(cs)V(cb) + &V^*(ts) V(tb) = \n\\delta _{sb}= 0 \\\\\n{\\cal O}(\\lambda ^4) & {\\cal O}(\\lambda ^2) & {\\cal O}(\\lambda ^2) \n\\end{array} \n\\label{TRI3} \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(td)V(cd) + &V^*(ts)V(cs) + &V^*(tb) V(cb) = \n\\delta _{ct}=0 \\\\\n{\\cal O}(\\lambda ^4) & {\\cal O}(\\lambda ^2) & {\\cal O}(\\lambda ^2) \n\\end{array} \n\\label{TRI4} \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(ud)V(ub) + &V^*(cd)V(cb) + &V^*(td) V(tb) = \n\\delta _{db}=0 \\\\\n{\\cal O}(\\lambda ^3) & {\\cal O}(\\lambda ^3) & {\\cal O}(\\lambda ^3) \n\\end{array} \n\\label{TRI5} \n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{ccc} \nV^*(td)V(ud) + &V^*(ts)V(us) + &V^*(tb) V(ub) = \n\\delta _{ut}=0 \\\\\n{\\cal O}(\\lambda ^3) & {\\cal O}(\\lambda ^3) & {\\cal O}(\\lambda ^3) \n\\end{array}\n\\label{TRI6} \n\\eeq \nwhere below each product of matrix elements I have noted \ntheir size in powers of $\\lambda $. \n\nWe see that the six triangles fall into three categories: \n\\begin{enumerate}\n\\item \nThe first two triangles are extremely `squashed': two sides are \nof order $\\lambda $, the third one of order $\\lambda ^5$ and their \nratio of order $\\lambda ^4 \\simeq 2.3 \\cdot 10^{-3}$; \nEq.(\\ref{TRI1}) and Eq.(\\ref{TRI2}) control the situation in \nstrange and charm decays; the relevant weak phases there \nare obviously tiny. \n\\item \nThe third and fourth triangles are still rather squashed, yet less so: \ntwo sides are of order $\\lambda ^2$ and the third one of order \n$\\lambda ^4$. \n\\item \nThe last two triangles have sides that are all of the same \norder, namely $\\lambda ^3$. All their angles are therefore \nnaturally large, i.e. $\\sim$ several $\\times$ $10$ degrees! Since to \nleading order in $\\lambda$ one has \n\\begin{equation}} \\def\\eeq{\\end{equation} \nV(ud) \\simeq V(tb) \\; , \\; V(cd) \\simeq - V(us) \\; , \\; \nV(ts) \\simeq - V(cb) \n\\eeq \nwe see that the triangles of Eqs.(\\ref{TRI5}, \\ref{TRI6}) \nactually coincide to that order. \n\\end{enumerate} \nThe sides of this triangle having naturally large angles are \ngiven by $\\lambda \\cdot V(cb)$, $V(ub)$ and \n$V^*(td)$; these are all quantities that control important \naspects of $B$ decays, namely CKM favoured and disfavoured \n$B$ decays and $B_d - \\bar B_d$ oscillations! \n\n\\subsubsection{Different, Yet Coherent Amplitudes!}\n$B^0 - \\bar B^0$ oscillations provide us with two different \namplitudes that by their very nature have to be coherent: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nB^0 \\Rightarrow \\bar B^0 \\rightarrow f \\leftarrow B^0 \n\\eeq \nOn general grounds one expects oscillations to be speedy for \n$B^0 - \\bar B^0$ (like for $K^0 - \\bar K^0$), yet slow for \n$D^0 - \\bar D^0$ \n\\footnote{$T^0 - \\bar T^0$ oscillations cannot \noccur since top quarks decay before they hadronize \n\\cite{RAPALLO}.}. \nExperimentally one indeed finds \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\Delta m(B_d)}{\\Gamma (B_d)} = 0.71 \\pm 0.06 \n\\label{OSCBD}\n\\eeq \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\Delta m(B_s)}{\\Gamma (B_s)} \\geq 10 \n\\label{OSCBS}\n\\eeq\nWhile Eq.(\\ref{OSCBD}) describes an almost optimal \nsituation the overly rapid pace of $B_s - \\bar B_s$ \noscillations will presumably cause experimental \nproblems. \n\nThe conditions are quite favourable also for {\\em direct} \nCP violation to surface. Consider a transition amplitude \n\\begin{equation}} \\def\\eeq{\\end{equation} \nT(B \\rightarrow f) = {\\cal M}_1 + {\\cal M}_2 = \ne^{i\\phi _1}e^{i\\alpha _1}|{\\cal M}_1| + \ne^{i\\phi _2}e^{i\\alpha _2}|{\\cal M}_2| \\; . \n\\eeq \nThe two partial amplitudes ${\\cal M}_1$ and ${\\cal M}_2$ are \ndistinguished by, say, their isospin -- as it was the case for \n$K \\rightarrow (\\pi \\pi )_{I=0,2}$ discussed before; $\\phi _1$, $\\phi _2$ \ndenote the phases in the {\\em weak} couplings and \n$\\alpha _1$, $\\alpha _2$ the phase shifts due to \n{\\em strong} final state interactions. For the CP conjugate reaction \none obtains \n\\begin{equation}} \\def\\eeq{\\end{equation} \nT(\\bar B \\rightarrow \\bar f) = \ne^{-i\\phi _1}e^{i\\alpha _1}|{\\cal M}_1| + \ne^{-i\\phi _2}e^{i\\alpha _2}|{\\cal M}_2| \\; . \n\\eeq \nsince under CP the weak parameters change into their \ncomplex conjugate values whereas the phase shifts remain \nthe same; for the strong forces driving final state \ninteractions conserve CP. The rate difference is then given by \n$$ \n\\Gamma (B \\rightarrow f) - \\Gamma (\\bar B \\rightarrow \\bar f) \\propto \n|T(B \\rightarrow f)|^2 - |T( \\bar B \\rightarrow \\bar f)|^2 = \n$$\n\\begin{equation}} \\def\\eeq{\\end{equation} \n= - 4 {\\rm sin}(\\phi _1 - \\phi _2) \\cdot \n{\\rm sin}(\\alpha _1 - \\alpha _2) \\cdot \n{\\cal M}_1 \\otimes {\\cal M}_2 \n\\eeq \nFor an asymmetry to arise in this way two conditions need to be \nsatisfied simultaneously, namely \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{l} \n\\phi _1 \\neq \\phi _2 \\\\ \n\\alpha _1 \\neq \\alpha _2\n\\end{array}\n\\eeq \nI.e., the two amplitudes ${\\cal M}_1$ and ${\\cal M}_2$ have to \ndiffer both in their weak and strong forces! The first condition \nimplies (within the Standard Model) that the reaction has to \nbe KM suppressed, whereas the second one require the intervention \nof nontrivial final state interactions. \n\nThere is a large number of KM suppressed channels in $B$ \ndecays that are suitable in this context: they receive significant \ncontributions from weak couplings with large phases -- \nlike $V(ub)$ in the Wolfenstein representation -- and there \nis no reason why the phase shifts should be small in general \n(although that could happen in some cases). \n\n\\subsubsection{Resume}\nLet me summarize the discussion just given and anticipate the \nresults to be presented below. \n\\begin{itemize}\n\\item \n{\\em Large} CP asymmetries are {\\em pre}dicted {\\em with} \nconfidence to occur in $B$ decays. If they are not found, there is \nno plausible deniability for the KM ansatz. \n\\item \nSome of these predictions can be made with high \n{\\em parametric} reliability. \n\\item \nNew theoretical technologies have emerged that will allow us to \ntranslate this {\\em parametric} reliability into \n{\\em numerical} precision. \n\\item \nSome of the observables exhibit a high and unambiguous \nsensitivity to the presence of New Physics since we are \ndealing with coherent processes with observables depending \n{\\em linearly} on New Physics amplitudes and where the \nCKM `background' is (or can be brought) \nunder theoretical control. \n\\end{itemize}\n\n\n\\subsection{General Phenomenology}\nDecay rates for CP conjugate channels can be expressed as follows: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array} {l} \n{\\rm rate} (B(t) \\rightarrow f) = e^{-\\Gamma _Bt}G_f(t) \\\\ \n{\\rm rate} (\\bar B(t) \\rightarrow \\bar f) = \ne^{-\\Gamma _Bt}\\bar G_{\\bar f}(t) \n\\end{array} \n\\label{DECGEN}\n\\eeq \nwhere CPT invariance has been invoked to assign the same lifetime \n$\\Gamma _B^{-1}$ to $B$ and $\\bar B$ hadrons. Obviously if \n\\begin{equation}} \\def\\eeq{\\end{equation}\n\\frac{G_f(t)}{\\bar G_{\\bar f}(t)} \\neq 1 \n\\eeq \nis observed, CP violation has been found. Yet one should \nkeep in mind that this can manifest itself in two (or three) \nqualitatively different ways: \n\\begin{enumerate} \n\\item \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{G_f(t)}{\\bar G_{\\bar f}(t)} \\neq 1 \n\\; \\; {\\rm with} \\; \\; \n\\frac{d}{dt}\\frac{G_f(t)}{\\bar G_{\\bar f}(t)} =0 \\; ; \n\\label{DIRECTCP1}\n\\eeq \ni.e., the {\\em asymmetry} is the same for all times of decay. This \nis true for {\\em direct} CP violation; yet, as explained later, it also \nholds for CP violation {\\em in the oscillations}. \n\\item \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{G_f(t)}{\\bar G_{\\bar f}(t)} \\neq 1 \n\\; \\; {\\rm with} \\; \\; \n\\frac{d}{dt}\\frac{G_f(t)}{\\bar G_{\\bar f}(t)} \\neq 0 \\; ; \n\\label{DIRECTCP2}\n\\eeq \nhere the asymmetry varies as a function of the time of decay. \nThis can be referred to as CP violation {\\em involving \noscillations} \\footnote{This nomenclature falls well short of \nShakespearean standards.}. \n\\end{enumerate} \n\nQuantum mechanics with its linear superposition principle makes \nvery specific statements about the possible time dependance of \n$G_f(t)$ and $\\bar G_{\\bar f}(t)$; yet before going into that \nI want to pose another homework problem: \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n{\\em Homework Problem \\# 4}: \n\\end{center}\nConsider the reaction \n\\begin{equation}} \\def\\eeq{\\end{equation} \ne^+ e^- \\rightarrow \\phi \\rightarrow \n(\\pi ^+\\pi ^-)_K (\\pi ^+\\pi ^-)_K \n\\eeq \nIts occurrance requires CP violation. For the {\\em initial} state -- \n$\\phi $ -- carries {\\em even} CP parity whereas the \n{\\em final} state with the two $(\\pi ^+\\pi ^-)$ combinations \nforming a P wave must be CP {\\em odd}: \n$(+1)^2 (-1)^l = -1$! Yet Bose statistics requiring identical \nstates to be in a symmetric configuration would appear to \nveto this reaction; for it places the two $(\\pi ^+\\pi ^-)$ states \ninto a P wave which is antisymmetric. What is the flaw in this \nreasoning? The same puzzle can be formulated in terms of \n\\begin{equation}} \\def\\eeq{\\end{equation} \ne^+ e^- \\rightarrow \\Upsilon (4S) \\rightarrow B_d \\bar B_d \\rightarrow \n(\\psi K_S)_B (\\psi K_S)_B \\; . \n\\eeq \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $\n\\end{center} \nA straightforward application of quantum mechanics yields \nthe general expressions \n\\cite{CARTER,BS,CECILIABOOK}:\n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{l}\nG_f(t) = |T_f|^2 \n\\left[ \n\\left( 1 + \\left| \\frac{q}{p}\\right| ^2|\\bar \\rho _f|^2 \\right) + \n\\left( 1 - \\left| \\frac{q}{p}\\right| ^2|\\bar \\rho _f|^2 \\right) \n{\\rm cos}\\Delta m_Bt \n+ 2 ({\\rm sin}\\Delta m_Bt) {\\rm Im}\\frac{q}{p} \\bar \\rho _f \n\\right] \\\\ \n\\bar G_{\\bar f}(t) = |\\bar T_{\\bar f}|^2 \n\\left[ \n\\left( 1 + \\left| \\frac{p}{q}\\right| ^2|\\rho _{\\bar f}|^2 \\right) + \n\\left( 1 - \\left| \\frac{p}{q}\\right| ^2|\\rho _{\\bar f}|^2 \\right) \n{\\rm cos}\\Delta m_Bt \n+ 2 ({\\rm sin}\\Delta m_Bt) {\\rm Im}\\frac{p}{q} \\rho _{\\bar f} \n\\right] \n\\end{array}\n\\eeq \nThe amplitudes for the instantaneous $\\Delta B=1$ \ntransition into a \nfinal state $f$ are denoted by \n$T_f = T(B \\rightarrow f)$ and $\\bar T_f = T(\\bar B \\rightarrow f)$ and \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\bar \\rho _f = \\frac{\\bar T_f}{T_f} \\; \\; , \n\\rho _{\\bar f} = \\frac{T_{\\bar f}}{\\bar T_{\\bar f}} \\; \\; , \n\\frac{q}{p} = \\sqrt{\\frac{M_{12}^* - \\frac{i}{2} \\Gamma _{12}^*}\n{M_{12} - \\frac{i}{2} \\Gamma _{12}}}\n\\eeq \nStaring at the general expression is not always very illuminating; \nlet us therefore consider three very simplified limiting cases: \n\\begin{itemize}\n\\item\n$\\Delta m_B = 0$, i.e. {\\em no} $B^0- \\bar B^0$ oscillations: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nG_f(t) = 2|T_f|^2 \\; \\; , \\; \\; \n\\bar G_{\\bar f}(t) = 2|\\bar T_{\\bar f}|^2 \n\\leadsto \\frac{\\bar G_{\\bar f}(t)}{G_{ f}(t)} = \n\\left|\n\\frac{\\bar T_{\\bar f}}{T_{ f}}\n\\right|^2 \\; \\; , \\frac{d}{dt}G_f (t) \\equiv 0 \\equiv \n\\frac{d}{dt}\\bar G_{\\bar f} (t) \n\\eeq \nThis is explicitely what was referred to above as {\\em direct} \nCP violation. \n\\item \n$\\Delta m_B \\neq 0$ \nand $f$ a flavour-{\\em specific} final state with {\\em no} \ndirect CP violation; i.e., \n$T_{f} = 0 = \\bar T_{\\bar f}$ and $\\bar T_f = T_{\\bar f}$ \n\\footnote{For a flavour-specific mode one has in general \n$T_f \\cdot T_{\\bar f} =0$; the more intriguing case arises \nwhen one considers a transition that requires oscillations \nto take place.}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array} {c} \nG_f (t) = \\left| \\frac{q}{p}\\right| ^2 |\\bar T_f|^2 \n(1 - {\\rm cos}\\Delta m_Bt )\\; \\; , \\; \\; \n\\bar G_{\\bar f} (t) = \\left| \\frac{p}{q}\\right| ^2 |T_{\\bar f}|^2 \n(1 - {\\rm cos}\\Delta m_Bt) \\\\ \n\\leadsto \n\\frac{\\bar G_{\\bar f}(t)}{G_{ f}(t)} = \\left| \\frac{q}{p}\\right| ^4 \n\\; \\; , \\; \\; \\frac{d}{dt} \\frac{\\bar G_{\\bar f}(t)}{G_{ f}(t)} \\equiv 0 \n\\; \\; , \\; \\; \\frac{d}{dt} \\bar G_{\\bar f}(t) \\neq 0 \\neq \n\\frac{d}{dt} G_ f(t)\n\\end{array} \n\\eeq \nThis constitutes CP violation {\\em in the \noscillations}. For the CP conserving decay into the flavour-specific \nfinal state is used merely to track the flavour identity of the \ndecaying meson. This situation can therefore be denoted also \nin the following way: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{{\\rm Prob}(B^0 \\Rightarrow \\bar B^0; t) - \n{\\rm Prob}(\\bar B^0 \\Rightarrow B^0; t)}\n{{\\rm Prob}(B^0 \\Rightarrow \\bar B^0; t) + \n{\\rm Prob}(\\bar B^0 \\Rightarrow B^0; t)} = \n\\frac{|q\/p|^2 - |p\/q|^2}{|q\/p|^2 + |p\/q|^2} = \n\\frac{1- |p\/q|^4}{1+ |p\/q|^4} \n\\eeq \n\n\\item \n$\\Delta m_B \\neq 0$ with $f$ now being a \nflavour-{\\em non}specific final state -- a final state {\\em common} \nto $B^0$ and $\\bar B^0$ decays -- of a special nature, namely \na CP eigenstate -- $|\\bar f\\rangle = {\\bf CP}|f\\rangle = \n\\pm |f\\rangle $ -- {\\em without} direct CP violation -- \n$|\\bar \\rho _f| = 1 = |\\rho _{\\bar f}| $: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array} {c} \nG_f(t) = 2 |T_f|^2 \n\\left[ 1 + ({\\rm sin}\\Delta m_Bt) \\cdot \n{\\rm Im} \\frac{q}{p} \\bar \\rho _f \n\\right] \\\\ \n\\bar G_f(t) = 2 |T_f|^2 \n\\left[ 1 - ({\\rm sin}\\Delta m_Bt )\\cdot \n{\\rm Im} \\frac{q}{p} \\bar \\rho _f \n\\right] \\\\ \n\\leadsto \n\\frac{d}{dt} \\frac{\\bar G_f(t) }{G_f(t)} \\neq 0\n\\end{array} \n\\eeq \nis the concrete realization of what was called CP violation \n{\\em involving oscillations}. \n\\end{itemize} \n\n\\subsubsection{CP Violation in Oscillations}\nUsing the convention blessed by the PDG \n\\begin{equation}} \\def\\eeq{\\end{equation} \nB = [\\bar b q] \\; \\; , \\; \\; \\bar B = [\\bar q b] \n\\eeq\nwe have \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array} {c} \nT(B \\rightarrow l^- X) = 0 = T(\\bar B \\rightarrow l^+ X) \\\\ \nT_{SL} \\equiv T(B \\rightarrow l^+ X) = T(\\bar B \\rightarrow l^- X) \n\\end{array} \n\\eeq \nwith the last equality enforced by CPT invariance. The \nso-called Kabir test can then be realized as follows: \n$$ \n\\frac\n{{\\rm Prob}(B^0 \\Rightarrow \\bar B^0; t) - \n{\\rm Prob}(\\bar B^0 \\Rightarrow B^0; t)} \n{{\\rm Prob}(B^0 \\Rightarrow \\bar B^0; t) + \n{\\rm Prob}(\\bar B^0 \\Rightarrow B^0; t)} = \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n= \\frac\n{{\\rm Prob}(B^0 \\Rightarrow \\bar B^0 \\rightarrow l^-X; t) - \n{\\rm Prob}(\\bar B^0 \\Rightarrow B^0 \\rightarrow l^+X; t)} \n{{\\rm Prob}(B^0 \\Rightarrow \\bar B^0 \\rightarrow l^-X; t) + \n{\\rm Prob}(\\bar B^0 \\Rightarrow B^0 \\rightarrow l^+X; t)} = \n\\frac{1 - |q\/p|^4}{1+|q\/p|^4} \n\\eeq \nWithout going into details I merely state the results here \n\\cite{CECILIABOOK}: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n1 - \\left| \\frac{q}{p} \\right| \\simeq \n\\frac{1}{2} {\\rm Im}\\left( \\frac{\\Gamma _{12}}\n{M_{12}} \\right) \\sim \n\\left\\{ \n\\begin{array}{ccc} \n10^{-3} & \\; {\\rm for} \\; & B_d=(\\bar bd) \\\\ \n10^{-4} & \\; {\\rm for} \\; & B_s =(\\bar bs) \\\\ \n\\end{array} \n\\right. \n\\eeq\ni.e., \n\\begin{equation}} \\def\\eeq{\\end{equation} \na_{SL} (B^0) \\equiv \\frac{\\Gamma (\\bar B^0(t) \\rightarrow l^+ \\nu X) - \n\\Gamma ( B^0(t) \\rightarrow l^- \\bar \\nu X)}\n{\\Gamma (\\bar B^0(t) \\rightarrow l^+ \\nu X) + \n\\Gamma ( B^0(t) \\rightarrow l^- \\bar \\nu X)} \\simeq \n\\left\\{ \n\\begin{array}{ccc} \n{\\cal O}(10^{-3}) & \\; {\\rm for} \\; & B_d \\\\ \n{\\cal O}(10^{-4}) & \\; {\\rm for} \\; & B_s \\\\ \n\\end{array} \n\\right. \n\\eeq \nThe smallness of the quantity $1-|q\/p|$ is primarily due to \n$|\\Gamma _{12}| \\ll |M_{12}|$ or $\\Delta \\Gamma _B \\ll \n\\Delta m_B$. Within the Standard Model this hierarchy \nis understood (semi-quantitatively at leaast) as due to the \nhierarchy in the GIM factors of the box diagram \nexpressions for $\\Gamma _{12}$ and $M_{12}$, namely \n$m_c^2\/M_W^2 \\ll m_t^2\/M_W^2$. \n\nFor $B_s$ mesons the phase between $\\Gamma _{12}$ and \n$M_{12}$ is further (Cabibbo) suppressed for reasons that \nare peculiar to the KM ansatz: for to leading order in the \nKM parameters quarks of the second and third family only \ncontribute and therefore \narg$(\\Gamma _{12}\/M_{12}) = 0$ to that order. If New \nPhysics intervenes in $B^0 - \\bar B^0$ oscillations, it would \nquite naturally generate a new phase between \n$\\Gamma _{12}$ and $M_{12}$; it could also reduce \n$M_{12}$. Altogether this CP asymmetry could get \nenhanced very considerably: \n\\begin{equation}} \\def\\eeq{\\end{equation} \na_{SL}^{New \\; Physics} (B^0) \\sim 1 \\% \n\\eeq \nTherefore one would be ill-advised to accept the somewhat \npessimistic KM predictions as gospel. \n\nSince this CP asymmetry does not vary with the time of decay, \na signal is not diluted by integrating over all times. It is, \nhowever, essential to `flavour tag' the decaying meson; i.e., \ndetermine whether it was {\\em produced} as a \n$B^0$ or $\\bar B^0$. This can be achieved in several ways \nas discussed later. \n \n\n\n\\subsubsection{Direct CP Violation}\n\nSizeable direct CP asymmetries arise rather naturally in \n$B$ decays. Consider \n\\begin{equation}} \\def\\eeq{\\end{equation} \nb \\rightarrow s \\bar u u \n\\eeq \nThree different processes contribute to it, namely \n\\begin{itemize}\n\\item \nthe tree process \n\\begin{equation}} \\def\\eeq{\\end{equation} \nb \\rightarrow u W^* \\rightarrow u (\\bar u s)_W \\; , \n\\eeq \n\\item \nthe penguin process with an internal top quark which is \npurely local (since $m_t > m_b$) \n\\begin{equation}} \\def\\eeq{\\end{equation} \nb \\rightarrow s g^* \\rightarrow s u \\bar u \\; , \n\\eeq \n\\item \nthe penguin reaction with an internal charm quark. Since \n$m_b > 2m_c + m_s$, this last operator is {\\em not} \nlocal: it contains an absorptive part that amounts to a \nfinal state interaction including a phase shift. \n\\end{itemize}\nOne then arrives at a guestimate \\cite{SONI,CECILIABOOK} \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\Gamma (b \\rightarrow s u \\bar u) - \n\\Gamma (\\bar b \\rightarrow \\bar s u \\bar u)}\n{\\Gamma (b \\rightarrow s u \\bar u) + \n\\Gamma (\\bar b \\rightarrow \\bar s u \\bar u)} \n\\sim {\\cal O}(\\% ) \n\\eeq \nInvoking quark-hadron duality one can expect \n(or at least hope) that this quark level analysis -- rather \nthan being washed out by hadronisation -- yields \nsome average asymmetry or describes the \nasymmetry for some inclusive subclass of nonleptonic \nchannels. I would like to draw the following lessons \nfrom these considerations: \n\\begin{itemize}\n\\item \nAccording to the KM ansatz the natural scale for direct \nCP asymmetries in the decays of beauty hadrons \n(neutral or charged mesons or baryons) is the \n$10^{-2}$ level -- not $10^{-6} \\div 10^{-5}$ as in \nstrange decays! \n\\item \nThe size of the asymmetry in {\\em individual} channels -- \nlike $B\\rightarrow K \\pi$ -- is shaped by the strong final state \ninteractions operating there. Those are likely to differ \nconsiderably from channel to channel, and at present we \nare unable to predict them since they reflect long-distance \ndynamics. \n\\item \nObservation of such an asymmetry (or lack thereof) will not provide \nus with reliable numerical information on the parameters of the \nmicroscopic theory, like the KM ansatz. \n\\item \nNevertheless comprehensive and detailed studies are an \nabsolute must!\n\\end{itemize} \nLater I will describe examples where the relevant long-distance \nparameters -- phase shifts etc. -- can be {\\em measured} \nindependantly. \n\n\\subsubsection{CP Violation Involving Oscillations}\nThe essential feature that a final state in this category has to \nsatisfy is that it can be fed both by $B^0$ and $\\bar B^0$ decays \n\\footnote{Obviously no such common channels can exist for \ncharged mesons or for baryons.}. However for convenience reasons \nI will concentrate on a special subclass of such modes, namely \nwhen the final state is a CP eigenstate. A more comprehensive \ndiscussion can be found in \\cite{CECILIABOOK,BOOK}. \n\nThree qualitative observations have to be made here: \n\\begin{itemize}\n\\item \nSince the final state is shared by $B^0$ and $\\bar B^0$ decays \none cannot even define a CP asymmetry unless one acquires \n{\\em independant} information on the decaying meson: was it \na $B^0$ or $\\bar B^0$ or -- more to the point -- was it originally \nproduced as a $B^0$ or $\\bar B^0$? There are several \nscenarios for achieving such {\\em flavour tagging}: \n\\begin{itemize} \n\\item \nNature could do the trick for us by providing us with \n$B^0$ - $\\bar B^0$ production asymmetries through, say, \nassociated production in hadronic collisions or the use of \npolarized beams in $e^+ e^-$ annihilation. Those production \nasymmetries could be tracked through decays that are \nnecessarily CP conserving -- like $\\bar B_d \\rightarrow \\psi K^- \\pi ^+$ vs. \n$B_d \\rightarrow \\psi K^+ \\pi ^-$. It seems unlikely, though, that such \na scenario could ever be realized with sufficient statistics. \n\n\\item \n{\\em Same Side Tagging}: One undertakes to repeat the success \nof the $D^*$ tag for charm mesons -- $D^{+*} \\rightarrow D^0 \\pi ^+$ \nvs. $D^{-*} \\rightarrow \\bar D^0 \\pi ^-$ -- through finding a conveniently \nplaced nearby resonance -- $B^{-**} \\rightarrow \\bar B_d \\pi ^-$ vs. \n$B^{+**} \\rightarrow B_d \\pi ^+$ -- or through employing correlations \nbetween the beauty mesons and a `nearby' pion (or kaon \nfor $B_s$) as pioneered by the CDF collaboration. This method can be calibrated by \nanalysing how well $B^0 - \\bar B^0$ oscillations are reproduced. \n \n\n\\item \n{\\em Opposite Side Tagging}: With electromagnetic and strong \nforces conserving the beauty quantum number, one can employ \ncharge correlations between the decay products (leptons and kaons) \nof the two beauty hadrons originally produced together. \n\n\\item \nIf the lifetimes of the two mass eigenstates of the neutral $B$ \nmeson differ sufficiently from each other, then one can wait \nfor the short-lived component to fade away relative to the \nlong-lived one and proceed in qualitative analogy to the $K_L$ \ncase. Conceivably this could become feasible -- or even \nessential -- for overly fast oscillating $B_s$ mesons \n\\cite{DUNIETZ}. \n\n\\end{itemize} \nThe degree to which this flavour tagging can be achieved is a crucial \nchallenge each experiment has to face. \n\n\\item \nThe CP asymmetry is largest when the two interfering amplitudes \nare comparable in magnitude. With oscillations having to provide \nthe second amplitude that is absent initially at time of production, \nthe CP asymmetry starts out at zero for decays that occur right after \nproduction and builds up for later decays. The (first) maximum \nof the asymmetry \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left| 1 - \\frac{1 - {\\rm Im}\\frac{q}{p}\\bar \\rho _f \n{\\rm sin}\\Delta m_Bt}\n{1 + {\\rm Im}\\frac{q}{p}\\bar \\rho _f \n{\\rm sin}\\Delta m_Bt}\n\\right| \n\\eeq \nis reached for \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{t}{\\tau _B} = \\frac{\\pi }{2} \\frac{\\Gamma _B}{\\Delta m_B} \n\\simeq 2 \n\\eeq \nin the case of $B_d$ mesons. \n\n\\item \nThe other side of the coin is that very rapid oscillations -- \n$\\Delta m_B \\gg \\Gamma _B$ as is the case for $B_s$ mesons -- \nwill tend to wash out the asymmetry or at least will severely \ntax the experimental resolution. \n\n\\end{itemize} \n\n\\subsubsection{Resume}\nThree classes of quantities each describe the three types of CP \nviolation: \n\\begin{enumerate} \n\\item \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left| \\frac{q}{p} \\right| \\neq 1 \n\\eeq \n\\item \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left| \n\\frac{T(\\bar B \\rightarrow \\bar f)}{T( B \\rightarrow f)} \n\\right| \\neq 1\n\\eeq \n\n\\item \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm Im} \\frac{q}{p} \\frac{T(\\bar B \\rightarrow \\bar f)}{T( B \\rightarrow f)} \n\\neq 0\n\\eeq \n\\end{enumerate}\nThese quantities obviously satisfy one necessary condition \nfor being observables: they are insensitive to the phase convention \nadopted for the anti-state. \n\n\\subsection{Parametric KM Predictions}\nThe triangle defined by \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\lambda V(cb) - V(ub) + V^*(td) = 0 \n\\eeq\nto leading order controls basic features of $B$ transitions. As \ndiscussed before, it has naturally large angles; it usually is called \n{\\em the} KM triangle. Its angles are given by KM matrix elements \nwhich are most concisely expressed in the Wolfenstein \nrepresentation: \n\\begin{equation}} \\def\\eeq{\\end{equation} \ne^{i\\phi _1} = - \\frac{V(td)}{|V(td)|} \\; \\; , \\; \\; \ne^{i\\phi _2} = \\frac{V^*(td)}{|V(td)|} \\frac{|V(ub)|}{V(ub)}\\; \\; , \\; \\; \ne^{i\\phi _3} = \\frac{V(ub)}{|V(ub)|} \n\\eeq \nThe various CP asymmetries in beauty decays are expressed in \nterms of these three angles. I will describe `typical' examples now. \n\n\\subsubsection{Angle $\\phi _1$}\nConsider \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\bar B_d \\rightarrow \\psi K_S \\leftarrow B_d\n\\eeq \nwhere the final state is an almost pure odd CP eigenstate. On the \nquark level one has two different reactions, namely one \ndescribing the direct decay process \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\bar B_d = [b \\bar d] \\rightarrow [c\\bar c] [s \\bar d] \n\\label{BDTREE}\n\\eeq \nand the other one involving a $B_d - \\bar B_d$ oscillation: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\bar B_d = [b \\bar d] \\Rightarrow B_d = [\\bar b d] \n\\rightarrow [c \\bar c] [ \\bar sd] \n\\label{BDOSC} \n\\eeq \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $ \\\\ \n{\\em Homework Problem \\# 5}: \n\\end{center}\nHow can the $[s \\bar d]$ combination in \nEq.(\\ref{BDTREE}) interfere with \n$[\\bar sd]$ in Eq.(\\ref{BDOSC})? \n\\begin{center} \n$\\spadesuit \\; \\; \\; \\spadesuit \\; \\; \\; \\spadesuit $\n\\end{center} \nSince the final state in $B\/\\bar B \\rightarrow \\psi K_S$ can carry \nisospin 1\/2 only, we have for the {\\em direct} \ntransition amplitudes: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array}{c}\nT(\\bar B_d \\rightarrow \\psi K_S) = V(cb)V^*(cs) \ne^{i\\alpha _{1\/2}} |{\\cal M}_{1\/2}| \\\\ \nT(B_d \\rightarrow \\psi K_S) = V^*(cb)V(cs) \ne^{i\\alpha _{1\/2}} |{\\cal M}_{1\/2}| \n\\end{array}\n\\eeq \nand thus \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\bar \\rho _{\\psi K_S} = \\frac{V(cb)V^*(cs)}{V^*(cb)V(cs)} \n\\eeq \nfrom which the hadronic quantities, namely the \nphase shift $\\alpha _{1\/2}$ and the hadronic matrix \nelement $|{\\cal M}_{1\/2}|$ -- both of which {\\em cannot} \nbe calculated in a reliable manner -- have dropped out. \nTherefore \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left| \\bar \\rho _{\\psi K_S} \\right| = \n\\left| \n\\frac{T(\\bar B_d \\rightarrow \\psi K_S}{T(B_d \\rightarrow \\psi K_S}\n\\right| = 1 \\; ; \n\\eeq \ni.e., there can be {\\em no direct} CP violation in this channel. \n\nSince $|\\Gamma _{12}| \\ll |M_{12}|$ one has \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{q}{p} \\simeq \\sqrt{\\frac{M^*_{12}}{M_{12}}} = \n\\frac{M^*_{12}}{|M_{12}|} \\simeq \n\\frac{V^*(tb)V(td)}{V(tb)V^*(td)}\n\\eeq \nwhich is a pure phase. Altogether one obtains \n\\footnote{The next-to-last (approximate) equality \nin Eq.(\\ref{SIN2PHI1}) holds in the Wolfenstein \nrepresentation, although the overall result is general.}\n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm Im} \\frac{q}{p} \\bar \\rho _{\\psi K_S} = \n{\\rm Im} \\left( \n\\frac{V^*(tb)V(td)} {V(tb)V^*(td)} \n\\frac{V(cb)V^*(cs)} {V^*(cb)V(cs)} \n \\right) \n\\simeq {\\rm Im} \\frac{V^2(td)}{|V(td)|^2} = \n-{\\rm sin}2\\phi _1 \n\\label{SIN2PHI1} \n\\eeq \nThat means \nthat to a very good approximation the observable \nIm $\\frac{q}{p} \\bar \\rho _{\\psi K_S}$, which is the amplitude \nof the oscillating CP asymmetry, is in general given by \n{\\em microscopic} parameters of the theory; within \nthe KM ansatz they combine to yield the angle $\\phi _1$ \n\\cite{BS}. \n\nSeveral other channels are predicted to exhibit a CP asymmetry \nexpressed by sin$2\\phi _1$, like $B_d \\rightarrow \\psi K_L$ \n\\footnote{Keep in mind that \nIm$\\frac{q}{p}\\bar \\rho _{\\psi K_L} =- \n{\\rm Im}\\frac{q}{p}\\bar \\rho _{\\psi K_S}$ holds because \n$K_L$ is mainly CP odd and $K_S$ mainly CP even.}, \n$B_d \\rightarrow D \\bar D$ etc. \n\n\\subsubsection{Angle $\\phi _2$}\nThe situation is not quite as clean for the angle $\\phi _2$. \nThe asymmetry in $\\bar B_d \\rightarrow \\pi ^+ \\pi ^-$ vs. \n$B_d \\rightarrow \\pi ^+ \\pi ^-$ is certainly sensitive to $\\phi _2$, \nyet there are two complications: \n\\begin{itemize}\n\\item \nThe final state is described by a superposition of {\\em two} \ndifferent isospin states, namely $I = 0$ and $2$. The \nspectator process contributes to both of them. \n\\item \nThe Cabibbo suppressed Penguin operator \n\\begin{equation}} \\def\\eeq{\\end{equation} \nb \\rightarrow d g^* \\rightarrow d u \\bar u \n\\eeq \nwill also contribute, albeit only to the $I=0$ amplitude. \n\\end{itemize} \n\nThe direct transition amplitudes are then expressed as follows: \n$$ \nT(\\bar B_d \\rightarrow \\pi ^+ \\pi ^-) = \nV(ub)V^*(ud)e^{i \\alpha _2}|{\\cal M}_2^{spect}| + \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n+ e^{i \\alpha _0}\\left( V(ub)V^*(ud) |{\\cal M}_0^{spect}| + \nV(tb)V^*(td) |{\\cal M}_0^{Peng}| \n\\right) \n\\eeq \n$$ \nT(B_d \\rightarrow \\pi ^+ \\pi ^-) = \nV^*(ub)V(ud)e^{i \\alpha _2}|{\\cal M}_2^{spect}| + \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n+ e^{i \\alpha _0}\\left( V^*(ub)V(ud) |{\\cal M}_0^{spect}| + \nV^*(tb)V(td) |{\\cal M}_0^{Peng}| \n\\right) \n\\eeq \nwhere the phase shifts for the $I=0,2$ states have been factored \noff. \n\n{\\em If} there were no Penguin contributions, we would have \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm Im} \\frac{q}{p} \\bar \\rho _{\\pi \\pi} = \n{\\rm Im} \\frac{V(td)V^*(tb)V(ub)V^*(ud)}\n{V^*(td)V(tb)V^*(ub)V(ud)} = \n- {\\rm sin}2\\phi _2 \n\\eeq \nwithout direct CP violation -- $|\\bar \\rho _{\\pi \\pi }| =1$ -- \nsince the two isospin amplitudes still contain the same weak \nparameters. The Penguin contribution changes the picture in \ntwo basic ways: \n\\begin{enumerate}\n\\item \nThe CP asymmetry no longer depends on $\\phi _2$ alone: \n$$ \n{\\rm Im} \\frac{q}{p} \\bar \\rho _{\\pi \\pi } \\simeq \n- {\\rm sin} 2\\phi _2 + \\left| \\frac{V(td)}{V(ub)} \\right| \n\\left[ {\\rm Im}\\left( e^{-i\\phi _2}\n\\frac{{\\cal M}^{Peng}}{{\\cal M}^{spect}}\\right) \n- {\\rm Im}\\left( e^{-3i\\phi _2}\n\\frac{{\\cal M}^{Peng}}{{\\cal M}^{spect}}\\right) \n\\right] + \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n+ {\\cal O}(|{\\cal M}^{Peng}|^2\/|{\\cal M}^{spect}|^2) \n\\label{PENGPOLL} \n\\eeq \nwhere \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal M}^{spect} = e^{i \\alpha _0}|{\\cal M}^{spect}_0| + \ne^{i \\alpha _2}|{\\cal M}^{spect}_2| \\; \\; , \\; \\; \n{\\cal M}^{Peng} = e^{i \\alpha _0}|{\\cal M}^{Peng}_0| \n\\eeq \n\\item \nA direct CP asymmetry emerges:\n\\begin{equation}} \\def\\eeq{\\end{equation} \n|\\bar \\rho _{\\pi \\pi}| \\neq 1 \n\\eeq \n\\end{enumerate}\nSince we are dealing with a Cabibbo suppressed Penguin operator, \nwe expect that its contribution is reduced relative to the \nspectator term: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left| \\frac{{\\cal M}^{Peng}}{{\\cal M}^{spect}} \n\\right| \n< 1 \\; , \n\\eeq \nwhich was already used in Eq.(\\ref{PENGPOLL}). \nUnfortunately this reduction might \nnot be very large. This concern is based on the observation \nthat the branching ratio for $\\bar B_d \\rightarrow K^- \\pi ^+$ appears to \nbe somewhat larger than for $\\bar B_d \\rightarrow \\pi ^+ \\pi ^-$ \nimplying that the Cabibbo favoured Penguin amplitude \nis at least not smaller than the spectator amplitude. \n\nVarious strategies have been suggested to unfold the \nPenguin contribution through a combination of additional \nor other \nmeasurements (of other $B \\rightarrow \\pi \\pi $ channels \nor of $B\\rightarrow \\pi \\rho$, $B \\rightarrow K\\pi$ etc.) and supplemented by \ntheoretical considerations like $SU(3)_{Fl}$ symmetry \n\\cite{PENGTRAP}. \nI am actually hopeful that the multitude of exclusive \nnonleptonic decays (which is the other side of the \ncoin of small branching ratios!) can be harnessed to \nextract a wealth of information on the strong dynamics that \nin turn will enable us to extract \nsin$2\\phi _2$ with decent accuracy. \n\n\\subsubsection{The $\\phi _3$ Saga}\nOf course it is important to determine $\\phi _3$ as accurately \nas possible. This will not be easy, and one better keep \na proper perspective. I am going to tell this saga now in \ntwo installments. \n\n{\\bf (I)} {\\em CP asymmetries involving $B_s - \\bar B_s$ \nOscillations}: In principle one can extract $\\phi _3$ from \nKM suppressed $B_s$ decays like one does $\\phi _2$ from \n$B_d$ decays, namely by measuring and analyzing the difference \nbetween the rates for, say, $\\bar B_s(t) \\rightarrow K_S \\rho ^0$ and \n$B_s(t) \\rightarrow K_S \\rho ^0$: \nIm$\\frac{q}{p}\\bar \\rho _{K_S\\rho ^0} \\sim {\\rm sin}2\\phi _3$. \nOne has to face the same complication, namely that in \naddition to the spectator term a \n(Cabibbo suppressed) Penguin amplitude contributes to $\\bar \\rho \n_{K_S\\rho ^0}$ with different weak parameters. Yet the situation \nis much more challenging due to the rapid \npace of the $B_s - \\bar B_s$ oscillations. \n\n\\noindent A more promising way might be to compare the rates for \n$\\bar B_s (t) \\rightarrow D_s^+ K^-$ with $B_s (t) \\rightarrow D_s^- K^+$ as a function \nof the time of decay $t$ since there is no Penguin contribution. \nThe asymmetry depends on sin$\\phi _3$ rather than \nsin$2\\phi _3$ \n\\footnote{Both $D_s^+K^-$ and $D_s^- K^+$ are final states common to \n$B_s$ and $\\bar B_s$ decays although they are not CP \neigenstates.}. \n\n{\\bf (II)} {\\em Direct CP Asymmetries}: The largish direct CP \nasymmetries sketched above for $B \\rightarrow K \\pi$ depend on \nsin$\\phi _3$ -- and on the phase shifts which in general are neither \nknown nor calculable. Yet in some cases they can be determined \nexperimentally -- as first described for \n$B^{\\pm} \\rightarrow D_{neutral} K^{\\pm}$ \n\\cite{WYLER}. There are \n{\\em four independant} rates that can be measured, namely \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Gamma (B^- \\rightarrow D^0 K^-) \\; , \\; \n\\Gamma (B^- \\rightarrow \\bar D^0 K^-) \\; , \\; \n\\Gamma (B^- \\rightarrow D_{\\pm} K^-) \\; , \\; \n\\Gamma (B^+ \\rightarrow D_{\\pm} K^+) \n\\eeq \nThe {\\em flavour eigenstates} $D^0$ and $\\bar D^0$ are defined \nthrough flavour specific modes, namely \n$D^0 \\rightarrow l^+ X$ and $\\bar D^0 \\rightarrow l^- X$, respectively; \n$D_{\\pm}$ denote the even\/odd CP eigenstates \n$D_{\\pm} = (D^0 \\pm \\bar D^0)\/\\sqrt{2}$ defined by \n$D_+ \\rightarrow K^+K^-, \\, \\pi ^+ \\pi ^-,$ etc., \n$D_- \\rightarrow K_S\\pi ^0, \\, K_S \\eta ,$ etc. \\cite{PAISSB}. \n\nFrom these four observables one can (up to a binary ambiguity) \nextract the four basic quantities, namely the moduli of the two \nindependant amplitudes ($|T(B^- \\rightarrow D^0 K^-)|$, \n$|T(B^- \\rightarrow \\bar D^0 K^-)|$), their strong phaseshift -- and \nsin$\\phi _3$, the goal of the enterprise! \n\n\\subsubsection{A Zero-Background Search for New \nPhysics: $B_s \\rightarrow \\psi \\phi , \\, D_s^+ D_s^-$}\nThe two angles $\\phi _1$ and $\\phi _2$ will be measured in \nthe next several years with decent or even good accuracy. I \nfind it unlikely that any of the direct measurements \nof $\\phi _3$ sketched above will yield a more precise \nvalue than inferred from simple trigonometry: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\phi _3 = 180^o - \\phi _1 - \\phi _2 \n\\label{180}\n\\eeq \nEq.(\\ref{180}) holds \nwithin the KM ansatz; of course the real goal is to uncover \nthe intervention of New Physics in $B_s$ transitions. It then \nmakes eminent sense to search for it in a reaction where \nKnown Physics predicts a practically zero result. \n$B_s \\rightarrow \\psi \\phi , \\, \\psi \\eta , \\, D_s \\bar D_s$ fit this bill \n\\cite{BS}: \nto leading order in the KM parameters the CP asymmetry has \nto vanish since on that level quarks of the second and third \nfamily only participate in $B_s - \\bar B_s$ oscillations -- \n$[s \\bar b] \\Rightarrow t^* \\bar t^* \\Rightarrow [b \\bar s]$ -- and \nin these direct decays -- $[b \\bar s] \\rightarrow c \\bar c s \\bar s$. Any \nCP asymmetry is therefore Cabibbo suppressed, i.e. $\\leq 4$\\% . \nMore specifically \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left. {\\rm Im} \\frac{q}{p}\\bar \\rho _{B_s \\rightarrow \\psi \\eta , \n\\psi \\phi , D_s \\bar D_s} \\right| _{KM} \\sim 2\\% \n\\eeq \nYet New Physics has a good chance to contribute to \n$B_s - \\bar B_s$ oscillations; if so, there is no reason for \nit to conserve CP and asymmetries can emerge that are easily \nwell in excess of 2\\% . New Physics scenarios with non-minimal \nSUSY or flavour-changing neutral currents could actually \nyield asymmetries of $\\sim 10 \\div 30 \\%$ \n\\cite{GABB} -- completely \nbeyond the KM reach! \n\n\n\\subsubsection{The HERA-B Menu}\nQuite often people in the US tend to believe that a restaurant that \npresents them with a long menu must be a very good one. The \nreal experts -- like the French and Italians -- of course know \nbetter: it is the hallmark of a top cuisine to concentrate on a \nfew very special dishes and prepare them in a spectacular fashion \nrather than spread one's capabilities too thinly. That is exactly the advice \nI would like to give the HERA-B collaboration, namely to focus \non a first class menu consisting of three main dishes and one side \ndish, namely \n\\begin{enumerate} \n\\item \nmeasure $\\Delta m(B_s)$ which within the Standard Model \nallows to extract $|V(td)|$ through \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\Delta m(B_d)}{\\Delta m(B_s)} \\simeq \n\\frac{Bf_{B_d}^2} {Bf_{B_s}^2} \\left| \\frac{V(td)}{V(ts)} \n\\right| ^2 \\; ; \n\\eeq \n\n\\item \ndetermine the rates for $\\bar B_d \\rightarrow \\psi K_S$ and \n$B_d \\rightarrow \\psi K_S$ to obtain the value of \nsin$2\\phi _1$; \n\n\\item \ncompare $\\bar B_s \\rightarrow \\psi \\phi , \\, D_s \\bar D_s$ with \n$B_s \\rightarrow \\psi \\phi , \\, D_s \\bar D_s$ \nas a clean search for New Physics and \n\n\\item \nas a side dish: measure the $B_s$ lifetime separately in \n$B_s \\rightarrow l \\nu D_s^{(*)}$ and $B_s \\rightarrow \\psi \\phi , \\, D_s \\bar D_s$ where \nthe former yields the algebraic average of the $B_{s, short}$ \nand $B_{s,long}$ lifetimes and the latter the $B_{s, short}$ \nlifetime. One predicts for them \\cite{URALTSEV}:\n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\tau (B_s \\rightarrow l \\nu D_s^{(*)}) - \\tau (B_s \\rightarrow \\psi \\phi \n,\\, D_s \\bar D_s)}\n{\\tau (B_s \\rightarrow l \\nu D_s^{(*)})} \\simeq 0.1 \\cdot \n\\left( \\frac{f_{B_s}}{200\\; {\\rm MeV}} \\right) ^2\n\\eeq \n\n\\end{enumerate} \nIf the HERA-B chefs succeed in preparing one of these \nmain dishes, then they have achieved three star status!\n\n\n\\subsection{Theoretical Technologies in Heavy Flavour Decays}\nOne other intriguing and gratifying aspect of heavy flavour \ndecays has become understood just over the last several \nyears, namely that the decays in particular of beauty hadrons \ncan be treated with a reliability and accuracy that before would \nhave seemed to be unattainable. These new theoretical \ntechnologies can be referred to as {\\em Heavy Quark Theory} \nwhich combines two basic elements, namely an \nasymptotic symmetry principle on one hand and a dynamical \ntreatment on the other, which tells us how the asymptotic limit \nis approached. The symmetry principle is Heavy Quark \nSymmetry stating that all sufficiently heavy quarks behave \nidentically under the strong interactions. The dynamical treatment \nis provided by $1\/m_Q$ expansions allowing us to express \nobservable transition rates through a series in inverse powers \nof the heavy quark mass. This situation is qualitatively similar \nto chiral considerations which start from the limit of chiral \ninvariance and describe the deviations from it through \nchiral perturbation theory. In both cases one has succeeded in \ndescribing nonperturbative dynamics in special cases. \n\nThe lessons we have learnt can be summarized as follows \n\\cite{HQEREV}: we have \n\\begin{itemize}\n\\item \nidentified the sources of the non-perturbative corrections; \n\\item \nfound them to be smaller than they could have been; \n\\item \nsucceeded in relating the basic quantities of the Heavy Quark \nTheory -- KM paramters, masses and kinetic energy of heavy quarks, \netc. -- to various a priori independant observables with a fair \namount of redundancy; \n\\item \ndeveloped a better understanding of incorporating perturbative \nand nonperturbative corrections without double-counting. \n\\end{itemize} \nIt has been shown that the heavy quark expansion has to be \nformulated in terms of short distance masses rather than \npole masses. One finds \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\begin{array} {c} \nm_b - m_c = 3.50 \\pm 0.04 \\; {\\rm GeV} \\\\ \nm_b(1 \\; {\\rm GeV}) = 4.64 \\pm 0.05 \\; {\\rm GeV} \n\\end{array} \n\\eeq \nThis information is then used to extract $|V(cb)|$ from the \nobserved semileptonic $B$ width with the result \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|V(cb)|_{incl} = 0.0412 \\cdot \n\\sqrt{\\frac{{\\rm BR}(B\\rightarrow l X)}{0.105}} \\cdot \n\\sqrt{\\frac{1.6 \\; {\\rm psec}}{\\tau _B}}\\cdot \n\\left( 1 \\pm 0.05 |_{theor} \\right) \n\\label{VCBIN}\n\\eeq \nAlternatively one can analyze the exclusive mode \n$B\\rightarrow l \\nu D^*$ and extrapolate to the kinematical \npoint of zero recoil to obtain \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|F_{D^*}(0) V(cb)| = 0.0339 \\pm 0.0014 \n\\eeq \nFrom Heavy Quark Theory one infers \\cite{HQEREV} \n\\begin{equation}} \\def\\eeq{\\end{equation} \nF_{D^*}(0) = 0.91 \\pm 0.06 \n\\eeq \nto arrive at \n\\begin{equation}} \\def\\eeq{\\end{equation} \n|V(cb)|_{excl} = 0.0377 \\pm 0.0016|_{exp} \\pm 0.002 |_{theor}\n\\label{VCBEX}\n\\eeq \nThe two determinations in Eqs.(\\ref{VCBIN}) and (\\ref{VCBEX}) \nare systematically very different both in their experimental and \ntheoretical aspects. Nevertheless they are quite consistent with \neach other with the experimental and theoretical uncertainties \nbeing very similar. A few years ago it would have seemed \nquite preposterous to claim such small theoretical uncertainties! \nI am actually confident that those can be reduced from the \npresent 5\\% level down to the 2\\% level in the foreseeable future. \n\n$|V(ub)|$ (or $|V(ub)\/V(cb)|$) is not known with an even remotely \nsimilar accuracy, and so far one has relied on models rather \nthan QCD proper to extract it from data. Yet we can be confident \nthat over the next ten years $|V(ub)|$ will be determined with a \ntheoretical uncertainty below 10\\% . It will be important to obtain \nit from systematically different semileptonic distributions and \nprocesses; Heavy Quark Theory provides us with the \nindispensable tools for combining the various analyses in a \ncoherent fashion. \n\nThis theoretical progress can embolden us to hope that in the end \neven $|V(td)|$ can be determined with good accuracy -- say \n$\\sim 10 \\div 15\\%$ -- from \n$\\Gamma (K^+ \\rightarrow \\pi ^+ \\nu \\bar \\nu )$, \n$\\Delta m(B_s)$ vs. $\\Delta m(B_d)$ or \n$\\Gamma (B \\rightarrow \\gamma \\rho \/\\omega )$ vs. \n$\\Gamma (B \\rightarrow \\gamma K^* )$ etc. \n\n\\subsection{KM Trigonometry}\nOne side of the triangle is exactly known since the base line can be \nnormalized to unity without affecting the angles: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n1 - \\frac{V(ub)}{\\lambda V(cb)} + \n\\frac{V^*(td)}{\\lambda V(cb)} = 0 \n\\eeq \nThe second side is known to some degree from semileptonic $B$ \ndecays: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\left| \n\\frac{V(ub)}{V(cb)} \n\\right| \n\\simeq 0.08 \\pm 0.03\n\\eeq\nwhere the quoted uncertainty is mainly theoretical and amounts \nto little more than a guestimate. In the Wolfenstein representation \nthis reads as \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\sqrt{\\rho ^2 + \\eta ^2} \\simeq 0.38 \\pm 0.11\n\\eeq \nThe area cannot vanish since $\\epsilon _K \\neq 0$. Yet at present \nnot much more can be said for certain. \n\nIn principle one would have enough observables -- namely \n$\\epsilon _K$ and $\\Delta m(B_d)$ in addition to \n$|V(ub)\/V(cb)|$ -- to determine the two KM parameters \n$\\rho$ and $\\eta$ in a {\\em redundant} way. In practise, though, \nthere are two further unknowns, namely the size of the \n$\\Delta S=2$ and $\\Delta B=2$ matrix elements, as expressed through \n$B_K$ and $B_B f_B^2$. For $m_t$ sufficiently large $\\epsilon _K$ \nis dominated by the top contribution: \n$d \\bar s \\Rightarrow t^* \\bar t^* \\Rightarrow s \\bar d$. The same holds \nalways for $\\Delta m(B_d)$. In that case things are simpler: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{|\\epsilon _K|}{\\Delta m(B_d)} \\propto \n{\\rm sin}2\\phi _1 \\simeq \n0.42 \\cdot UNC \n\\label{SIN2BETA} \n\\eeq \nwith the factor $UNC$ parametrising the uncertainties \n\\begin{equation}} \\def\\eeq{\\end{equation} \nUNC \\simeq \\left( \\frac{0.04}{|V(cb)|}\\right) \n\\left( \\frac{0.72}{x_d}\\right) \\cdot \n\\left( \\frac{\\eta _{QCD}^{(B)}}{0.55}\\right) \\cdot \n\\left( \\frac{0.62}{\\eta _{QCD}^{(K)}}\\right) \\cdot \n\\left( \\frac{2B_B}{3B_K}\\right) \\cdot \n\\left( \\frac{f_B}{160\\, {\\rm MeV}}\\right) ^2 \n\\eeq\nwhere \n$x_d \\equiv \\Delta m(B_d)\/\\Gamma (B_d)$; \n$\\eta _{QCD}^{(B)}$ and $\\eta _{QCD}^{(K)}$ denote the \nQCD radiative corrections for ${\\cal H}(\\Delta B=2)$ and \n${\\cal H}(\\Delta S=2)$, respectively; $B_B$ and $B_K$ \nexpress the expectation value of ${\\cal H}(\\Delta B=2)$ or \n${\\cal H}(\\Delta S=2)$ in units of the `vacuum saturation' \nresult which is given in terms of the decay constants \n$f_B$ and $f_K$ (where the latter is known). The main \nuncertainty is obviously of a theoretical nature related to \nthe hadronic parameters $B_B$, $B_K$ and $f_B$; as discussed \nbefore, state-of-the-art theoretical technologies yield \n$B_B \\simeq 1$, $B_K \\simeq 0.8 \\pm 0.2$ and \n$f_B \\simeq 180 \\pm 30 \\, {\\rm MeV}$ where the latter range \nmight turn out to be anything but conservative! \nEq.(\\ref{SIN2BETA}) represents an explicit illustration that some CP \nasymmetries in $B^0$ decays are huge. \n\nFor $m_t \\simeq 180$ GeV the $c \\bar c$ and $c\\bar t + t \\bar c$ \ncontributions to $\\epsilon _K$ are still sizeable; nevertheless \nEq.(\\ref{SIN2BETA}) provides a good approximation. Furthermore \nsin$2\\phi _1$ can still be expressed reliably as a function of the \nhadronic matrix elements: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm sin}2\\phi _1 = f(B_Bf_B^2\/B_K) \n\\eeq \nIt will become obvious why this is relevant. \n\nThe general idea is, of course, to construct the triangle as accurately \nas possible and then probe it; i.e. search for inconsistencies that \nwould signal the intervention of New Physics. A few remarks on that \nwill have to suffice here. \n\nAs indicated before we can expect the value of \n$|V(ub)\/V(cb)|$ to be known to better than 10\\% and hope for \n$|V(td)|$ to be determined with decent accuracy as well. \nThe triangle \nwill then be well determined or even overdetermined. Once the \nfirst asymmetry in $B$ decays that can be interpreted reliably -- \nsay in $B_d \\rightarrow \\psi K_S$ -- has been measured and $\\phi _1$ \nbeen determined, the triangle is fully constructed from \n$B$ decays alone. Furthermore one has arrived at the first \nsensitive consistency check of the triangle: one \ncompares the measured value of sin$2\\phi _1$ with \nEq.(\\ref{SIN2BETA}) to infer which value of \n$B_Bf_B^2$ is thus required; this value is inserted into the \nStandard Model expression for $\\Delta m(B_d)$ together \nwith $m_t$ to see whether the experimental result is \nreproduced. \n\nA host of other tests can be performed that are highly sensitive to \n\\begin{itemize}\n\\item \nthe presence of New Physics and \n\\item \nto some of their salient dynamical features. \n\\end{itemize}\nDetails can be found in the ample literature on that subject. \n\n\n\n\n\\section{Oscillations and CP Violation in Charm Decays -- \nThe Underdog's Chance for Fame}\nIt is certainly true that \n\\begin{itemize}\n\\item \n$D^0-\\bar D^0$ oscillations proceed very slowly in the \nStandard Model and \n\\item \nCP asymmetries in $D$ decays are small or even tiny within \nthe KM ansatz. \n\\end{itemize}\nYet the relevant question quantitatively is: how slow and how small? \n\n\\subsection{$D^0-\\bar D^0$ Oscillations}\nBounds on $D^0 - \\bar D^0$ oscillations are most cleanly \nexpressed through `wrong-sign' semileptonic decays: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nr_D = \\frac{\\Gamma (D^0 \\rightarrow l^-X)}{\\Gamma (D^0 \\rightarrow l^+X)} \n\\simeq \\frac{1}{2} \\left( x_D^2 + y_D^2\\right) \n\\eeq \nwith $x_D = \\Delta m_D\/\\Gamma _D$, \n$y_D = \\Delta \\Gamma _D\/2\\Gamma _D$. It is often stated that \nthe Standard Model predicts \n\\begin{equation}} \\def\\eeq{\\end{equation} \nr_D \\leq 10^{-7} \\; \\; \\hat = \\; \\; \nx_D, \\, y_D \\leq 3 \\cdot 10^{-4} \n\\eeq \nI myself am somewhat flabbergasted by the boldness of such \npredictions. For one should keep the following in mind for \nproper perspective: there are quite a few channels that can drive \n$D^0 - \\bar D^0$ \noscillations -- like $D^0 \\Rightarrow K\\bar K , \\; \\pi \\pi \\Rightarrow \n\\bar D^0$ or $D^0 \\Rightarrow K^- \\pi ^+ \\Rightarrow \n\\bar D^0$ -- and \nthey branching ratios on the $({\\rm few})\\times 10^{-3}$ \nlevel \n\\footnote{For the $K^- \\pi ^+$ mode this represents the average of \nits Cabibbo allowed and doubly Cabibbo suppressed incarnations.}. \nIn the limit of $SU(3)_{Fl}$ symmetry all these contributions have \nto cancel of course. Yet there are sizeable violations of $SU(3)_{Fl}$ \ninvariance in $D$ decays, and one should have little confidence in an \nimperfect symmetry to ensure that a host of channels with branching \nratios of order few$\\times 10^{-3}$ will cancel as to render \n$x_D, \\, y_D \\leq 3\\cdot 10^{-4}$. To say it differently: \nThe relevant question in this context is {\\em not} whether \n$r_D \\sim 10^{-7} \\div 10^{-6}$ is a possible or even reasonable \nStandard Model estimate, but whether \n$10^{-6} \\leq r_D \\leq 10^{-4}$ can {\\em reliably be ruled out}! I \ncannot see how anyone could make such a claim with the \nrequired confidence. \n\nThe present experimental bound is \n\\begin{equation}} \\def\\eeq{\\end{equation} \nr_D |_{exp} \\leq 3.4 \\cdot 10^{-3} \\; \\; \\hat = \\; \\; \nx_D, \\, y_D \\leq 0.1 \n\\eeq \nto be compared with a {\\em conservative} Standard Model bound \n\\begin{equation}} \\def\\eeq{\\end{equation} \nr_D |_{SM} < 10^{-4} \\; \\; \\hat = \\; \\; y_D, \\, x_D|_{SM}\n\\leq 10^{-2} \\; \n\\eeq \nNew Physics on the other hand can enhance $\\Delta m_D$ \n(though not \n$\\Delta \\Gamma _D$) very considerably up to \n\\begin{equation}} \\def\\eeq{\\end{equation} \nx_D |_{NP} \\sim 0.1 \\; , \n\\eeq \ni.e. the present experimental bound. \n\n\\subsection{CP Violation involving $D^0 - \\bar D^0$ Oscillations}\nOne can discuss this topic in close qualitative analogy to \n$B$ decays. First one considers final states that are CP \neigenstates like $K^+K^-$ or $\\pi ^+ \\pi ^-$ \n\\cite{BSDDBAR}: \n$$ \n{\\rm rate}(D^0(t) \\rightarrow K^+ K^-) \\propto e^{-\\Gamma _D t} \n\\left( \n1+ ({\\rm sin}\\Delta m_Dt) \\cdot {\\rm Im}\\frac{q}{p}\n\\bar \\rho _{K^+K^-} \n\\right) \n\\simeq \n$$\n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\simeq e^{-\\Gamma _D t} \n\\left( \n1+ \\frac{\\Delta m_Dt}{\\Gamma _D}\\cdot \n\\frac{t}{\\tau _D} \n\\cdot {\\rm Im}\\frac{q}{p} \\bar \\rho _{K^+K^-} \n\\right) \n\\eeq \nWith $x_D|_{SM} \\leq 10^{-2}$ and \nIm$\\frac{q}{p} \\bar \\rho _{K^+K^-}|_{KM} \\sim {\\cal O}(10^{-3})$ one \narrives at an asymmetry of around $10^{-5}$, i.e. for all practical \npurposes zero, since it presents the product of two \nvery small numbers. \nYet with New Physics one conceivably has $x_D|_{NP} \\leq 0.1$, \nIm$\\frac{q}{p} \\bar \\rho _{K^+K^-}|_{NP} \\sim {\\cal O}(10^{-1})$ \nleading to an asymmetry that could be as large as of order 1\\%. \nLikewise one should compare the doubly Cabibbo suppressed transitions \\cite{BIGIBERK,NIR}\n$$ \n{\\rm rate}(D^0(t) \\rightarrow K^+ \\pi ^-) \\propto \ne^{-\\Gamma _{D^0} t} {\\rm tg}^4\\theta _C|\\hat \\rho _{K\\pi }|^2 \\cdot \n$$ \n$$ \n\\cdot \\left[ 1 - \\frac{1}{2}\\Delta \\Gamma _D t + \n\\frac{(\\Delta m_Dt)^2}{4{\\rm tg}^4\\theta _C|\\hat \\rho _{K\\pi }|^2} \n+ \\frac{\\Delta \\Gamma _Dt}\n{2{\\rm tg}^2\\theta _C|\\hat \\rho _{K\\pi }|}\n{\\rm Re}\\left( \n\\frac{p}{q}\\frac{\\hat \\rho _{K\\pi }}{|\\hat \\rho _{K\\pi }|} \n\\right) \n- \\right. \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation}\n\\left. - \\frac{\\Delta m_Dt}{{\\rm tg}^2\\theta _C|\\hat \\rho _{K\\pi }|} \n{\\rm Im}\\left( \n\\frac{p}{q}\\frac{\\hat \\rho _{K\\pi }}{|\\hat \\rho _{K\\pi }|} \n\\right) \n\\right] \n\\eeq \n$$ \n{\\rm rate}(\\bar D^0(t) \\rightarrow K^- \\pi ^+) \\propto \ne^{-\\Gamma _{D^0} t} {\\rm tg}^4\\theta _C|\\hat{\\bar \\rho }_{K\\pi }|^2 \\cdot \n$$ \n$$ \n\\cdot \\left[ 1 - \\frac{1}{2}\\Delta \\Gamma _D t + \n\\frac{(\\Delta m_Dt)^2}{4{\\rm tg}^4\\theta _C\n|\\hat{\\bar \\rho}_{K\\pi }|^2} \n+ \\frac{\\Delta \\Gamma _Dt}\n{2{\\rm tg}^2\\theta _C|\\hat{\\bar \\rho }_{K\\pi }|}\n{\\rm Re}\\left( \n\\frac{p}{q}\\frac{\\hat{\\bar \\rho }_{K\\pi }}\n{|\\hat{\\bar \\rho }_{K\\pi }|} \n\\right) \n+\\right. \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation}\n\\left. + \\frac{\\Delta m_Dt}{{\\rm tg}^2\\theta _C\n|\\hat{\\bar \\rho }_{K\\pi }|} \n{\\rm Im}\\left( \n\\frac{p}{q}\\frac{\\hat{\\bar \\rho }_{K\\pi }}{|\\hat{\\bar \\rho }_{K\\pi }|} \n\\right) \n\\right] \n\\eeq\nwhere \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm tg}^2\\theta _C \\cdot \\hat \\rho _{K\\pi } \\equiv \n\\frac{T(D^0 \\rightarrow K^+ \\pi ^-)}{T(D^0 \\rightarrow K^- \\pi ^+)}\\; \\; , \\; \\; \n{\\rm tg}^2\\theta _C \\cdot \\hat{\\bar \\rho }_{K\\pi } \\equiv \n\\frac{T(\\bar D^0 \\rightarrow K^- \\pi ^+)}{T(\\bar D^0 \\rightarrow K^+ \\pi ^-)}\\; ; \n\\eeq \nin such New Physics scenarios \none would expect a considerably enhanced asymmetry \nof order $1\\%\/{\\rm tg}^2 \\theta _C \\sim 20\\%$ -- at the cost \nof smaller statistics. \n\nEffects of that size would unequivocally signal the intervention \nof New Physics! \n\n\n\\subsection{Direct CP Violation}\nAs explained before a direct CP asymmetry requires the presence \nof two coherent amplitudes with different weak and different \nstrong phases. Within the Standard Model (and the \nKM ansatz) such effects can occur in Cabibbo suppressed \n\\footnote{The effect could well reach the $10^{-3}$ \nand exceptionally the $10^{-2}$ level.}, yet not \nin Cabibbo allowed or doubly Cabibbo suppressed modes. There \nis a subtlety involved in this statement. Consider for example \n$D^+ \\rightarrow K_S \\pi ^+$. At first sight it appears to be \na Cabibbo allowed mode described by a single amplitude without \nthe possibility of an asymmetry. However \\cite{YAMAMOTO} \n\\begin{itemize} \n\\item \ndue to $K^0 - \\bar K^0$ mixing the final state \ncan be reached also through a doubly Cabibbo suppressed \nreaction, and the two amplitudes necessarily interfere; \n\\item \nbecause of the CP violation in the $K^0 - \\bar K^0$ complex \nthere is an asymmetry that can be predicted on general grounds \n$$ \n\\frac{\\Gamma (D^+ \\rightarrow K_S \\pi ^+) - \\Gamma (D^- \\rightarrow K_S \\pi ^+)} \n{\\Gamma (D^+ \\rightarrow K_S \\pi ^+) + \\Gamma (D^- \\rightarrow K_S \\pi ^+)} \n\\simeq - 2 {\\rm Re}\\, \\epsilon _K \\simeq - 3.3 \\cdot 10^{-3} \n\\simeq \n$$ \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\simeq \n\\frac{\\Gamma (D^+ \\rightarrow K_L \\pi ^+) - \\Gamma (D^- \\rightarrow K_L \\pi ^+)} \n{\\Gamma (D^+ \\rightarrow K_L \\pi ^+) + \\Gamma (D^- \\rightarrow K_L \\pi ^+)} \n \\; ; \n\\label{DIRECTCPEPS}\n\\eeq \n\\item\n If New Physics contributes to the doubly Cabibbo suppressed \namplitude $D^+ \\rightarrow K^0 \\pi ^+$ (or $D^- \\rightarrow \\bar K^0 \\pi ^-$) then \nan asymmetry could occur quite conceivably on the few percent \nscale; \n\\item \nsuch a manifestation of New Physics would be unequivocal; against \nthe impact of $\\epsilon _K$, Eq.(\\ref{DIRECTCPEPS}) it \ncould be distinguished not only through the size of the asymmetry, \nbut also how it surfaces in $D^+ \\rightarrow K_L \\pi ^+$ vs. \n$D^- \\rightarrow K_L \\pi ^-$: if it is New Physics one has \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\frac{\\Gamma (D^+ \\rightarrow K_S \\pi ^+) - \\Gamma (D^- \\rightarrow K_S \\pi ^+)} \n{\\Gamma (D^+ \\rightarrow K_S \\pi ^+) + \\Gamma (D^- \\rightarrow K_S \\pi ^+)} \n= - \n\\frac{\\Gamma (D^+ \\rightarrow K_L \\pi ^+) - \\Gamma (D^- \\rightarrow K_L \\pi ^+)} \n{\\Gamma (D^+ \\rightarrow K_L \\pi ^+) + \\Gamma (D^- \\rightarrow K_L \\pi ^+)} \n\\eeq \ni.e., the CP asymmetries in $D \\rightarrow K_S \\pi$ and $D \\rightarrow K_L \\pi$ \ndiffer in sign -- in contrast to Eq.(\\ref{DIRECTCPEPS}). \n\\end{itemize} \n\n\\section{Baryogenesis in the Universe}\n\\subsection{The Challenge}\n\nOne of the most intriguing aspects of big bang \ncosmology is to `understand' nucleosynthesis, i.e. to reproduce the \nabundances observed for the nuclei in the universe as \n{\\em dynamically} generated rather than merely dialed as \ninput values. This challenge has been met successfully, in \nparticular for the light nuclei, and actually so much so that \nit is used to obtain information on dark matter in the universe, \nthe number of neutrinos etc. It is natural to ask whether such \na success could be repeated for an even more basic quantity, \nnamely the baryon number density of the universe which is defined \nas the difference in the abundances of baryons and antibaryons: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Delta n_{Bar} \\equiv n_{Bar} - n_{\\overline{Bar}} \n\\eeq \nQualitatively one can summarize the observations through \ntwo statements: \n\\begin{itemize}\n\\item \nThe universe is not empty. \n\\item \nThe universe is almost empty. \n\\end{itemize}\nMore quantitatively one finds\n\\begin{equation}} \\def\\eeq{\\end{equation} \nr_{Bar} \\equiv \\frac{\\Delta n_{Bar}}{n_{\\gamma}} \\sim {\\rm few} \n\\times 10^{-10} \n\\label{BNUMUNI}\n\\eeq \nwhere $n_{\\gamma}$ denotes the number density of photons in the \ncosmic background radiation. Actually we know more, namely \nthat at least in our corner of the universe there are \npractically no primary antibaryons: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nn_{\\overline{Bar}} \\ll n_{Bar} \\ll n_{\\gamma}\n\\label{TWOHIER}\n\\eeq\nIt is conceivable that in other \nneighbourhoods antimatter dominates and that the universe \nis formed by a patchwork quilt of matter and antimatter \ndominated regions with the whole being \nmatter-antimatter symmetric. Yet it is widely held \nto be quite unlikely -- primarily because no mechanism has been \nfound by which a matter-antimatter symmetric universe following \na big bang evolution can develop sufficiently large regions \nwith non-vanishing baryon number. While there will be \nstatistical fluctuations, they can be nowhere near large \nenough. Likewise for dynamical effects: baryon-antibaryon \nannihilation is by far not sufficiently effective to create pockets \nwith the observed baryon number, Eq.(\\ref{BNUMUNI}). For the \nnumber density of {\\em surviving} baryons can be estimated as \n\\cite{DOLGOV1} \n\\begin{equation}} \\def\\eeq{\\end{equation} \nn_{Bar} \\sim \\frac{n_{\\gamma}}{\\sigma _{annih}m_N M_{PL}} \n\\simeq 10^{-19}n_{\\gamma} \n\\eeq\nwhere $\\sigma _{annih}$ denotes the cross section of nucleon \nannihilation, $m_N$ and $M_{Pl}$ the nucleon and Planck mass, \nrespectively. Hence we conclude for the universe as a whole \n\\begin{equation}} \\def\\eeq{\\end{equation} \n0 \\neq \\frac{n_{Bar}}{n_{\\gamma}} \\simeq \n\\frac{\\Delta n_{Bar}}{n_{\\gamma}} \\sim {\\cal O}(10^{-10}) \n\\eeq \nwhich makes more explicit the meaning of the statement \nquoted above that the universe has been observed to be \nalmost empty, but not quite. Understanding this double \nobservation is the challenge we are going to address now. \n\\subsection{The Ingredients}\nThe question is: under which condition can one have a situation \nwhere the baryon number of the universe that vanishes at the initial time -- \nwhich for all practical purposes is the Planck time -- develops \na non-zero value later on: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Delta n_{Bar}(t = t_{Pl}\\simeq 0) = 0 \n\\; \\; \\stackrel{?}{\\Longrightarrow} \\; \\; \n\\Delta n_{Bar}(t = `today') \n\\neq 0 \n\\eeq \nOne can and should actually go one step further in the task one is \nsetting for oneself: explaining the observed baryon number as \ndynamically generated \n{\\em no matter what its initial value was!} \n\nIn a seminal paper that appeared in 1967 Sakharov \nlisted the three ingredients that are essential for the feasibility \nof such a program \\cite{SAKH,DOLGOV2} : \n\\begin{enumerate}\n\\item \nSince the final and initial baryon number differ, there have to be baryon number violating transitions: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\cal L}(\\Delta n_{Bar} \\neq 0) \\neq 0 \n\\eeq \n\\item \nCP invariance has to be broken. Otherwise for every baryon number \nchanging transition $N \\rightarrow f$ there is its CP conjugate one \n$\\bar N \\rightarrow \\bar f$ and no net baryon number can be generated. \nI.e., \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Gamma (N \\; \\; \\stackrel{{\\cal L}(\\Delta n_{Bar} \\neq 0)}\n{\\longrightarrow} \\; \\; f ) \n\\; \\; \\; \\neq \\; \\; \\; \n\\Gamma (\\bar N \\; \\; \\stackrel{{\\cal L}(\\Delta n_{Bar} \\neq 0)}\n{\\longrightarrow} \\; \\; \\bar f ) \n\\eeq \nis needed. \n\\item \nUnless one is willing to entertain thoughts of CPT violations, the \nbaryon number and CP violating transitions have to proceed \nout of thermal equilibrium. For in thermal equilibrium time \nbecomes irrelevant globally and CPT invariance reduces to \nCP symmetry which has to be avoided, see above: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n{\\rm CPT \\; invariance} \\; \\; \\; \n\\stackrel{thermal \\; equilibrium}\n{\\Longrightarrow} \\; \\; \\; {\\rm CP \\; invariance} \n\\eeq \n\\end{enumerate} \nIt is important to keep in mind that these three conditions have \nto be satisfied {\\em simultaneously}. The other side of the coin is, \nhowever, the following: once a baryon number has been \ngenerated through the concurrance of these three effects, \nit can be washed out again by these same effects.\n\n\\subsection{GUT Baryogenesis}\nSakharov's paper was not noticed (except for \\cite{KUZMIN}) \nfor several years until the concept of Grand Unified Theories \n(=GUTs) emerged starting in 1974 \\cite{PATI}; \nfor those naturally provide all three necessary ingredients: \n\n\\begin{enumerate} \n\\item \nBaryon number changing reactions have to exist in GUTs. For placing \nquarks and leptons into common representations of the underlying \ngauge groups -- the hallmark of GUTs -- means that gauge interactions exist changing baryon and lepton numbers. \nThose gauge bosons are generically \nreferred to as $X$ bosons and have two couplings to fermions \nthat violate baryon and\/or lepton number: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nX \\leftrightarrow qq \\; , \\; \\; q \\bar l \n\\label{XDEC} \n\\eeq \n\n\\item \nThose models are sufficiently complex to allow for several \npotential sources of CP violation. Since $X$ bosons have (at least) \ntwo decay channels open CP asymmetries can arise \n\\begin{eqnarray} \n\\Gamma (X \\rightarrow qq) = (1+\\Delta _q) \\Gamma _q \n&,& \\; \\; \n\\Gamma (X \\rightarrow q\\bar l) = (1-\\Delta _l) \\Gamma _l \\\\ \n\\Gamma (\\bar X \\rightarrow \\bar q \\bar q) = \n(1-\\Delta _q) \\Gamma _q \n&,& \\; \\; \n\\Gamma (\\bar X \\rightarrow \\bar q l) = (1+\\Delta _l) \\Gamma _l \n\\end{eqnarray} \nwhere \n\\begin{eqnarray} \n{\\bf \\rm CPT} \\; \\; &\\Longrightarrow & \\; \\; \n\\Delta _q \\Gamma _q = \\Delta _l \\Gamma _l \\\\ \n{\\bf \\rm CP} \\; \\; &\\Longrightarrow & \\; \\; \\Delta _q = 0 = \\Delta _l \\\\ \n{\\bf \\rm C} \\; \\; &\\Longrightarrow & \\; \\; \\Delta _q = 0 = \\Delta _l \n\\end{eqnarray} \n\n\\item \nGrand Unification means that a phase transition takes \nplace around an energy scale $M_{GUT}$. For temperatures \n$T$ well above the transition point -- $T \\gg M_{GUT}$ -- \nall quanta are relativistic with a number density \n\\begin{equation}} \\def\\eeq{\\end{equation} \nn(T) \\propto T^3\n\\label{NREL} \n\\eeq \nFor temperatures around the phase transition -- \n$T \\sim M_{GUT}$ -- some of the quanta, in particular those \ngauge bosons generically referred to as $X$ bosons aquire \na mass $M_X \\sim {\\cal O}(M_{GUT})$ and their \nequilibrium number \ndensity becomes Boltzmann suppressed: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nn_X(T) \\propto (M_XT)^{\\frac{3}{2}} {\\rm exp}\n\\left( - \\frac{M_X}{T}\\right) \n\\label{BOLTZ} \n\\eeq \nMore $X$ bosons will decay according to Eq.(\\ref{XDEC}) \nthan be regenerated from $qq$ and $q \\bar l$ collisions \nultimately bringing the number of $X$ bosons down to the \nlevel described by Eq.(\\ref{BOLTZ}). Yet that will take some time; \nthe expansion in the big bang cosmology leads to a cooling rate \nthat is so rapid that thermal equilibrium cannot be maintained \nthrough the phase transition. That means that $X$ bosons \ndecay -- and in general interact -- out of thermal \nequilibrium \\cite{DOLGOV2}. \n\\end{enumerate} \nTo the degree that the back production of $X$ \nbosons in $qq$ and $q\\bar l$ collisions can be ignored one finds \nas an order-of-magnitude estimate \n\\begin{equation}} \\def\\eeq{\\end{equation} \nr_{Bar} \\sim \\frac{\\frac{4}{3}\\Delta _q \\Gamma _q - \n\\frac{2}{3}\\Delta _l \\Gamma _l}{\\Gamma _{tot}} \\frac{n_X}{n_0} \n= \\frac{\\frac{2}{3}\\Delta _q \\Gamma _q }{\\Gamma _{tot} }\\frac{n_X}{n_0}\n\\label{GUTRESULT} \n\\eeq \nwith $n_X$ denoting the initial number density of $X$ \nbosons and $n_0$ the number density of the light decay \nproducts \n\\footnote{Due to thermalization effects one can have \n$n_0 \\gg 2n_X$.}. \nThe three essential conditions for baryogenesis are thus \nnaturally realized around the GUT scale in big bang cosmologies, as \ncan be read off from Eq.(\\ref{GUTRESULT}): \n\\begin{itemize} \n\\item \n$\\Gamma _q \\neq 0$ representing baryon number violation; \n\\item \n$\\Delta _q \\neq 0$ reflecting CP violation and \n\\item \nthe absence of the back reaction due to an absence of thermal \nequilibrium. \n\\end{itemize} \n\nThe fact that this problem can be formulated in GUT models \nand answers obtained that are very roughly in the right \nballpark is a highly attractive feature of GUTs, in particular \nsince this was {\\em not} among the original motivations \nfor constructing such theories. \n\nOn the other hand it would be highly misleading to claim \nthat baryogenesis has been understood. There are \nserious problems in any attempt to have baryogenesis \noccur at a GUT scale: \n\\begin{itemize}\n\\item \nA baryon number generated at such high temperatures is \nin grave danger to be washed out or diluted in the subsequent \nevolution of the universe. \n\\item \nVery little is known about the dynamical actors operating at \nGUT scales and their characteristics -- and that is putting it mildly. Actually \neven in the future we can only hope to obtain some \nslices of indirect information on them. \n\\end{itemize} \nOf course it would be premature to write-off baryogenesis at \nGUT scales, yet it might turn out that it is best \ncharacterised as a proof of principle -- namely that the \nbaryon number of the universe can be understood as dynamically \ngenerated -- rather than as a semi-quantitative realization. \n\\subsection{Electroweak Baryogenesis}\nBaryogenesis at the electroweak scale \n\\cite{RUBAKOV} \nis the most actively analyzed scenario at present. For it possesses several highly attractive features: \n\\begin{itemize}\n\\item \nWe know that dynamical landscape fairly well.\n\\begin{itemize}\n\\item \nIn particular CP violation has been found to exist there. \n\\item \nA well-studied phase transition, namely the spontaneous breaking \\begin{equation}} \\def\\eeq{\\end{equation} \nSU(2)_L\\times U(1) \\; \\; \\Longrightarrow U(1)_{QED}\n\\eeq \ntakes place. \n\\end{itemize}\n\\item \nFuture experiments will certainly probe that dynamical regime \nwith ever increasing sensitivity, both by searching for the \non-shell production of new quanta -- like SUSY and\/or Higgs \nstates -- and the indirect impact through quantum corrections on \nrare decays and CP violation. \n\\end{itemize}\nHowever at this point the reader might wonder: \"What \nabout the third required ingredient, baryon number violation? At the \nelectroweak scale?\" It is often not appreciated that \nthe electroweak forces of the Standard Model by themselves \nviolate baryon number, though in a very subtle way. \nWe find here what is called an anomaly: the baryon \nnumber current is conserved on the classical, \nyet {\\em not} on the quantum level: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\partial _{\\mu} J_{\\mu}^{Bar} = \n\\partial _{\\mu}\\sum _q (\\bar q_L \\gamma _{\\mu}q_L) = \n\\frac{g^2}{16 \\pi ^2} G_{\\mu \\nu} \\tilde G_{\\mu \\nu} \\neq 0 \n\\label{ANOM} \n\\eeq \nwhere $G_{\\mu \\nu}$ denotes the electroweak field strength \ntensor \n\\begin{equation}} \\def\\eeq{\\end{equation} \nG_{\\mu \\nu} = \\partial _{\\mu} A_{\\nu} - \\partial _{\\nu} A_{\\mu} \n+ g[A_{\\mu}, A_{\\nu}] \n\\eeq \nand $\\tilde G_{\\mu \\nu} $ its dual: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\tilde G_{\\mu \\nu} = \\epsilon _{\\mu \\nu \\alpha \\beta }\nG_{\\alpha \\beta} \n\\eeq \nThe right hand side of Eq.(\\ref{ANOM}) can be written as \nthe divergence of a current \n\\begin{equation}} \\def\\eeq{\\end{equation} \nG_{\\mu \\nu} \\tilde G_{\\mu \\nu} = \\partial _{\\mu} K_{\\mu} \\; , \n\\; \\; K_{\\mu} = 2 \\epsilon _{\\mu \\nu \\alpha \\beta } \n\\left( A_{\\nu} \\partial _{\\alpha} A_{\\beta} + \n\\frac{2}{3} ig A_{\\nu}A_{\\alpha}A_{\\beta} \n\\right) \n\\label{KCURRENT} \n\\eeq \nA total derivative is usually unobservable since partial \nintegration allows to express its contribution \nthrough a surface integral at infinity. The field strength \ntensor $G_{\\mu \\nu}$ indeed vanishes at infinity -- but \nnot necessarily the gauge potential $A_{\\mu}$. \nTo be more specific: The field configuration at infinity is \nthat of a ground state for which $G_{\\mu \\nu} = 0$ \nholds. Yet that property does not define it uniquely: \nground states get differentiated by the value of their \n$K$ charge, i.e. the space integral \nof $K_0$, the zeroth component of the current $K_{\\mu}$ \nconstructed from their gauge field configuration. \nThis integral reflects differences in the gauge topology \nof the ground states and therefore is called the \n{\\em topological charge}. \nWhile this charge is irrelevant for abelian gauge theories \nwhere the \nlast term in Eq.(\\ref{KCURRENT}) necessarily vanishes, \nit becomes relevant for non-abelian theories. \nWe have encountered \nthis phenonemon already in our discussion of the \nStrong CP Problem that is driven by the axial quark current \nnot being conserved in the strong interactions of QCD. It is often \nreferred to as `Chiral' Anomaly since it breaks chiral invariance, \nor `Triangle' Anomaly since it is produced by a triangular fermion \nloop diagram or `Adler-Bell-Jackiw' Anomaly named after the authors who discovered it. \n\nThe concrete impact of the triangle anomaly on the physics \ndepends on the specifics of the theory: here \nbecause of the chiral \nnature of the weak interactions it induces baryon number \nviolation. \nEq.(\\ref{ANOM}) and Eq.(\\ref{KCURRENT}) show that the difference \n$J_{\\mu}^{Bar} - K_{\\mu}$ is conserved. The transition from one \nground state to another which represents a tunneling \nphenomenon is thus accompanied by a change in \nbaryon number. Elementary quantum mechanics tells \nus that this baryon number violation is described as a \nbarrier penetration and exponentially suppressed at low \ntemperatures or energies \\cite{THOOFTBAR}: \n$Prob(\\Delta n_{Bar}\\neq 0) \\propto \n{\\rm exp}(-16 \\pi ^2\/g^2) \\sim {\\cal O}(10^{-160})$ -- \na suppression that reflects the tiny size of the \nweak coupling. \n\nThere is a corresponding anomaly for the lepton number \ncurrent implying that lepton number is violated as \nwell with the selection rule \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Delta n_{Bar} - \\Delta n_{lept} = 0 \\; . \n\\label{B-L}\n\\eeq\nThis is usually referred to by saying that $B-L$, the difference between baryon and lepton number, is still conserved. \n\nAt sufficiently high energies this huge suppression of baryon \nnumber changing transition rates will evaporate since \nthe transition between different ground states can be \nachieved classically through a motion {\\em over} \nthe barrier. The question then is at which energy scale this \nwill happen and how quickly baryon number violation will \nbecome operative. Some semi-quantitative observations can \nbe offered and answers given \n\\cite{RUBAKOV2,DOLGOV2}. \n\nThere are special field configurations -- called sphalerons -- \nthat carry the topological $K$ charge. In the Standard Model they \ninduce effective multistate interactions among left-handed \nfermions that change baryon and lepton number by three \nunits each: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Delta n_{Bar} = \\Delta n_{lept} = 3 \n\\eeq \nAt high energies where the weak bosons $W$ and $Z$ \nare massless, the height of the transition barrier between \ndifferent groundstates vanishes likewise and the change \nof baryon number can proceed in an unimpeded way and \npresumably faster than the universe expands. Thermal \nequilibrium is then maintained and any baryon \nasymmetry existing before this era is actually washed out \n\\footnote{To be more precise, only $B+L$ is erased \nwithin the Standard Model whereas $B-L$ remains \nunchanged.}! \nRather than generate a baryon number sphalerons act to \ndrive the universe back to matter-antimatter \nsymmetry at this point in its evolution. \n\nAt energies below the phase transition, i.e. in the broken phase \nof $SU(2)_L \\times U(1)$ baryon number is conserved for \nall practical purposes as pointed out above. \n\nThe value of \n$\\Delta n_{Bar}$ as observed today can thus be generated \nonly in the transition from the unbroken high energy to the \nbroken low energy phase. With $\\Delta n_{Bar}\\neq 0$ \nprocesses operating there the issue now turns to the \nstrength of the phase transition: is it relatively smooth \nlike a second order phase transition or violent like a \nfirst order one? Only the latter scenario can support \nbaryogenesis. \n\nA large amount of interesting theoretical work has been \non the thermodynamics of the Standard Model in an \nexpanding universe. Employing perturbation theory \nand lattice studies one has arrived at the following result: \nfor light Higgs masses up to around 70 GeV, the phase \ntransition is first order, for larger masses it is second \norder \n\\cite{SHAP}. Since no such light Higgs states have been observed \nat LEP, one infers that the phase transition is second order \nthus apparently foreclosing baryogenesis occurring at the \nelectroweak scale. \n\nWe have concentrated here on the questions of thermal \nequilibrium and baryon number while \ntaking CP violation for \ngranted since it is known to operate at the electroweak scale. \nYet most authors -- with the exception of some notable \nheretics -- agree that the KM ansatz is not at all \nup to {\\em this} task: it fails by several orders of \nmagnitude. On the other hand New Physics \nscenarios of CP violation -- in particular \nof the Higgs variety -- can reasonably be called upon to perform \nthe task. \n\n\\subsection{Leptogenesis Driving Baryogenesis}\nIf the electroweak phase transition is indeed a \nsecond order one, \nsphaleron mediated reactions cannot drive baryogenesis \nas just discussed and they will wipe out any \npre-existing $B+L$ number. Yet if at some high \nenergy scales a lepton number is generated the very \nefficiency of these sphaleron processes can \n{\\em communicate} this asymmetry to the baryon \nsector through them maintaining conservation of \n$B-L$. \n\nThere are various ways in which such scenarios can \nbe realized. The simplest one is to just add \n{\\em heavy right-handed Majorana} neutrinos \nto the Standard Model. This is highly attractive \nin any case since it enables us \nto implement the see-saw mechanism for explaining \nwhy the observed neutrinos are (practically) massless; \nit is also easily embedded into $SO(10)$ GUTs. \n\nThe basic idea is the following \\cite{FUKUGITA}: \n\\begin{itemize} \n\\item \nA primordial lepton asymmetry is generated at high \nenergies well above the electroweak phase transition: \n\\begin{itemize} \n\\item \nSince a Majorana neutrino $N$ is its own CPT mirror image, \nits dynamics necessarily violate lepton number. It will \npossess at least the following classes of decay channels: \n\\begin{equation}} \\def\\eeq{\\end{equation} \nN \\rightarrow l \\bar H \\; , \\; \\; \\bar l H\n\\eeq \nwith $l$ and $\\bar l$ denoting a light charged or neutral lepton \nor anti-lepton and $H$ and $\\bar H$ a Higgs or anti-Higgs field, \nrespectively. \n\\item \nA CP asymmetry will in general arise \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\Gamma (N \\rightarrow l \\bar H) \\neq \n\\Gamma (\\bar N \\rightarrow \\bar l H)\n\\eeq \nthrough a KM analogue in the neutrino mass \nmatrix (which can be quite different from the \nmass matrix for charged leptons). \n\\item \nThese neutrino decays are sufficiently slow as to occur \nout of thermal equilibrium around the energy scale \nwhere the Majorana masses emerge. \n\\end{itemize} \n\\item \nThe resulting lepton asymmetry is transferred into a \nbaryon number through sphaleron mediated processes \nin the unbroken high energy phase of \n$SU(2)_L \\times U(1)$: \n\\begin{equation}} \\def\\eeq{\\end{equation} \n\\langle \\Delta n_{lept}\\rangle = \n\\frac{1}{2}\\langle \\Delta n_{lept} + \\Delta n_{Bar}\\rangle + \n\\frac{1}{2}\\langle \\Delta n_{lept} - \\Delta n_{Bar}\\rangle \n\\Longrightarrow \n\\frac{1}{2}\\langle \\Delta n_{lept} - \\Delta n_{Bar}\\rangle \n\\eeq \n\\item \nThe baryon number thus generated survives through the \nsubsequent evolution of the universe. \n\\end{itemize} \n\n\n\n\\subsection{Wisdom -- Conventional and Otherwise}\nWe understand how nuclei were formed in the universe \ngiven protons and neutrons. Obviously it would be \neven more fascinating if we could understand how \nthese baryons were generated in the first place. \nWe do not possess a specific and \nquantitative theory successfully describing baryogenesis. \nHowever leaving it at that statement would -- we believe -- \nmiss the main point. We have learnt which kinds of \ndynamical ingredients are neccessary for baryogenesis to \noccur in the universe. We have seen that these ingredients \ncan be realized naturally: \n\\begin{itemize} \n\\item \nGUT scenarios for baryogenesis \nprovide us with a proof of principle that such a program \ncan be realized. In practical terms however they suffer from \nvarious shortcomings: \n\\begin{itemize} \n\\item \nSince the baryon number is generated \nat the GUT scales, very little is and not much more might \never be known about that dynamics. \n\\item \nIt \nappears quite likely that a baryon number produced at such \nhigh scales is subsequently washed out. \n\\end{itemize} \n\\item \nThe highly fascinating proposal of baryogenesis at the electroweak \nphase transition has attracted a large degree of attention -- \nand deservedly so: \n\\begin{itemize} \n\\item \nA baryon number emerging from \nthis phase transition would be in no danger of being \ndiluted substantially. \n\\item \nThe dynamics involved here \nis known to a considerable degree and will be probed \neven more with ever increasing sensitivity over the \ncoming years. \n\\end{itemize} \nHowever it seems that the electroweak phase transition is \nof second order and thus not \nsufficiently violent. \n\\item \nA very intriguing variant turns some of the vices of sphaleron \ndynamics into virtues by attempting to understand \nthe baryon number of the universe as a reflection of a \n{\\em primary} lepton asymmetry. \nThe required new dynamical entities \n-- Majorana neutrinos and their decays -- obviously would \nimpact on the universe in other ways as well. \n\\end{itemize} \nThe challenge to understand baryogenesis has already inspired \nour imagination, \nprompted the development of some very intriguing \nscenarios and thus has initiated many \nfruitful studies -- and in the end we might even be \nsuccessful in meeting it! \n\n\n\\section{The Cathedral Builders' Paradigm}\n\\subsection{The Paradigm}\n\nThe dynamical ingredients for numerous and multi-layered \nmanifestations of CP and T violations do exist or are likely to exist. Accordingly one searches \nfor them in many phenomena, namely in \n\\begin{itemize}\n\\item \nthe neutron electric dipole moment probed with ultracold \nneutrons at ILL in Grenoble, France; \n\\item \nthe electric dipole moment of electrons studied through the \ndipole moment of atoms at Seattle, Berkeley and Amherst in the US; \n\\item \nthe transverse polarization of muons in \n$K^- \\rightarrow \\mu ^- \\bar \\nu \\pi ^0$ at KEK in Japan; \n\\item \n$\\epsilon ^{\\prime}\/\\epsilon _K$ as obtained from $K_L$ \ndecays at FNAL and CERN and soon at DA$\\Phi$NE in Italy; \n\\item \nin decay distributions of hyperons at FNAL; \n\\item \nlikewise for $\\tau$ leptons at CERN, the beauty factories and BES \nin Beijing; \n\\item \nCP violation in the decays of charm hadrons produced \nat FNAL and the beauty factories; \n\\item \nCP asymmetries in beauty decays at DESY, at the beauty \nfactories at Cornell, SLAC and KEK, at the FNAL collider and \nultimately at the LHC. \n\n\\end{itemize} \nA quick glance at this list already makes it clear \nthat frontline research on this topic \nis pursued at all high energy labs in the world -- and then some; \ntechniques from several different branches of physics -- \natomic, nuclear and high energy physics -- are harnessed in \nthis endeavour together with a wide range of set-ups. \nLastly, experiments are performed at the lowest temperatures \nthat can be realized on earth -- ultracold neutrons -- and at the \nhighest -- in collisions produced at the LHC. And all of that dedicated \nto one profound goal. \nAt this point I can explain what I mean by the term \n\"Cathedral Builders' Paradigm\". \nThe building of cathedrals required interregional collaborations, \nfront line technology (for the period) from many different fields \nand commitment; it had to be based on solid foundations -- and \nit took time. The analogy to the ways and needs of high energy \nphysics are obvious -- but it goes deeper than that. \nAt first sight a cathedral looks \nlike a very complicated and confusing structure with something \nhere and something there. Yet further scrutiny reveals that \na cathedral is more appropriately \ncharacterized as a complex rather than a complicated \nstructure, one that is multi-faceted and multi-layered -- \nwith a coherent theme! One cannot (at least for \nfirst rate cathedrals) remove any of its elements \nwithout diluting (or even destroying) its technical soundness and \nintellectual message. Neither can one in our efforts to come to grips \nwith CP violation! \n\n\\subsection{Summary}\n\\begin{itemize} \n\\item \nWe know that CP symmetry is not exact in nature since \n$K_L \\rightarrow \\pi \\pi $ proceeds and presumably because we \nexist, i.e. because the baryon number of the universe does \n{\\em not} vanish. \n\\item \nIf the KM mechanism is a significant actor in $K_L \\rightarrow \\pi \\pi$ \ntransitions then there must be large CP asymmetries in the decays \nof beauty hadrons. In $B^0$ decays they \nare naturally measured in units of 10 \\%! \n\\item \nSome of these asymmetries are predicted with high parametric \nreliability. \n\\item \nNew theoretical technologies will allow us to translate such \nparametric reliability into quantitative accuracy. \n\\item \nAny significant difference between certain KM predictions for the \nasymmetries and the data reveals the intervention of New Physics. \nThere will be no `plausible deniability'. \n\\item \nWe can expect 10 years hence the theoretical uncertainties in \nsome of the \npredictions to be reduced below 10 \\% . \n\\item \nI find it likely that deviations from the KM predictions will show up \non that level. \n\\item \nYet to exploit this discovery potential to the fullest one will have to \nharness the statistical muscle provided by beauty production \nat hadronic colliders. \n\\end{itemize}\n\n\n\\subsection{Outlook}\nI want to start with a statement about the past: \n{\\em The comprehensive study of kaon and hyperon physics \nhas been instrumental in guiding us to the Standard Model.} \n\\begin{itemize}\n\\item \nThe $\\tau -\\theta $ puzzle led to the realization that parity is not \nconserved in nature. \n\\item \nThe observation that the production rate exceeded the decay rate \nby many orders of magnitude -- this was the origin of the \nname `strange particles' -- was explained through postulating \na new quantum number -- `strangeness' -- conserved by the strong, \nthough not the weak forces. This was the beginning of the second \nquark family. \n\\item \nThe absence of flavour-changing neutral currents was incorporated \nthrough the introduction of the quantum number `charm', which \ncompleted the second quark family. \n\\item \nCP violation finally led to postulating yet another, the third \nfamily. \n\\end{itemize}\nAll of these elements which are now essential pillars of the Standard \nModel were New Physics at {\\em that} time! \n\nI take this historical \nprecedent as clue that a detailed, comprehensive and thus \nneccessarily long-term program on beauty physics will lead to a \nnew paradigm, a {\\em new} Standard Model! \n\nCP violation is a fundamental as well as mysterious phenomenon \nthat we have not understood yet. This is not surprising: after all \naccording to the KM mechanism CP violation enters through the \nquark mass matrices; it thus relates it to three central \nmysteries of the Standard Model: \n\\begin{itemize}\n\\item \nHow are fermion masses generated? \n\\footnote{Or more generally: how are masses produced in \ngeneral? For in alternative models CP violation enters through \nthe mass matrices for gauge bosons and\/or Higgs bosons.} \n\\item \nWhy is there a family structure?\n\\item \nWhy are there three families rather than one?\n\\end{itemize} \nIn my judgement it would be unrealistic to expect that these \nquestions can be answered through pure thinking. I strongly \nbelieve we have to appeal to nature through experimental efforts to \nprovide us with more pieces that are surely missing in the \npuzzle. CP studies are essential in obtaining the full dynamical \ninformation contained in the mass matrices or -- in the language \nof v. Eichendorff's poem quoted in the beginning, \"to find the \nmagic word\" that will decode nature's message for us. \n\nConsiderable progress has been made in theoretical engineering \nand developing a comprehensive CP phenomenology from \nwhich I conclude: \n\\begin{itemize}\n\\item \n$B$ decays constitute an almost ideal, certainly optimal and unique \nlab. Personally I believe that even if no deviation \nfrom the KM predictions were uncovered, we would find \nthat the KM parameters, in particular the angles of the \nKM triangle, carry special values that would give us \nclues about New Physics. Some very interesting \ntheoretical work is being done about how GUT dynamics in \nparticular of the SUSY (or Supergravity) variety operating \nat very high scales would shape the observable \nKM parameters. \n\\item \nA comprehensive analysis of charm decays with special emphasis on \n$D^0 - \\bar D^0$ oscillations and CP violation is a moral \nimperative! Likewise for $\\tau$ leptons. \n\\item \nA vigorous research program must be pursued for light \nfermion systems, namely in the decays of kaons and hyperons \nand in electric dipole moments. After all it is conceivable \nof course that no CP asymmetries are found \nin $B$ decays on a measurable level. \nThen we would know that the KM ansatz is {\\em not} a \nsignificant actor in $K_L \\rightarrow \\pi \\pi$, that New Physics drives it -- \nbut what kind of New Physics would it be? Furthermore even if \nlarge CP asymmetries were found in $B$ decays, it could \nhappen that the signals of New Physics are obscured by \nthe large `KM background'. This would not be the case \nif electric dipole moments were found or a transverse \npolarization of muons in $K_{\\mu 3}$ decays. \n\\item Close feedback between experiment and theory will be \nessential. \n\n\\end{itemize}\n As the final summary: insights about Nature's \nGrand Design that can be obtained from a \ncomprehensive and detailed program of CP studies \n\\begin{itemize}\n\\item \nare of fundamental importance, \n\\item \ncannot be obtained any other way and \n\\item \ncannot become obsolete! \n\n\n\\end{itemize} \n \n\n\n\\bigskip \n\n{\\bf Acknowledgments:} \\hspace{.4em} \nThis work was supported by the National Science Foundation under\nthe grant numbers PHY 92-13313 and PHY 96-05080.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nWe consider the dynamics of the Heisenberg-Ising spin-1\/2 chain with anisotropy parameter $\\Delta$, also known as the XXZ spin-1\/2 chain,\non the one-dimensional lattice $\\mathbb{Z}$ with domain wall initial conditions. We start with an initial state of $N$ up-spins at the\nsites $\\{1,2,\\ldots, N\\}$ in a sea of down-spins; and by utilizing ideas from coordinate Bethe Ansatz \\cite{Bethe, Gaudin, Suth, YY} to solve the\nSchr\\\"odinger equation, we find the quantum state $\\Psi_N(t)$ at time $t$ is\n\\[ \\Psi_N(t)=\\sum_X \\psi_N(X,t) e_X, \\]\nwhere the sum is over all $X=\\{x_1 0$ that satisfy the following inequalities $\\max\\{2|\\D|^{-1} , 2(1+2|\\D|)\\} < R < \\max\\{4|\\D|^{-1}, 4(1 + 2|\\D|)\\} < R'\/2$.\n\nWe expect this series expansion to to give rise to a series expansion of a Fredholm determinant in the infinite time limit. In fact, we may deform the contours in the previous formula to the steepest descent to contours in an effort to obtain the infinite time limit by a saddle point analysis. The result is given by our \\textsc{Conjecture 8.1}. Aside from technical details of certain bounds and approximations, there are some terms that we still can't control after the saddle point analysis. Recent results \\cite{BK,CdLV, St1, St1a}, based on numerical, hydrodynamic and analytical arguments are inconclusive in the appropriate scaling, i.e.~$t^{1\/2}$ versus $t^{1\/3}$, for the fluctuations of the one-point function in the infinite time limit. Based on our conjecture, we expect the location of the left-most particle to be at $-2t$ with fluctuations on the order of $t^{1\/3}$ but the limiting distribution is still unclear.\n\n\n\n\n\n\\section{XXZ Quantum Spin-$\\frac{1}{2}$ Hamiltonian}\\label{s:XXZ}\nThe definition of the quantum spin chain Hamiltonian on the infinite lattice $\\mathbb{Z}$ requires some explanation since there is the problem of making sense of infinite tensor products in the construction of a Hilbert space of states. The general construction uses the Gelfand-Naimark-Segal (GNS) construction; but in the case considered here, there is an elementary treatment \n\\cite{NSS} which we now describe.\n\\par\n Let $\\mathcal{H}_{0}=\\mathbb{C}$. For each positive integer $N$ we define\n\\[ \\mathcal{X}_N:=\\left\\{X=\\{x_1,\\ld,x_N\\}\\in\\mathbb{Z}^N: x_1<\\cdots e_X&\\textrm{if}\\>\\>\\>j\\in X=\\{x_1,\\ld,x_N\\},\\\\\n\t\t\t\t\t\t\t\t-\\> e_X & \\textrm{otherwise}.\\end{array}\\right.\\\\ \n\\s_j^{+} e_X&=&\\left\\{\\begin{array}{ll} 0 &\\textrm{if}\\>\\>\\> j\\in \\{x_1,\\ld,x_N\\},\\\\\ne_{X^{+}} & \\textrm{where}\\>\\>\\> X^+=\\{x_1,\\ld,x_k, j, x_{k+1},\\ld, x_N\\}, \\> x_k\\>\\> j\\notin X=\\{x_1,\\ld,x_N\\}\\\\\ne_{X^{-}}& \\textrm{where}\\>\\>\\>X^{-}=\\{x_1,\\ld,x_{k-1},x_{k+1},\\ld,x_N\\}, \\>\\>\\> j=x_k\\end{array}\\right.\\end{eqnarray}\n\nIn words, $\\s_j^{+}:\\mathcal{H}_N\\rightarrow\\mathcal{H}_{N+1}$ acts as the identity except at the site $j$ where\nit takes $\\downarrow \\leadsto \\uparrow$ and annihilates a $\\uparrow$ state. Similarly, $\\s^{-}_j:\\mathcal{H}_{N}\\rightarrow\\mathcal{H}_{N-1}$ acts as the identity except at the site $j$\nwhere it takes $\\uparrow\\leadsto \\downarrow$ and annihilates a $\\downarrow$ state. By definition \n$\\s_j^3\\Omega=-\\Omega$, $\\s_j^{-}\\Omega=0$ and $\\s_j^{+}\\Omega=e_{\\{j\\}}$. We also recall the \nPauli operators $\\s_j^1=\\s_j^{+}+\\s_j^{-}$ and $\\s_j^2=-i\\s_j^{+}+i\\s_j^{-}$.\nDefine\n \\[ h_{j,j+1}=\\frac{1}{2}\\left(\\s_j^1\\s_{j+1}^1\n+\\s_{j}^2\\s_{j+1}^2+\\D(\\s_j^3\\s_j^3-1)\\right) =\\s_j^{+}\\s_{j+1}^{-}+\\s_j^{-}\\s_{j+1}^{+}+\\frac{\\D}{2}(\\s_j^3\\s_{j+1}^3-1)\\]\nand\n\\begin{equation} H_{XXZ}=\\sum_{j\\in\\mathbb{Z}} h_{j,j+1} \\label{XXZ}.\\end{equation}\nThe operator $H_{XXZ}$ is the \\textit{Heisenberg-Ising spin-$\\frac{1}{2}$ chain Hamiltonian}; or more briefly, the $XXZ$ spin Hamiltonian.\nIt's clear from the above definitions that $H_{XXZ}:\\mathcal{H}_N\\rightarrow\\mathcal{H}_N$. Since the number of particles is conserved under the dynamics of $H_{XXZ}$, we can work in a sector $\\mathcal{H}_N$.\n\\par\nA state $\\Psi_N=\\Psi_N(t)\\in\\mathcal{H}_N$ can be represented by\n\\begin{equation} \\Psi_N(t)=\\sum_{X\\in\\mathcal{X}_N} \\psi_N(X,t) e_X.\\label{BigPsi}\\end{equation}\nThe initial condition is $\\Psi(0)=e_Y$, $Y=\\{y_1,\\ld, y_N\\}\\in\\mathcal{X}_N$, so that $\\psi_N(X;0)=\\dl_{X,Y}$. The dynamics is determined by the Schr\\\"odinger equation\n\\begin{equation} \\textrm{i}\\, \\frac{\\pl\\Psi_N}{\\pl t}=H_{XXZ}\\Psi_N.\\label{SchroEqn}\\end{equation}\nThe Hamiltonian $H_{XXZ}$ is self-adjoint and so by Stone's theorem there\nexists a unitary operator \\newline $U=\\exp(-\\textrm{i} t H_{XXZ})$ such that $\\Psi_N(t)=U(t)\\Psi_N(0)$. We have\n\\[ \\langle \\Psi_N(t),\\Psi_N(t)\\rangle =\\sum_{X\\in \\mathcal{X}_N} \\left\\vert \\psi_N(X;t)\\right\\vert^2=1.\\]\n\n \n\n\nThe goal is to describe the dynamics $\\Psi_{DW}(t)$ starting from the \\textit{domain wall} (DW) initial state \n\\[ e_{\\mathbb{N}}=\\vert\\cd \\downarrow\\da\\underset{0}{\\downarrow}\\underset{1}{\\uparrow}\\uparrow\\ua\\cd\\rangle.\\]\nOne immediately sees the difficulty in that $e_{\\mathbb{N}}$ is not an element of $\\mathcal{H}_N$ for any \n$N$.\\footnote{Presumably,\none could construct a domain wall Hilbert space $\\mathcal{H}_{DW}$ by replacing the state $\\Omega$ by\n$e_{\\mathbb{N}}$. Unfortunately, we do not know how to proceed with a Bethe Ansatz solution in this space.} If\n$X_m(t)$ denotes the position of the $m$th particle on the left, we define\n\\[ \\mathbb{P}_{\\mathbb{N}}(X_m(t)=x)=\\lim_{N\\rightarrow\\iy} \\mathbb{P}_{\\{1,\\ldots,N\\}}(X_m(t)=x).\\]\n\n\\section{Bethe Ansatz Solution $\\Psi_N(t)$}\\label{s:bethe_ansatz}\nThis section closely follows \\cite{TW1, YY}. We first note that \n\\begin{eqnarray}\nh_{j,j+1}\\lvert \\cd \\underset{j}{\\uparrow}\\underset{j+1}{\\uparrow}\\cd\\rangle&=&0,\\label{h1}\\\\\nh_{j,j+1}\\lvert \\cd \\underset{j}{\\downarrow}\\underset{j+1}{\\downarrow}\\cd\\rangle&=&0,\\label{h2}\\\\\nh_{j,j+1}\\lvert \\cd \\underset{j}{\\uparrow}\\underset{j+1}{\\downarrow}\\cd\\rangle&=&\n-\\D \\lvert \\cd \\underset{j}{\\uparrow}\\underset{j+1}{\\downarrow}\\cd\\rangle +\n\\lvert \\cd\\underset{j}{\\downarrow}\\underset{j+1}{\\uparrow}\\cd\\rangle,\\label{h3}\\\\\n h_{j,j+1}\\lvert \\cd \\underset{j}{\\downarrow}\\underset{j+1}{\\uparrow}\\cd\\rangle&=&\n-\\D \\lvert \\cd \\underset{j}{\\downarrow}\\underset{j+1}{\\uparrow}\\cd\\rangle +\n\\lvert \\cd\\underset{j}{\\uparrow}\\underset{j+1}{\\downarrow}\\cd\\rangle.\\label{h4}\n\\end{eqnarray}\n\n\\subsection{$N=1$}\nLet $\\Psi_1(t)=\\sum_{x_1}\\psi_1(x_1;t) e_{\\{x_1\\}}$, then\n\\begin{eqnarray*} H_{XXZ}\\Psi_1(t)&=&\\sum_{j} h_{j,j+1}\\sum_{x_1}\\psi_1(x_1;t) e_{\\{x_1\\}}\\\\\n&=& \\sum_j\\psi_1(j;t) h_{j,j+1} e_{\\{j\\}}+\\psi_1(j+1;t)h_{j,j+1}e_{\\{j+1\\}}\\\\\n&=& \\sum_j \\psi_1(j;t)\\left[ -\\D e_{\\{j\\}}+e_{\\{j+1\\}}\\right]+\\psi_1(j+1;t)\n\\left[-\\D e_{\\{j+1\\}}+e_{\\{j\\}}\\right]\\\\\n&=&\\sum_j \\left[ -2\\D\\psi_1(j;t)+\\psi_1(j-1;t)+\\psi_1(j+1;t)\\right] e_{\\{j\\}}\n\\end{eqnarray*}\nThus the coordinates $\\psi_1(x;t)$ must satisfy\n\\[ i\\frac{\\pl\\psi_1(x;t)}{\\pl t}= \\psi_1(x-1;t)+\\psi_1(x+1;t)-2\\D \\psi_1(x;t),\\>\\>x\\in\\mathbb{Z}, t\\ge 0,\\]\nwith initial condition\n\\[ \\psi_1(x;0)=\\dl_{x,y}.\\]\nThe solution is\n\\begin{equation} \\psi_1(x;t)=\\frac{1}{2\\pi i}\\,\\int_{\\mathcal{C}_r}\\xi^{x-y-1} e^{-i t\\ve(\\xi)}d\\xi\\, =e^{2i\\D t} (-i)^{x-y} \\, J_{x-y}(2t)\\label{psi1}\\end{equation}\nwhere\n\\[ \\ve(\\xi)=\\xi+\\frac{1}{\\xi}-2\\D\\]\nand $J_\\nu(z)$ is the Bessel function of order $\\nu$.\n\\par\nIt is obvious probabilistically, and is easily verified analytically, that\n\\[ \\sum_{x\\in\\mathbb{Z}} \\vert \\psi_1(x;t)\\vert^2 =1 \\]\nfor all $y\\in\\mathbb{Z}$. We also have (setting $y=0$)\n\\[ \\sum_{x\\in \\mathbb{Z}} x \\vert\\psi_1(x;t)\\vert^2=0,\\>\\> \\sum_{x\\in\\mathbb{Z}} x^2\\vert\\psi_1(x;t)\\vert^2=t,\n\\>\\>\\textrm{and}\\>\\>\\sum_{x\\in\\mathbb{Z}} x^4\\vert\\psi_1(x;t)\\vert^2=t^2+3 t^4.\n\\]\n\\subsection{$N=2$}\nFor $N=2$ we set\n\\[ \\Psi_2(t)=\\sum_{x_1x_1+1$ the action of $H_{XXZ}$ on $\\Psi_2$ is the same as above in each coordinate $x_i$:\n\\begin{equation} i \\frac{\\pl \\psi_2(x_1,x_2;t)}{\\pl t}=\\psi_2(x_1-1,x_2;t)+\\psi_2(x_1+1;x_2;t)+\\psi_2(x_1,x_2-1;t)+\\psi_2(x_1,x_2+1)-4\\D\\psi_2(x_1,x_2;t).\n\\label{n2a}\\end{equation}\nIf $x_2=x_1+1$ then due to (\\ref{h1}) and (\\ref{h2}) there are terms missing with the result that\n\\begin{equation} i \\frac{\\pl \\psi_2(x_1,x_2;t)}{\\pl t}=\\psi_2(x_1-1,x_2;t)+\\psi_2(x_1,x_2+1)-2\\D\\psi_2(x_1,x_2;t).\\label{n2b}\\end{equation}\nWe now require (\\ref{n2a}) to hold for all $(x_1,x_2)\\in\\mathbb{Z}^2$ and to satisfy the boundary condition\n\\[ \\psi_2(x_1,x_1;t)+\\psi_2(x_1+1,x_1+1;t)-2\\D\\psi_2(x_1,x_1+1;t)=0,\\>\\> x_1\\in\\mathbb{Z}.\\]\nIf this last boundary condition is satisfied then in region $\\mathcal{X}_2$ equation (\\ref{n2b}) is satisfied.\n\n\nDefine \\cite{YY} (the Yang-Yang $S$-matrix)\n\\begin{equation} S_{21}(\\xi_2,\\xi_1)=-\\frac{1+\\xi_1\\xi_2-2\\D\\xi_2}{1+\\xi_1\\xi_2-2\\D\\xi_1},\\quad \\xi_1,\\xi_2\\in\\mathbb{C}.\\label{YY}\\end{equation}\nWith this choice of $S$,\n\\[ \\left\\{ \\xi_1^{x_1-y_1-1}\\xi_2^{x_2-y_2-1}+S_{21}(\\xi_2,\\xi_1) \\xi_2^{x_1-y_2-1}\n\\xi_1^{x_2-y_1-1}\\right\\} e^{-it(\\ve(\\xi_1)+\\ve(\\xi_2))}\\]\nsatisfies (\\ref{n2a}) and the boundary condition (\\ref{n2b}); however it does not satisfy the initial condition.\nWe take\\footnote{ Here and later all differentials $d\\xi$ and $d\\zeta$ incorporate the factor $(2\\pi \\textrm{i})^{-1}$.} ``linear combinations''\n\\begin{equation} \\psi_2(x_1,x_2;t)=\\int_{\\mathcal{C}_r}\\int_{\\mathcal{C}_r}\\left\\{ \\xi_1^{x_1-y_1-1}\\xi_2^{x_2-y_2-1}+S_{21}(\\xi_2,\\xi_1) \\xi_2^{x_1-y_2-1}\n\\xi_1^{x_2-y_1-1}\\right\\} e^{-it(\\ve(\\xi_1)+\\ve(\\xi_2))}\\, d\\xi_1 d\\xi_2.\\label{psi2}\\end{equation}\nA residue calculation shows that in the physical region\\footnote{$x_1\\>\\t=\\frac{p}{q},\\>\\>2\\D=\\frac{1}{\\sqrt{pq}},\\>\\>\nS^{XXZ}_{\\b\\a}(\\xi_{\\b},\\xi_{\\a})=S_{\\b\\a}^{ASEP}(\\xi_{\\b}',\\xi_{\\a}'),\\>\\>\n\\ve^{XXZ}(\\xi)=\\frac{1}{\\sqrt{pq}}\\ve^{ASEP}(\\xi').\\]\n\\par\nThus given the ASEP result \\cite{TW1, TW3} and the above identifications, we have\n\\par\n\\textsc{Theorem 1.} \nFor $\\s\\in\\mathcal{S}_N$, define\n\\begin{equation} A_\\s(\\xi)=\\prod\\left\\{S_{\\b\\a}(\\xi_\\b,\\xi_\\a): \\{\\b,\\a\\}\\>\\> \\textrm{is an inversion in}\\>\\> \\s\\right\\}, \\label{e:A_coeff}\\end{equation}\nthen the solution to (\\ref{SchroEqn}) satisfying the initial condition $\\psi_N(X;0)=\\delta_{X,Y}$ is\n\\begin{equation} \\psi_N(X;t)=\\sum_{\\s\\in\\mathcal{S}_N}\\int_{\\mathcal{C}_r}\\cd\\int_{\\mathcal{C}_r} A_\\s(\\xi) \\prod_i\\xi_{\\s(i)}^{x_i} \\prod_i \\left(\\xi_i^{-y_{i}-1}\\,\n\\textrm{e}^{-\\textrm{i} t \\ve(\\xi_i)}\\right)\n d\\xi_1\\cd d\\xi_N\\label{psi_N}\\end{equation}\nwhere $\\mathcal{C}_r$ is a circle centered at zero with radius $r$ so small that all the poles of $A_\\s$ lie outside of\n$\\mathcal{C}_r$.\n\nAdditionally, we have a contour integral formula with large contours instead of small contours as above in \\textsc{Theorem 1}. Below, we will use a combination of the small and large contour formulas.\n\n\\textsc{Theorem 1a.} \nFor $\\s\\in\\mathcal{S}_N$, define\n\\[ A_\\s(\\xi)=\\prod\\left\\{S_{\\b\\a}(\\xi_\\b,\\xi_\\a): \\{\\b,\\a\\}\\>\\> \\textrm{is an inversion in}\\>\\> \\s\\right\\},\\]\nthen the solution to (\\ref{SchroEqn}) satisfying the initial condition $\\psi_N(X;0)=\\delta_{X,Y}$ is\n\\begin{equation} \\psi_N(X;t)=\\sum_{\\s\\in\\mathcal{S}_N}\\int_{\\mathcal{C}_R}\\cd\\int_{\\mathcal{C}_R} A_\\s(\\xi) \\prod_i\\xi_{\\s(i)}^{x_i} \\prod_i \\left(\\xi_i^{-y_{i}-1}\\,\n\\textrm{e}^{-\\textrm{i} t \\ve(\\xi_i)}\\right)\n d\\xi_1\\cd d\\xi_N\\label{psi_N_large}\\end{equation}\nwhere $\\mathcal{C}_R$ is a circle centered at zero with radius $R$ so large that all the poles of $A_\\s$ lie inside of $\\mathcal{C}_R$.\n\nThe proof of this statement is an adaptation of the arguments in \\cite{TW1} that give the proof of \\textsc{Theorem 1}. For completeness, we give the proof of \\textsc{Theorem 1a} in \\Cref{s:appendix_a}.\n\n\n\n \\section{Probability $\\mathcal{P}_Y(x,m;t)$}\\label{s:one_point}\nIf the initial state is $e_Y\\in\\mathcal{H}_N$, $Y\\in\\mathcal{X}_N$, then\nat time $t$ the system is in state $\\Psi_N(t)=\\sum_{X\\in\\mathcal{X}_N} \\psi_N(X;t) e_X$ where\n$\\psi_N(X;t)$ is given by (\\ref{psi_N}) or (\\ref{psi_N_large}). The\nquantity\n\\[ \\left\\vert\\langle e_X,\\Psi_N(t)\\rangle\\right\\vert^2=\\left\\vert\\psi_N (X;t)\\right\\vert^2 ,\\>\\> X\\in\\mathcal{X}_N,\\]\nis the probability that the system is in state $e_X$ at time $t$. \n\\par\nDenote by $\\mathcal{P}_Y(x,m;t)$ the probability that at time $t$ the state has the $m$th particle from the left at position $x$\ngiven initially the state is $Y$. \nLet $X=\\{x_1,x_2,\\ldots, x_N\\}\\in\\mathcal{X}_N$, $1\\le m\\le N$, and define the projection operator\n\\begin{equation} P_{x,m}e_X=\\left\\{\\begin{array}{cc}e_X&\\textrm{if}\\>\\> x_m=x,\\\\\n0& \\textrm{otherwise}.\\end{array}\\right.\\end{equation} \nThen the outcome of the measurement yielding ``the $m$th spin from the left is at position $x$ at\ntime $t$'' is that the system is now in state\n\\[\\Psi_N(x,m;t):= P_{x,m}\\Psi_N(t)=\\sum_{\\substack{X\\in\\mathcal{X}_N \\\\xi_m=x}} \\psi_N(X;t) e_X.\\]\nThus the probability of this outcome is\n\\begin{equation} \\mathcal{P}_Y(x,m;t):=\\langle\\Psi_N(x,m;t),\\Psi_N(x,m;t)\\rangle=\n\\sum_{\\substack{X\\in\\mathcal{X}_N \\\\xi_m=x}}\\left\\vert\\psi_N(X;t)\\right\\vert^2. \\label{1ptProb}\\end{equation}\n\n\\subsection{Distribution of left-most particle}\nWe now restrict to the case $m=1$, i.e.\\ $\\mathcal{P}_Y(x,1;t)$.\nLet\n\\[ x_1=x,\\> x_2=x+v_1,\\>\\ldots, x_N=x+v_1+v_2+\\cdots+v_{N-1},\\> v_i\\ge 1,\\]\nand note that $\\overline{\\Psi_N(x;t)}=\\Psi_N(x;-t)$. Then, using (\\ref{psi_N}) for $\\Psi(x;t)$ and (\\ref{psi_N_large}) for $\\Psi(x;-t)$ with $R \\, r < 1$, followed by performing the\ngeometric sums (since $R\\,r <1$, the summations may be brought inside) \n\\begin{eqnarray*}\n\\mathcal{P}_Y(x,1;t)&=&\\sum_{\\substack{X\\in\\mathcal{X}_N \\\\xi_1=x}} \\psi_N(X;t)\\psi_N(X;-t)\\\\\n&=&\\sum_{\\s,\\mu\\in \\mathcal{S}_N}\\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r} \\sum_{v_i\\ge 1} A_\\s(\\xi) A_\\mu(\\zeta)\\,\n(\\xi_{\\s(2)}\\zeta_{\\mu(2)})^{v_1} (\\xi_{\\s(3)}\\zeta_{\\mu(3)})^{v_1+v_2}\\cdots \n(\\xi_{\\s(N)}\\zeta_{\\mu(N)})^{v_1+\\cdots+v_{N-1}}\\\\\n&&\\times\n\\prod_j (\\xi_j\\zeta_j)^{x-y_j-1} \\textrm{e}^{-\\textrm{i} t(\\ve(\\xi_j)-\\ve(\\zeta_j))}\\,d\\zeta_1\\cdots d\\zeta_N d\\xi_1\\cdots d\\xi_N\\\\\n&\\hspace{-15ex} =&\\hspace{-10ex}\\sum_{\\s,\\mu\\in \\mathcal{S}_N}\\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r} A_\\s(\\xi) A_\\mu(\\zeta)\n\\frac{\\xi_{\\s(2)}\\zeta_{\\mu(2)} \\xi_{\\s(3)}^2\\zeta_{\\mu(3)}^2\\cdots\n \\xi_{\\s(N)}^{N-1} \\zeta_{\\mu(N)}^{N-1}}{(1-\\xi_{\\s(2)}\\zeta_{\\mu(2)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\n(1-\\xi_{\\s(3)}\\zeta_{\\mu(3)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\\cdots (1-\\xi_{\\s(N)}\\zeta_{\\mu(N)})}\\\\\n&&\\times\\prod_j (\\xi_j\\zeta_j)^{x-y_j-1} \\textrm{e}^{-\\textrm{i} t(\\ve(\\xi_j)-\\ve(\\zeta_j))}\\,d\\zeta_1\\cdots d\\zeta_N d\\xi_1\\cdots d\\xi_N \n\\end{eqnarray*}\nIn the formulas above, we have $2N$ contour integrals with the contour $\\mathcal{C}_r$ for the first $N$ contours and the contours $\\mathcal{C}_R$ for the following $N$ contours.\nNow, at the analogous step in ASEP, an identity\\footnote{See (1.6) in \\cite{TW1}.} was\nderived that simplified the sum over $\\mathcal{S}_N$\n resulting in a single multidimensional integral.\\footnote{See Theorem 3.1 in \\cite{TW1}.}\n Now we have a \\textit{double sum} over $\\mathcal{S}_N$ and we need a new identity. Fortunately\n such an identity has been discovered by Cantini, Colomo, and Pronko \\cite{CCP}.\n Let\n \\begin{equation} d(x,y):=\\frac{1}{(1-x \\,y)(x+y -2\\Delta\\, x\\, y)}\\>\\>\\textrm{and}\\>\\> \nD_N(\\xi,\\zeta)=\\det\\left(d(\\xi_i,\\zeta_j)\\vert_{1\\le i,j\\le N}\\right),\\label{IK}\\end{equation}\nthen\n\\begin{eqnarray}\n\\sum_{\\s,\\mu\\in\\mathcal{S}_N} A_\\s(\\xi) A_{\\mu}(\\zeta) \\frac{\\xi_{\\s(2)}\\zeta_{\\mu(2)} \\xi_{\\s(3)}^2\\zeta_{\\mu(3)}^2\\cdots\n \\xi_{\\s(N)}^{N-1} \\zeta_{\\mu(N)}^{N-1}}{(1-\\xi_{\\s(2)}\\zeta_{\\mu(2)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\n(1-\\xi_{\\s(3)}\\zeta_{\\mu(3)}\\cdots \\xi_{\\s(N)}\\zeta_{\\mu(N)})\\cdots (1-\\xi_{\\s(N)}\\zeta_{\\mu(N)})}\\nonumber\\\\\n=\\frac{(1-\\prod_j \\xi_j\\zeta_j) \\, \n\\prod_{i,j=1}^N(\\xi_i+\\zeta_j-2\\Delta \\xi_i\\zeta_j)}{\n\\prod_{iD_N(\\xi,\\zeta)&&\\label{1ptIden}\n\\end{eqnarray}\nRemarks:\\vspace{-4ex}\n\\begin{itemize}\n\\item The identity (\\ref{1ptIden}) is Proposition 6 of \\cite{CCP} (with a change of notation). The identity (\\ref{1ptIden}) also appears in a more general setting of (spin) Hall-Littlewood functions in \\cite{P, WZJ}, which specializes to the ASEP case as shown in Corollary 7.1 in \\cite{P}.\n\\item\n In Appendix B of \\cite{CCP}, the authors show that\n (\\ref{1ptIden}) reduces to (1.6) of \\cite{TW1} \nin the limit $\\xi_j\\rightarrow \\sqrt{\\frac{q}{p}}\\,\\xi_j$ and $\\zeta_j\\rightarrow \\sqrt{\\frac{p}{q}}$.\n\\item The determinant $D_N(\\xi,\\zeta)$ ``is nothing but the well-known \\textit{Izergin-Korepin\ndeterminant} \\cite{Iz, Ko} in disguise'' \\cite{W}. \n\\end{itemize}\n\\par\nWe thus have\n\\vspace{-2ex}\n\\begin{equation} \\hspace{-5ex}\\mathcal{P}_Y(x,1;t)=\\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r}\\frac{(1-\\prod_j \\xi_j\\zeta_j) \\, \n\\prod_{i,j=1}^N(\\xi_i+\\zeta_j-2\\Delta \\xi_i\\zeta_j)}{\n\\prod_{iD_N(\\xi,\\zeta)\n\\prod_j (\\xi_j\\zeta_j)^{x-y_j-1} \\textrm{e}^{-\\textrm{i} t(\\ve(\\xi_j)-\\ve(\\zeta_j))}\\,d^N\\zeta d^N\\xi\n\\label{detRep}\\end{equation}\nThe factor $(1-\\prod_j \\xi_j\\zeta_j)$ is eliminated if we consider \n\\begin{equation} \\mathcal{F}_N(x,t):=\\mathbb{P}_Y(X_1(t)\\ge x)=\\sum_{n=x}^\\iy\\mathcal{P}_Y(n,1;t)\\label{cFn}\\end{equation}\n\\par\n\nFrom \\cite{CCP}\n\\begin{equation} \\prod_{1\\le j,k\\le N} (\\xi_j+\\zeta_k-2 \\D \\xi_j \\zeta_k)\\,\\cdot\\, D_N(\\xi,\\zeta)\n=\\frac{\\D_N(\\xi)\\D_N(\\zeta)}{\\prod_{j,k} (1-\\xi_j\\zeta_k)} Q_{N}(\\xi,\\zeta) \\label{Wfn} \\end{equation}\nwhere $Q_N$ is a ``polynomial of degree $N-1$ in each variable, separately symmetric under permutations\nof the variables within each set'' \\cite{CCP, W}.\\footnote{For example\n\\begin{eqnarray*} Q_1(\\xi,\\zeta)&=&1,\\\\\nQ_2(\\xi,\\zeta)&=&4 \\Delta ^2 \\zeta _1 \\zeta _2 \\xi _1\n \\xi _2-2 \\Delta \\zeta _1 \\zeta\n _2 \\xi _1-2 \\Delta \\zeta _1\n \\zeta _2 \\xi _2-2 \\Delta \\zeta\n _1 \\xi _1 \\xi _2-2 \\Delta \\zeta\n _2 \\xi _1 \\xi _2+\\zeta _1 \\zeta\n _2 \\xi _1 \\xi _2+\\zeta _1 \\zeta\n _2+\\xi _1 \\xi _2+1,\n \\end{eqnarray*}\n $Q_3$ in expanded form has 459 terms, and $Q_4$ has 60,820 terms.} Here $\\D_N(\\xi)$ is the Vandermonde product $\\prod_{1\\le j\\> \n\\psi_j(\\zeta)=\\zeta^{x-y_j-1}\\textrm{e}^{\\textrm{i} t\\ve(\\zeta)}\\]\nand\n\\begin{equation} K(j,k)=\\frac{\\phi_j(\\xi_j) \\psi_k(\\zeta_k)}{1-\\xi_j\\zeta_k}\\label{K}\\end{equation}\nThus\n\\begin{equation} \n\\mathcal{F}_N(x,t)\\big\\vert_{\\D=0}=\n \\int_{\\mathcal{C}_R}\\cdots\\int_{\\mathcal{C}_r} \\det(K)\\, d^N\\zeta \\,d^N\\xi= \\int_{\\mathcal{C}_r}\\cdots\\int_{\\mathcal{C}_r} \\det(K)\\, d^N\\zeta \\,d^N\\xi\\hspace{5ex}\\label{Kdet}\\end{equation}\n \\par\n For the second identity, we deformed the contours from $\\mathcal{C}_R$ to $\\mathcal{C}_r$ for all the $\\zeta$-variables. When we deform the contours, we don't cross any poles since the poles, given by $1- \\xi_j \\zeta_k=0$, are located outside of the contour $\\mathcal{C}_R$ since we have taken $R\\, r <1$. Additionally, note that the variable $\\xi_j$ appears only in row $j$ and $\\zeta_k$ appears only in column $k$. It follows that the multiple integral\n is gotten by integrating each $K(j,k)$ with respect to $\\xi_j$, $\\zeta_k$.\n Therefore the multiple integral (\\ref{Kdet}) equals the determinant with $j,k$ entry\n \\[ K_N(j,k)=\\int_{\\mathcal{C}_r}\\int_{\\mathcal{C}_r} \\frac{\\phi_j(\\xi)\\psi_k(\\zeta)}{1-\\xi\\zeta}\\,d\\zeta d\\xi \\]\n \\par\n We consider step initial condition, so that $y_j=j$. In preparation\n for taking the limit as $N\\rightarrow\\iy$, we make the replacements\n $j\\rightarrow j+1$, $k\\rightarrow k+1$, so that the indicies run for $0$ to $N-1$ rather than\n $1$ to $N$. Then, in preparation for eventual steepest descent, we make\n the substitutions $\\xi\\rightarrow\\textrm{i}\\,\\xi$, $\\zeta\\rightarrow\\zeta\/\\textrm{i}$. Aside from the factor $\\textrm{e}^{\\textrm{i}\\pi(j-k)\/2}$, which will not\n affect the determinant, the kernel becomes\n \\[ L_N(j,k)=\\int_{\\mathcal{C}_r}\\int_{\\mathcal{C}_r}\\frac{\\xi^{x-j-2}\\,\\zeta^{x-k-2}}{1-\\xi\\zeta} \n \\textrm{e}^{t (\\theta(\\xi)+\\theta(\\zeta))}\\, d\\zeta d\\xi,\\]\n where we have set $\\theta(\\xi)=\\xi-1\/\\xi$. We write the above as\n \\[ \\sum_{\\ell=0}^\\iy\\int_{\\mathcal{C}_r}\\int_{\\mathcal{C}_r} \\xi^{x-j+\\ell-2}\\zeta^{x-k+\\ell-2}\\,\\textrm{e}^{t(\\theta(\\xi)+\\theta(\\zeta))}\\,d\\zeta d\\xi.\\]\n \\par\n We may take all integrations over the unit circle $\\mathcal{C}_1$ and in the $\\zeta$-integral make the substitution\n $\\zeta\\ra1\/\\zeta$. We obtain\n \\[ L_N(j,k)=\\sum_{\\ell=0}^\\iy \\int_{\\mathcal{C}_1}\\int_{\\mathcal{C}_1} \\xi^{x-j+\\ell-2}\\zeta^{-x+k-\\ell}\\,\\textrm{e}^{t(\\theta(\\xi)-\\theta(\\zeta))}\\, d\\zeta d\\xi.\\]\n In Toeplitz terms this is the operator\n \\[ P_N T(a) T(a^{-1}) P_N,\\]\n where $P_N$ is the projection from $\\ell^2(\\mathbb{Z}^+)$\\footnote{$\\mathbb{Z}^+$ denotes\n the set of nonnegative integers.} to $\\ell^2([0,\\ldots,N-1])$ and where $a$ is the symbol\n \\[ a(\\xi)=\\xi^{x-1} \\textrm{e}^{t\\,\\theta(\\xi)}.\\]\n It it known (see, e.g.\\ \\S5.1 in \\cite{BS}) that $T(a) T(a^{-1})$ is of the form $I$+trace class and so $\\det(K_N)$ has\n the limit $\\det(T(a) T(a^{-1}))$ on $\\ell^2(\\mathbb{Z}^+)$.\\footnote{One can show that for $x>1$ the determinant\n of the product is zero.}\n \\par\n By a well-known identity, $T(a) T(a^{-1})=I-H(a) H(\\tilde{a}^{-1})$, where $H(a)$ denotes\n the Hankel operator and $\\tilde{a}(\\xi)=a(\\xi^{-1})$. In this case $\\tilde{a}=a^{-1}$ and the square\n of $H(a)$ has kernel\\footnote{Recall that the $i,j$-entry of $H(f)$ is $f_{i+j+1}=\\int \\xi^{-i-j-2} f(\\xi)\\,d\\xi$.}\n \\[ L(j,k)=\\sum_{\\ell=0}^\\iy \\int_{\\mathcal{C}_1}\\int_{\\mathcal{C}_1} \\xi^{x-j-\\ell-3} \\zeta^{x-k-\\ell-3} \\,\\textrm{e}^{t(\\theta(\\xi)+\\theta(\\zeta))}\n \\, d\\zeta d\\xi,\\]\n and we are interested in $\\det(I-L)$. The substitutions $\\xi\\ra1\/\\xi$, $\\zeta\\rightarrow 1\/\\zeta$ give\n \\begin{equation} L(j,k)=\\sum_{\\ell=0}^\\iy \\int_{\\mathcal{C}_1}\\int_{\\mathcal{C}_1}\\xi^{-x+j+\\ell+1} \\zeta^{-x+k+\\ell+1}\\,\\textrm{e}^{-t(\\theta(\\xi)+\\theta(\\zeta))}\\, d\\zeta d\\xi.\\label{Lexpand}\\end{equation}\n If we take our integrals over $\\mathcal{C}_r$ and sum we obtain\n \\begin{equation} L(j,k)=\\int_{\\mathcal{C}_r}\\int_{\\mathcal{C}_r}\\frac{\\xi^{-x+j+1}\\zeta^{-x+k+1}\\,\\textrm{e}^{-t(\\theta(\\xi)+\\theta(\\zeta))}}{1-\\xi\\zeta}\\ d\\zeta d\\xi\n \\label{L}\\end{equation}\n The kernel $L(j,k)$ is known as the \\textit{discrete Bessel kernel} \\cite{Bo} (see also Chapter 8 in \\cite{BDS}) due to the following representation.\n Using the Bessel generating function\n \\[\\exp(t\\theta(\\xi))=\\sum_{n=-\\iy}^\\iy \\xi^n J_n(2t) \\] \n in (\\ref{Lexpand}) and the identity, $\\nu\\neq \\mu$,\n \\begin{equation} \\sum_{n=0}^\\iy J_{\\nu+n}(t) J_{\\mu+n}(t)= \\frac{t}{2(\\nu-\\mu)}\\left[J_{\\nu-1}(t) J_{\\mu}(t)-J_{\\nu}(t) \n J_{\\mu-1}( t)\\right]\\label{besselIden}\\end{equation}\nwe find\n\\[ L(j,k) =t\\, \\frac{J_{j-x+1}(2t) J_{k-x+2}(2t)-J_{j-x+2}(2t) J_{k-x+1}(2 t)}{j-k}\\]\nFor $j=k$ one lets $\\mu\\rightarrow\\nu$ in (\\ref{besselIden}) to find\n\\[L(j,j)=\\sum_{n=0}^\\iy J_{\\nu+n}(2t)^2 =t\\,\\left[ J_\\nu(2t) \\frac{\\pl J_\\mu}{\\pl\\mu}\\big\\vert_{\\mu=\\nu-1} -J_{\\nu-1}(2t) \n\\frac{\\pl J_\\mu}{\\pl \\mu}\\big\\vert_{\\mu=\\nu-1}\\right],\\>\\nu=-x+j+1.\\] \nFor $x\\le 1$ and domain wall initial condition $Y=\\mathbb{N}$, we have the Toeplitz representation \n\\begin{equation} \\mathbb{P}_{\\mathbb{N}}(X_1(t)\\ge x)\\big\\vert_{\\D=0}=\\det(I-L)_{\\ell^2(\\{1-x, 2-x, \\dots\\})}=\\textrm{e}^{-t^2} \\det\\left(I_{j-k}(2 t)\\right)\n \\Big\\vert_{j,k=0,\\ldots,-x}\n\\label{BOiden}\\end{equation}\nwhere the last equality\\footnote{$I_\\nu(z)$ is the modified Bessel function of order $\\nu$.} was proved in \\cite{BOk}.\n \\par\n If $\\mathcal{L}(t)$ denotes the length of the longest increasing subsequence of a random permutation of size $\\mathcal{N}$ where\n $\\mathcal{N}$ is a Poisson random variable with parameter $t^2$, then \\cite{BDJ, BDS, G}\n \\[ \\mathbb{P}(\\mathcal{L}(t)\\le n) =\\textrm{e}^{-t^2} \\det(I_{j-k}(2t))_{j,k=0,\\ldots,n-1}\\]\n \n\\textsc{Theorem 3.} For $x \\leq 1$ and domain wall initial conditions $Y = \\mathbb{N}$, we have\n\\begin{equation}\n\\mathbb{P}_{\\mathbb{N}}(X_1(t)\\ge x)\\big\\vert_{\\D=0}=\\mathbb{P}(\\mathcal{L}(t)\\le 1-x)\n\\end{equation}\nwhere $\\mathcal{L}(t)$ denotes the length of the longest increasing subsequence of a random permutation of size $\\mathcal{N}$ so that $\\mathcal{N}$ is a Poisson random variable with parameter $t^2$.\n\n\\subsection{Asymptotics}\nFrom the classic work of Baik, Deift, and Johnasson \\cite{BDJ} (see also Chapter 9 in \\cite{BDS}), we know that the limiting distribution of\n$\\mathcal{L}(t)$ is\n\\begin{equation} \\lim_{t\\rightarrow\\iy}\\mathbb{P}\\left(\\frac{\\mathcal{L}(t)-2 t}{t^{1\/3}}\\le x\\right) =F_2(x)\\end{equation}\nwhere $F_2$ is the $\\beta=2$ TW distribution \\cite{TW0a, TW0b}.\nIn the present problem, $\\D=0$, we can therefore conclude \nthat the left-most particle for domain wall initial condition $Y=\\mathbb{N}$ has the limiting distribution\n\\begin{equation} \\lim_{t\\rightarrow\\iy} \\mathbb{P}\\left(\\frac{X_1(t)+2t}{t^{1\/3}}\\ge -y\\right)= F_2(y).\\label{TWF2}\\end{equation}\n\n\n\n\\section{Steepest descent curve}\\label{s:steepest_descent}\n\n\\subsection{Spectral functions}\nWe introduce a pair of functions \n\\begin{equation}\\label{e:spectral_function}\n G(\\xi) = x \\log \\xi - i t (\\xi + \\xi^{-1}), \\quad H(\\zeta) = -x \\log \\zeta - i t (\\zeta + \\zeta^{-1}),\n\\end{equation}\nwhich we call the \\emph{spectral functions}. Note that the spectral functions appear in the integrand of the formula for $\\mathcal{F}_N(x,t)$ given by (\\ref{Fprob2}). In particular, we have\n\\begin{equation}\n (\\xi_j \\zeta_j)^{x} e^{- i t (\\varepsilon(\\xi_j) - \\varepsilon(\\zeta_j))} = \\exp \\left \\{ G(\\xi_j) - H(\\zeta_j)\\right\\}.\n\\end{equation}\nIn the following, we will deform the contours in the contour integral formula for $\\mathcal{F}_N$ given by (\\ref{Fprob2}) so that the real part of the difference of the spectral function is negative, $\\mathrm{Re}(G- H) <0$. Thus, making $\\mathcal{F}_N$ suitable for asymptotic analysis.\n\n\\subsection{Critical points}\nThe steepest descent contours in the contour integral formula $\\mathcal{F}_N$ given by (\\ref{Fprob1}) are determined by the critical points of the spectral functions. We have\n\\begin{equation}\n G'(\\xi) = \\frac{- i t \\xi^2 + x \\xi + i t}{\\xi^2}, \\quad H'(\\zeta) = \\frac{-i t \\zeta^2 - x \\zeta + it }{\\zeta^2}.\n\\end{equation}\nso that the critical points are given by\n\\begin{equation}\n \\xi = \\frac{x \\pm \\sqrt{x^2 -4 t^2}}{2 i t},\\quad \\zeta = \\frac{-x \\pm \\sqrt{x^2 -4 t^2}}{2 i t}.\n\\end{equation}\nNote that each function, $G$ and $H$, has a double critical point when $x = \\pm 2 t$ and the critical point are\n\\begin{equation}\n \\xi_0 = \\begin{cases} - i , \\quad x= 2 t \\\\ i , \\quad x = -2t \\end{cases}, \\quad \\zeta_0 = \\begin{cases} i , \\quad x= 2 t \\\\ -i , \\quad x = -2t \\end{cases},\n\\end{equation}\nrespectively. Physically, we expect the point $x=-2t$ to correspond to the left-edge of the up-spins and the point $x= 2t$ to correspond to the right-edge of the up-spins. Thus, we restrict our attention to the critical point given by $x= -2t$ and take $(\\xi_0, \\zeta_0)= (i, -i)$.\n\n\\subsection{Steepest descent curve: local}\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[scale=0.3]{local_SD_curve.pdf}\n \\caption{The curves show the directions of large increase of magnitude for the real part of $G$ (i.e.~steepest descent) near the critical point, which is centered to be at the origin. We label the curves so that the sign of $Re\\, G$ is clear.}\n \\label{f:local_SD}\n\\end{figure}\n\nThe steepest descent curve depends locally on the value of the third derivative for the double critical points. In particular, we have\n\\begin{equation}\n G^{(3)}(- 2t) = 2 i t, \\quad H^{(3)}(- 2 t) = 2 i t.\n\\end{equation}\nThis means that the steepest descent curves for both spectral functions are locally the same. That is,\n\\begin{equation}\\label{e:Taylor}\n G(\\xi) - G(\\xi_0) = \\frac{i t}{3} (\\xi - \\xi_0)^3 + \\mathcal{O}((\\xi-\\xi_0)^4), \\quad H(\\zeta) - H(\\zeta_0) = \\frac{i t}{3} (\\zeta - \\zeta_0)^3 + \\mathcal{O}((\\zeta-\\zeta_0)^4) \n\\end{equation}\nA diagram for the local steepest descent curve is give in Figure \\ref{f:local_SD}.\n\n\n\n\\subsection{Steepest descent curve: global}\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[h!]{0.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{global_SD_Curve2a.png}\n \\caption{}\n \n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[h!]{0.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{global_SD_Curve2b.png}\n \\caption{}\n \n \\end{subfigure}\n \\caption{The level curves for the spectral functions, $G$ and $H$, with $\\xi_0 = i$, $\\zeta_0=-i$, and $x = -2 t$. The component corresponding to the steepest descent path is denoted by red arrows with the direction along the steepest descent.}\n \\label{f:global_SD_2}\n\\end{figure}\n\nGlobally, we obtain the steepest descent curve by an implicit equation through the Cauchy-Riemann equations for holomorphic functions. In short, we have that the direction of greatest change in the real part of a holomorphic function is the same direction of no change in the imaginary part of a the same holomorphic function (away from singular points). Therefore, the steepest descent curve is given by the level curves of the imaginary part of the spectral functions:\n\\begin{equation}\n \\{\\xi \\in \\mathbb{C} : Im\\, G(\\xi) = Im\\, G(\\xi_0)\\}, \\quad \\{\\zeta \\in \\mathbb{C} : Im\\, H(\\zeta) = Im\\, H(\\zeta_0)\\}\n\\end{equation}\nwith $(\\xi_0 , \\zeta_0) = ( i, -i)$. The level curves will have multiple components, but we are only interested in the components that (i) are closed curves in $\\mathbb{C} \\cup \\{ \\infty\\}$ and (ii) $Re \\, G(\\xi) - G(\\xi_0) \\leq 0$ and $Re \\, H(\\xi) - H(\\xi_0) \\geq 0$.\\par\n\nThe steepest descent curves are components of the curves given by\n\\begin{equation}\\label{e:steepest_descent}\n\\begin{split}\n &2 \\tan^{-1}\\left(\\frac{y}{x} \\right) + \\frac{x^3 + x y^2 +x}{x^2 +y^2} = \\begin{cases} \\pi, &\\quad x>0 \\\\ -\\pi, &\\quad x<0 \\end{cases},\\\\\n & 2 \\tan^{-1}\\left(\\frac{v}{u} \\right) - \\frac{u^3 + u v^2 +u}{u^2 +v^2} = \\begin{cases} -\\pi, &\\quad u>0 \\\\ \\pi, &\\quad u<0 \\end{cases}\n\\end{split}\n\\end{equation}\nwith $\\xi = x + i y$ and $\\zeta = u + i v$. We plot these curves using \\emph{Mathematica}, see Figure \\ref{f:global_SD_2}. The components of the level curves corresponding to the steepest descent path is determined by the local behaviour determined in Figure \\ref{f:local_SD}.\n\n\n\\subsection{Steep descent curve}\n\nWe want a friendly version of the steepest descent curves given implicitly by (\\ref{e:steepest_descent}) or, rather, a more explicit version. We introduce the \\emph{steep descent contours} given by three segments on three regions: in the region near the critical points, we take straight lines coming emanating from the critical point at angles $\\pm\\pi\/6$ and $\\pm5\\pi\/6$; in an intermediate region, we take horizontal lines emanating from the end points of the straight lines in region near the critical point; in the region far away from the critical point, we take a segment of the circle $\\mathcal{C}_R$ that connects with the horizontal lines. We use these contours so that we may explicitly determine the location of the poles when we deform to these \\emph{steep descent contours}. Although these contours don't follow the path of steepest descent for the real part of the spectral function, we show below that we still have the main property that $Re\\,\\{ G(\\xi) - G(\\xi_0)\\} \\leq 0$ and $Re \\, \\{H(\\zeta) - H(\\zeta_0)\\} \\geq0$ along these steep descent contours.\n\nWe now give a precise definition for the steep descent contours. We give a piece-wise description based on the proximity to the critical points. Let $\\mathcal{B}(z, r)$ be a ball centered at $z\\in \\mathbb{C}$ of radius $r>0$ and $\\mathcal{B}(z,r)^c$ be its complement. Then, we take the components \n\\begin{equation}\\label{e:contour_parts}\n \\begin{split}\n \\Gamma_{\\pm}^{(1)}&=\\{\\pm i + x e^{ \\pm\\pi i\/6} \\mid 0 \\leq x \\} \\cap \\mathcal{B}(\\pm i, 1),\\\\\n \\Gamma_{\\pm}^{(2)}&=\\{\\pm i + x e^{ \\pm 5\\pi i\/6} \\mid 0 \\leq x \\} \\cap \\mathcal{B}(\\pm i, 1)\\\\\n \\Gamma_{\\pm}^{(3)}&=\\{\\pm i + e^{\\pm \\pi i\/6} +x \\mid 0 \\leq x \\} \\cap \\mathcal{B}(\\pm i, 1)^c \\cap \\mathcal{B}(0, R_{\\pm}),\\\\\n \\Gamma_{\\pm}^{(4)}&=\\{\\pm i + 1 e^{\\pm 5 \\pi i\/6} -x \\mid 0 \\leq x \\} \\cap \\mathcal{B}(\\pm i, 1)^c \\cap \\mathcal{B}(0, R_{\\pm})\\\\\n \\Gamma_{\\pm}^{(5)}&= \\mathcal{C}_R \\cap \\{z\\in \\mathbb{C} \\mid Im \\, \\{z\\} \\leq (\\pm1)Im\\, \\{\\pm i + e ^{\\pm \\pi i \/6}\\} \\}\n \\end{split},\n\\end{equation}\nwith radii $R_{\\pm} > \\sqrt{3}$. The bound on the radii is chosen so that the horizontal segments of the contours are non-trivial. Then, the steep descent contours are given by\n\\begin{equation}\\label{e:steep_contours}\n \\Gamma_{k} = \\Gamma_k^{(1)} \\cup \\Gamma_k^{(2)} \\cup \\Gamma_k^{(3)} \\cup \\Gamma_k^{(4)} \\cup \\Gamma_k^{(5)}\n\\end{equation}\nfor $k=\\pm$. See Figure \\ref{f:global_SD_3}.\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[h]{0.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{SDConttour4a.pdf}\n \\caption{}\n \n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[h]{0.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{SDContour3a.pdf}\n \\caption{}\n \n \\end{subfigure}\n \\caption{The components of the $\\Gamma_{\\pm}$ contours.}\n \\label{f:global_SD_3}\n\\end{figure}\n\n\\textsc{Lemma 6.1} Let $x=-2t$ and take the contours $\\Gamma_k$, $k=\\pm$, given by (\\ref{e:contour_parts}) and (\\ref{e:steep_contours}). Additionally, take $t^{-\\alpha} \\leq T \\ll 1$ with $1\/4 < \\alpha < 1\/3$. Then, we have\n\\begin{equation}\\label{e:bound1}\n Re\\,\\{ G(\\xi) - G(\\xi_0)\\} \\leq 0, \\quad Re \\, \\{H(\\zeta) - H(\\zeta_0)\\} \\geq 0 \n\\end{equation}\nif $\\xi \\in \\Gamma_{+}$ and $\\zeta \\in \\Gamma_{-}$. Moreover, if $\\xi \\in \\Gamma_{+} \\cap \\mathcal{B}(\\textrm{i}, t^{-\\alpha})^c$ and $\\zeta \\in \\Gamma_{-} \\cap \\mathcal{B}(-\\textrm{i}, t^{-\\alpha})^c$, we have\n\\begin{equation}\\label{e:bound2}\n Re\\,\\{ G(\\xi) - G(\\xi_0)\\} < - c_1(T)\\, t^{1- 3\\alpha}, \\quad Re \\, \\{H(\\zeta) - H(\\zeta_0)\\} > c_2(T)\\, t^{1-3\\alpha}, \n\\end{equation}\nfor some constants $c_1(T), c_2(T)>0$ that depend only on $T$.\n\n\\begin{proof}\nWe prove the bounds by showing that derivative of the real part of the functions are monotone along the different segments of the contours $\\Gamma_{\\pm}$ as parameterized in (\\ref{e:contour_parts}). Since $G(\\xi) - G(\\xi_0) = 0$ for $\\xi = \\xi_0$ and $H(\\zeta) - H(\\zeta_0) = 0$ for $\\zeta = \\zeta_0$, the first bounds (\\ref{e:bound1}) then follow by monotonicity. Moreover, since the real part of the functions are monotone, we establish the bounds (\\ref{e:bound2}) by bounding the real part of the functions on the boundary of the segment $\\Gamma_{\\pm} \\cap \\mathcal{B}(\\pm \\textrm{i}, t^{-\\alpha})$. \n\nThe arguments for both functions are the same, except for some negative signs here and there. So, we focus solely on the case for the $G$ function. Additionally, the arguments are fairly routine and standard. So, we just sketch the main idea needed for the bounds.\n\nTake $\\xi \\in \\Gamma_{+}^{(1)} \\cup \\Gamma_{+}^{(2)}$. In this case, we have $\\xi =\\textrm{i} + x e^{\\pi i \/6}$ or $\\xi = \\textrm{i} +x e^{5\\pi i \/6}$, with $0 \\leq x \\leq 1$ since $\\Gamma_{+}^{(1)} \\cup \\Gamma_{+}^{(2)} \\subset \\mathcal{B}(i, 1)$. Then, we may write the real part of the $G$ function explicitly and show that it is monotone by taking its derivative. For instance, we have\n\\begin{equation}\n \\frac{d}{dx} Re\\,\\{G(i + x e^{\\pi i \/6} ) - G(i)\\} = \\frac{t}{2} \\left(1 - \\frac{2 +4 x}{1 +x + x^2} + \\frac{1 + 4x +x^2}{(1 + x+x^2)^2} \\right).\n\\end{equation}\nOne may now check that the derivative is zero when $x=0$ and negative if $0 < x < 1 + \\sqrt{3}$. Thus, the bound (\\ref{e:bound1}) follows for this segment.\n\nTake $\\xi \\in \\Gamma_{+}^{(3)} \\cup \\Gamma_{+}^{(4)}$. In this case, we have $\\xi = i + e^{\\pi i \/6} + x$ or $\\xi = i+ e^{5\\pi i \/6} -x$, with $x$ non-negative and bounded since $\\Gamma_{+}^{(3)} \\cup \\Gamma_{+}^{(4)} \\subset \\mathcal{B}(0, R_{+})$. Then, we may write the real part of the $G$ function explicitly and show that it is monotone by taking its derivative. For instance, we have\n\\begin{equation}\n \\frac{d}{dx} Re\\,\\{G(i + e^{\\pi i \/6} + x) - G(i)\\} = -t \\left(1 - \\frac{3(\\sqrt{3} + 2 x)}{2(3 + \\sqrt{3}\\, x + x^2)^2} \\right).\n\\end{equation}\nForm this, one may show that the derivative is strictly negative for all $x \\geq 0$. The bound (\\ref{e:bound1}) follows for this segment.\n\nTake $\\xi \\in \\Gamma_{+}^{(5)}$. In this case, we have $\\xi = R_{+}\\, e^{i \\theta}$, with $- \\pi\/2 \\leq \\theta \\leq \\phi_1 < \\pi\/2$ and $\\pi\/2 < \\phi_2 \\leq \\theta \\leq 3\\pi\/2$ for some constants $\\phi_1$ and $\\phi_2$ since $\\Gamma_{+}^{(5)} \\subset \\{z \\in \\mathbb{C} \\mid Im\\, \\{z\\} \\leq Im\\, \\{ i + e^{\\pi i \/6}\\} \\}$. In this case, we have\n\\begin{equation}\n Re\\,\\{G(\\xi) - G(i)\\} = -2 t\\,\\log R_{+} + t(R_{+}+R_{+}^{-1}) \\sin \\theta.\n\\end{equation}\nSince $R_{+}>1$, one may then show that this function is monotone on $\\theta$ for each of the segments $- \\pi\/2 \\leq \\theta \\leq \\phi_1 < \\pi\/2$ and $\\pi\/2 < \\ph_2 \\leq \\theta \\leq 3\\pi\/2$. The bound (\\ref{e:bound1}) follows for this segment.\n\nThe bound (\\ref{e:bound2}), now that we have established that the function is monotone along all the segments of the contours, follows by evaluating the function on the boundary of the segment $\\Gamma_{+} \\cap \\mathcal{B}(i, t^{-\\alpha})$. That is, we evaluate the function at the points $\\xi = \\xi_0 + t^{-\\alpha} \\, e^{ \\pi \\textrm{i}\/6}$ and $\\xi = \\xi_0 + t^{-\\alpha} e^{5 \\pi \\textrm{i}\/6 }$. In particular, we use the Taylor expansion\n\\begin{equation}\n G(\\xi) - G(\\xi_0) = -\\frac{1}{3}x^3 t^{1- 3 \\alpha} + \\mathcal{O}(t^{1-4\\alpha})\n\\end{equation}\nto approximate the function at the desired points. Since $t^{-\\alpha}< T \\ll 1$, we obtain the bound (\\ref{e:bound2}). \n\\end{proof}\n\n\n\\section{Contour Deformations}\\label{s:contour_deformations}\n\n\\subsection{Small to Large Contour deformations}\n\nWe deform the contours in the probability function for the left-most particle given by (\\ref{Fprob1}). In particular, we deform the contours $\\mathcal{C}_r$, for the $\\zeta$-variables, to some contour $\\mathcal{C}_{R'}$ with a large radius $R'>0$. Let\n\\begin{equation}\\label{e:deformation_contour}\n \\Omega(\\xi) := \\mathcal{C}^{(0)} \\cup -\\mathcal{C}^{(1)} \\cup -\\mathcal{C}^{(2)} \\cup \\cdots \\cup -\\mathcal{C}^{(N)} \n\\end{equation}\nbe the union of $(N+1)$ circles so that $-\\mathcal{C}^{(j)}$, for $j=1, \\dots, N$, is a negatively oriented circle centered at $\\xi_j^{-1}$ with radius $ r' >0$ and $\\mathcal{C}^{(0)}$ is a positively oriented circle centered at the origin with radius $R'>0$. We give precise conditions on the radii in the statement of \\textsc{Lemma 7.1} below. Then, as we deform the $\\mathcal{C}_r$ contour, we will encounter poles at $\\zeta_i = \\xi_j^{-1}$ for $i, j =1, \\dots, N$. As a result, we obtain the contour $\\Omega(\\xi)$ when we deform the contour $\\mathcal{C}_r$ to $\\mathcal{C}_{R'}$. This result and the proof for the contour deformations, given by \\textsc{Lemma 7.1} below, is similar to the contour deformation in \\cite{CDP}.\n\n\\textsc{Lemma 7.1}\nFor $\\Delta \\neq 0$, $\\mathcal{F}_N(x,t)=\\mathbb{P}_Y(X_1(t)\\ge x)$ equals\n\\begin{equation}\n\\int_{\\mathcal{C}_R}\\cdots\\int_{\\Omega(\\xi)} \\frac{\\prod_{j,k}(\\xi_j+\\zeta_k-2\\D\\xi_j\\zeta_k)}{\\prod_{j0$ and the contour $\\Omega(\\xi)$ for the $\\zeta$-variables is given by (\\ref{e:deformation_contour}) with radii $R' >0$ and $r' = 1\/(2 R)$, so that the radii satisfy the following inequalities $\\max\\{2|\\D|^{-1}, 2(1+2|\\D|)\\}< R< \\max\\{ 4|\\D|^{-1}, 4(1 + 2|\\D|)\\} < R' \/2$.\n\n\\begin{proof}\nWe take formula (\\ref{Fprob1}) with radius $R$ as given in the conditions in the Lemma and radius $r>0$ so that $\\max\\{ 4|\\D|^{-1}, 4(1 + 2|\\D|)\\}0$, with $R'$ satisfying the conditions given in the Lemma. We begin by deforming the contour for $\\zeta_N$, then the contour for $\\zeta_{N-1}$, and continue successively until we deform the contour for $\\zeta_1$. When we deform the contour for the $\\zeta_n$ variable, we encounter three types of poles\n\\begin{equation}\n (a)\\, 1 - \\xi_i \\zeta_n= 0; \\quad (b)\\, 1 + \\zeta_i \\zeta_n -2 \\Delta \\zeta_i = 0, \\, i < n;\\quad (c)\\, 1 +\\zeta_n \\zeta_j - 2 \\Delta \\zeta_n, \\, n < j\n\\end{equation}\nfor any $i, j =1, \\dots, N$. The contribution for a type $(a)$ pole is given by the contour integral with respect to the variable $\\zeta_n$ with contour $- \\mathcal{C}^{(i)}$, i.e.~a negatively oriented circle centered at $\\xi_i^{-1}$ with radius $r'>0$ as given in the conditions of the Lemma. Note that the only pole, with respect to the variable $\\zeta_n$, inside the contour $-\\mathcal{C}^{(i)}$ is given by $\\zeta_n = \\xi_i^{-1}$ because $r'$ is chosen to be small enough. The result then follows by showing that the type $(b)$ and $(c)$ poles contribute no residue.\n\nAssume we have already deformed the $\\zeta_j$ variables for $j >n$ so that $\\zeta_i \\in \\mathcal{C}_r$ for $i n$. We then deform the contour for the $\\zeta_n$ variable. Below, we consider the residue contribution from the type $(b)$ and $(c)$ poles.\n\n\\noindent \\textbf{Case (b).} We compute the residue at\n\\begin{equation}\n \\zeta_n = (2 \\Delta \\zeta_{\\ell} - 1)\/\\zeta_{\\ell}\n\\end{equation}\nfor $\\ell r \\\\\n \\zeta_{\\ell} = \\frac{1}{2 \\Delta - \\zeta_j} \\quad &\\Rightarrow \\quad \\zeta_n = 2 \\Delta - \\zeta_{\\ell}^{-1} = \\zeta_j\\\\\n \\zeta_{\\ell} = 2 \\Delta - \\zeta_i^{-1} \\quad &\\Rightarrow\\quad \\left| 2\\Delta - \\zeta_i^{-1}\\right| >r\n \\\\\n \\zeta_n = 2\\Delta - \\zeta_i^{-1} \\quad &\\Rightarrow \\quad \\zeta_{\\ell} = \\zeta_i\\\\\n \\zeta_{\\ell} = \\frac{2\\D - \\zeta_j}{4\\D^4- 2 \\D \\zeta_j -1} \\quad &\\Rightarrow \\quad \\left| \\frac{2\\D - \\zeta_j}{4\\D^2- 2 \\D \\zeta_j -1}\\right|>r\\\\\n \\zeta_{\\ell}^{y_n - y_{\\ell}} \\quad &\\Rightarrow \\quad y_n- y_{\\ell}\\geq 1.\n \\end{split}\n\\end{equation}\nWe use the assumptions on the radii $R, R', r'>0$ given in the statement of the Lemma and the condition on the radius $r>0$ fixed at the beginning of the proof to establish the inequalities above. For the first two inequalities, it suffices to have $R , r^{-1} > 1 + 2 |\\D|$. For the third inequality, we have to consider two cases $\\zeta_j \\in \\mathcal{C}^{(0)}$ or $\\zeta_j \\in \\mathcal{C}^{(k)}$ with $k\\neq 0$. In the first case when $\\zeta_j \\in \\mathcal{C}^{(0)}$, we have that $|\\zeta_j| = R'$ and we use the bounds $R'>16 |\\D|$ and $R' > 8 |\\D|^{-1}$ that follow from the condition on the statement of the Lemma. In the second case when $\\zeta_j\\in\\mathcal{C}^{(k)}$ with $k \\neq 0$, we have that $|\\zeta_j| \\leq (3\/2)R^{-1}$ and we use the bound $(3\/2)R^{-1} < |\\D|$ that follows from the statement of the Lemma. Then, in all the cases above except for the second and fourth case, the poles lie outside the contour $\\mathcal{C}_r$, meaning that there is no residue contribution. In the second and fourth cases, the determinant term $D_N(\\xi, \\zeta)$ vanishes because two columns in the matrix of the determinant are equal to each other since two $\\zeta$ variables are equal to each other. In the last case, there is no pole since the exponent is positive. Then, the pole from the denominator and the zero from the determinant cancel out, meaning that these cases don't produce a residue. \n\nTherefore, by computing the integral with respect to the $\\zeta_{\\ell}$ variable, we have that the residues from the type $(b)$ poles vanish.\n\n\\noindent\\textbf{Case (c).} We compute the residue at \n\\begin{equation}\n \\zeta_n = \\frac{1}{2 \\D - \\zeta_{\\ell}}\n\\end{equation}\nwith $n <\\ell$. The result is a $(2N-1)$-fold contour integral with the same integrand, say $I_N(\\xi, \\zeta; t)$, except that the term $1 + \\zeta_{n} \\zeta_{\\ell} - 2 \\Delta \\zeta_{n}$ is replaced by $2\\D-\\zeta_{\\ell}$ and the variable $\\zeta_n$ is evaluated at $1\/(2\\D - \\zeta_{\\ell})$ for the rest of the terms. \n\nIn this case, we have have $\\zeta_{\\ell} \\in \\Omega(\\xi)$ since $\\ell >n$. Thus, we have two possibilities: (i) $\\zeta_{\\ell} \\in \\mathcal{C}^{(0)}= \\mathcal{C}_{R'}$ , or (ii) $\\zeta_{\\ell} \\in - \\mathcal{C}^{(k)}$ for some $k =1, \\dots, N$ (i.e.~a negatively oriented small circle of radius $r'$ centered at $\\xi_k^{-1}$). In the first case, we will not cross a pole in the contour deformation and there will be no residue to consider. In the second case, the pole will cancel out with a zero from the numerator and, again, there will be no residue to consider. We give more details below. \n\nIn the first case, when $\\zeta_{\\ell} \\in \\mathcal{C}_{R'}$, we have $\\zeta_n = 1\/(2 \\Delta - \\zeta_{\\ell})$. This pole lies inside the contour $\\mathcal{C}_r$ since $R'\\,r > 2 $ and $r < (1 + 2 |\\D|)^{-1} $. Thus, we don't cross this pole when we deform the $\\mathcal{C}_r$ contour to $\\mathcal{C}^{(0)} = \\mathcal{C}_{R'}$.\n\nIn the second case, when $\\zeta_\\ell \\in - \\mathcal{C}^{(k)}$, we first compute the residue at $\\zeta_{\\ell} = \\xi_k^{-1}$. We obtain an $(2N-1)$-fold contour integral with the same integrand, say $I_N(\\xi, \\zeta; t)$, except that the determinant $D_N(\\xi, \\zeta)$ is replaced the same determinant with the $k^{th}$ row and the ${\\ell}^{th}$ removed and multiplied by the factor $(1 + \\xi_k^2 - 2\\Delta \\xi_k)^{-1}$, and the rest of the terms are the same with the variable $\\zeta_{\\ell}$ evaluated at $\\xi_k^{-1}$.\n\nWe now deform the contour for $\\zeta_{n}$ to the contour $\\mathcal{C}^{(0)}$. After taking the $\\zeta_{\\ell} = \\xi_k^{-1}$ residue, it turns out that the terms giving rise to the pole $\\zeta_n= 1\/(2\\Delta - \\zeta_{\\ell})$ becomes\n\\begin{equation}\n (1 + \\zeta_n \\zeta_{\\ell} - 2\\Delta \\zeta_n) = \\xi_k^{-1}(\\xi_k + \\zeta_n - 2\\Delta \\xi_k \\zeta_n).\n\\end{equation}\nNote that this term also appears in the numerator of the integrand, meaning that this term cancels out and there is no residue in this case.\n\nTherefore, when we deform the contour for the $\\zeta_n$ to infinity, we don't cross any type $(c)$ poles. Moreover, this, along with the argument for the type $(b)$ poles, means that we only cross the poles due to the type $(a)$ poles. This establishes the result.\n\n\\end{proof}\n\n\\subsection{Series Expansion}\n\nWe write the contour formula (\\ref{Fprob2}) as a summation by expanding the integrals over the contour $\\Omega(\\xi)$, given by (\\ref{e:deformation_contour}), as a summations of $N+1$ integrals. We introduce some notation to encode the different terms in the summation.\n\nTake the set of all maps from the index set $\\{ 1, \\dots, N\\}$ to the set $\\{0, 1, \\dots, N\\}$ and denote it by\n\\begin{equation}\\label{e:maps}\n \\mathcal{T} := \\{\\tau : \\{1, \\dots, N\\} \\rightarrow \\{0, 1, \\dots, N\\} \\} = \\mathrm{Hom}\\left(\\{1, \\dots, N\\}, \\{0,1, \\dots, N\\}\\right).\n\\end{equation}\nIn the following, a map $\\tau \\in \\mathcal{T}$ will correspond to a term with contours $\\mathcal{C}^{(\\tau(k))}$, given by (\\ref{e:deformation_contour}), for the $\\zeta_k$ variable and $k=1, \\dots, N$. Moreover, we will show that some contour integrals will vanish for certain $\\tau \\in \\mathcal{T}$. We consider the set of maps that map injectively to the elements $\\{1, \\dots, N\\}$ in the image and the cardinality of the preimage $\\sigma^{-1}(0)$ is fixed; \n\\begin{equation}\\label{e:maps_cont}\n \\mathcal{T}_n := \\{\\tau \\in \\mathcal{T} \\mid |\\tau^{-1}(0)| =n; |\\tau^{-1}(k)| \\leq 1,\\, k =1, \\dots, N \\}.\n\\end{equation}\n\n\\textsc{Lemma 7.2}\nFor $\\D \\neq 0$, $\\mathcal{F}_N(x,t)=\\mathbb{P}_Y(X_1(t)\\ge x)$ equals\n\\begin{equation}\n\\sum_{n=0}^N \\sum_{\\tau \\in \\mathcal{T}_n}\\oint_{\\mathcal{C}_R}\\cdots\\oint_{\\mathcal{C}_R} \\oint_{\\mathcal{C}^{(\\tau(1))}} \\cdots \\oint_{\\mathcal{C}^{(\\tau(N))}}I_N(\\xi, \\zeta; x, t)\\, d^N\\zeta \\,d^N\\xi\\hspace{5ex}\\label{Fprob3}\\end{equation}\nwhere the integrand $I_N(\\xi, \\zeta; x, t)$ is the same integrand as in (\\ref{Fprob2}), the summation is take over the set of maps $\\mathcal{T}_n$ given by (\\ref{e:maps_cont}), the contour $\\mathcal{C}_R$ is a circle centered at zero with radius $R>0$, the contours $\\mathcal{C}^{(\\tau(k))}$ are given by (\\ref{e:deformation_contour}) with radii $r' = R^{-1}\/2, R' >0$ so that the radii satisfy the bounds $\\max\\{2 |\\D|^{-1}, 2(1+2|\\D|) \\} < R < \\max \\{4 |\\D|^{-1}, 4(1+2|\\D|) \\}< R'\/2$ \n\n\\begin{proof}\nWe take the contour formula (\\ref{Fprob2}) from \\textsc{Lemma7.1}. We then expand the integrals over the contours $\\Omega(\\xi)$ as a sum of $N+1$ integrals with contours given by the right side of (\\ref{e:deformation_contour}). The result is a summation over the set of maps $\\mathcal{T}$ given by (\\ref{e:maps}),\n\\begin{equation}\\label{e:sum1}\n \\mathcal{F}_N(x,t) = \\sum_{\\tau \\in \\mathcal{T}} \\oint_{\\mathcal{C}_R}\\cdots \\oint_{\\mathcal{C}_R} \\oint_{\\mathcal{C}^{(\\tau(1))}} \\cdots \\oint_{\\mathcal{C}^{(\\tau(N))}}I_N(\\xi, \\zeta; x, t)\\, d^N\\zeta \\,d^N\\xi.\n\\end{equation}\nThe result of this lemma follows by showing that some terms vanish, i.e.~if $\\tau \\notin \\mathcal{T}_n$ the corresponding contour integral will vanish. Below, we show that a term in the summation vanishes if $\\tau(j) = \\tau(k) >0$ with $j \\neq k$.\n\nTake $\\tau \\in \\mathcal{T}$ with $\\tau(j') = \\tau(k') =\\ell >0$ with $n \\neq m$ and $j', k'=1, \\dots, N$. We show that the term in the summation (\\ref{e:sum1}) with this $\\tau \\in \\mathcal{T}$ vanishes by taking the integrals with respect to the variables $\\zeta_{j'}$ and $\\zeta_{k'}$. We take the integral with respect to the $\\zeta_{j'}$ and $\\zeta_{k'}$ variables by taking the residues at the poles given by $\\zeta_{j'} = \\xi_{\\ell}^{-1}$ and $\\zeta_{k'} = \\xi_{\\ell}^{-1}$. Note that the poles given by $\\zeta_{j'} = \\xi_{\\ell}^{-1}$ and $\\zeta_{k'} = \\xi_{\\ell}^{-1}$ correspond to the $(\\ell, j')$-entry and the $(\\ell, k')$-entry of the matrix for the $D_N(\\xi, \\zeta)$ determinant. First, we take the residue at $\\zeta_{j'} = \\xi_\\ell^{-1}$, the determinant transforms as follows\n\\begin{equation}\\label{e:transform1}\n D_N(\\xi, \\zeta) = \\det \\left( \\frac{1}{(1 - \\xi_j \\zeta_k)(\\xi_j+ \\zeta_k - 2 \\Delta \\xi_j \\zeta_k)} \\right)_{j, k=1}^N \\longrightarrow \\frac{(-1)^{\\tau(j')-j'-1}}{1+\\xi_{\\ell}^2 - 2\\Delta \\xi_{\\ell}} \\det \\left( \\frac{1}{(1 - \\xi_j \\zeta_k)(\\xi_j+ \\zeta_k - 2 \\Delta \\xi_j \\zeta_k)} \\right)_{j \\neq \\ell, k \\neq j'}.\n\\end{equation}\nFor the rest of the factors in the integrand, one evaluates $\\zeta_{j'} = \\xi_\\ell^{-1}$ when we take the residue at $\\zeta_{j'} = \\xi_{\\ell}^{-1}$. One may check that this doesn't introduce any poles with respect to the $\\zeta_{j'}$ variable inside the $\\mathcal{C}^{(\\ell)}$ contour. Then, the residue at $\\zeta_{j'} = \\xi_{\\ell}^{-1}$ doesn't have a pole at $\\zeta_{k'} = \\xi_{\\ell}^{-1}$ since the pole at $\\zeta_{k'} = \\xi_{\\ell}^{-1}$ is removed when we take the residue and no other pole is introduced. Thus, by taking the residue at $\\zeta_{k'} = \\xi_{\\ell}^{-1}$ after taking the residue at $\\zeta_{j'} = \\xi_{\\ell}^{-1}$, we have that the term vanishes. That is,\n\\begin{equation}\\label{e:vanish_term1}\n \\oint_{\\mathcal{C}_R}\\cdots \\oint_{\\mathcal{C}_R} \\oint_{\\mathcal{C}^{(\\tau(1))}} \\cdots \\oint_{\\mathcal{C}^{(\\tau(N))}}I_N(\\xi, \\zeta; x, t)\\, d^N\\zeta \\,d^N\\xi = 0\n\\end{equation}\nif $\\tau(n) = \\tau(m) =\\ell >0$ with $n \\neq m$ and $n, m=1, \\dots, N$.\n\n\n\nThe result of the lemma then follows by taking the summation representation given by (\\ref{e:sum1}) and noting that the terms with $\\tau \\notin \\mathcal{T}_n$ for some $n =0, 1, \\dots, N$ vanish due to the identities (\\ref{e:vanish_term1}).\n\\end{proof}\n\n\\subsection{Residue Computations}\n\nWe compute the contour integrals with respect to the $\\zeta_k$ variables with $\\tau(k)\\neq 0$ for each of the terms in the series expansion given by (\\ref{Fprob3}). First, we introduce some notation to represent the resulting integrand after the residue computations.\n\nFix $\\tau \\in \\mathcal{T}_{N-M}$ with $0 \\leq M \\leq N$ and $\\mathcal{T}_{N-M}$ given by (\\ref{e:maps_cont}). Then, define the following sets\n\\begin{equation}\\label{e:index_sets}\n K_1 := \\tau^{-1}(0), \\quad K_2 := K_1^{c} = \\{k_1 < \\cdots < k_M \\}, \\quad J_2=\\tau \\left( K_2 \\right) = \\{\\tau_1 = \\tau(k_1), \\dots, \\tau_{M}=\\tau(k_M) \\}, \\quad J_1 = J_2^c.\n\\end{equation}\nWe also introduce the following functions\n\\begin{equation}\\label{e:integrands}\n \\begin{split}\n I_N(\\xi, \\zeta; \\tau)&=\\frac{\\prod_{j\\in J_1, k \\in K_1}(\\xi_j + \\zeta_k - 2\\D \\xi_j \\zeta_k)D_N(\\xi, \\zeta; \\tau)}{\\prod_{\\substack{j0$ so that $\\max\\{2 |\\D|^{-1}, 2(1 + 2|\\D|)\\} < R < \\max\\{4 |\\D|^{-1}, 4(1 + 2|\\D|)\\} < R'\/2$.\n\n\\begin{proof}\nThe result is a direct consequence of \\textsc{Lemma 7.2} and \\textsc{Lemma 7.3}.\n\\end{proof}\n\n\\subsection{Deformation to Steep Descent Contours}\n\nWe take the series expansion formula (\\ref{Fprob4}) and deform the contours to the steep descent contours given by (\\ref{e:steep_contours}).\n\nLet $\\widehat{\\Gamma}$ be a positive oriented rectangle centered at zero, with length equal to $2 L = 2 \\sqrt{R^2-1}$ and height equal to $2$, and two half-circle bumps as indicated on Figure \\ref{f:rec_contour}. The bump centered at $\\textrm{i}$ has radius $\\epsilon_1$ and the bump centered at $\\textrm{i} + 2\\D$ has radius $\\epsilon_2$ so that $0 < \\epsilon_2 \\ll \\epsilon_1 \\ll 1$. \n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[scale=0.35]{RecContour3a.png}\n \\caption{The contour $\\widehat{\\Gamma}$.}\n \\label{f:rec_contour}\n\\end{figure}\n\n\n\\textsc{Lemma 7.5} Fix $\\tau \\in \\mathcal{T}_{N-M}$, with $0 \\leq M \\leq N$, take the notation from (\\ref{e:index_sets}). Then, for $\\D \\neq 0$, we have\n\\begin{equation}\n \\begin{split}\n &\\oint_{\\mathcal{C}_R}\\cdots \\oint_{\\mathcal{C}_R} \\oint_{\\mathcal{C}_{R'}} \\cdots \\oint_{\\mathcal{C}_{R'}} I_N(\\xi, \\zeta; \\tau) f(\\xi, \\zeta; \\tau)\\, d^J \\zeta\\, d^N \\xi= \\oint_{\\Gamma_{+}} \\cdots \\oint_{\\Gamma_{-}} I_{N}(\\xi, \\zeta; \\tau)\\, \\left( \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi\\right) d^{K_1}\\zeta\\, d^{J_1}\\xi\n \\end{split}\n\\end{equation}\nwhere the integrand is given by (\\ref{e:integrands}), the differentials $d^S \\xi$ or $d^S \\zeta$ are $|S|$-fold differential over the variables $\\xi_s$ or $\\zeta_s$ with $s\\in S$, the contours $\\Gamma_{\\pm}$ are given by (\\ref{e:steep_contours}) with $R_{+} = R$ and $R_{-} = R'$ so that $\\xi_{j} \\in \\Gamma_{+}$ and $\\zeta_{k} \\in \\Gamma_{-}$ for $j\\in J_1$ and $k \\in K_1$, the contour $\\widehat{\\Gamma}$ is given by Figure \\ref{f:rec_contour}, the contours $\\mathcal{C}_R$ and $\\mathcal{C}_{R'}$ are circles centered at zero with radii $R,R' >0$ so that $\\max\\{2|\\D|^{-1}, 2(1 + 2|\\D|)\\} < R < \\max\\{4|\\D|^{-1}, 4(1+2 |\\D|)\\}< R'\/2$.\n\n\n\\begin{proof}\nWe obtain the result by deforming the contours and showing that we don't cross any poles. We begin by deforming the contour, for the $\\xi_j$ variables with $j \\in J_2$, from $\\mathcal{C}_R$ to $\\widehat{\\Gamma}$. Then, for the $\\xi_j$ variables with $j \\in J_1$, we deform the contours $\\mathcal{C}_R$ to the contours $\\Gamma_{+}$. Finally, for the $\\zeta_k$ variables, we deform the contours $\\mathcal{C}_{R'}$ to the contours $\\Gamma_{-}$.\n\nConsider the integral with respect to $\\xi_{\\ell} \\in \\mathcal{C}_R$ and $\\ell \\in J_2$. We deform the contour $\\mathcal{C}_R$ to the contour $\\widehat{\\Gamma}$. Note that the factor $I_N(\\xi,\\zeta; \\tau)$ is independent of the $\\xi_{\\ell}$ variable. Then, the only possible poles are given by\n\\begin{equation}\\label{e:poles1}\n 1 + \\xi_{\\ell} \\xi_k -2\\D \\xi_{\\ell}= 0 , \\quad \\xi_{\\ell} + \\zeta_k - 2\\D =0.\n\\end{equation}\n\nIn the first case of (\\ref{e:poles1}), the location of the pole is given by $(2\\D - \\xi_k)^{-1}$ with $\\xi_k \\in \\mathcal{C}_R$ or $\\xi_k \\in \\widehat{\\Gamma}$, depending on the index and if the contour for the variable has been deformed. If $\\xi_k \\in \\mathcal{C}_R$, the location of the pole $(2\\D - \\xi_k)^{-1}$ clearly lies inside the unit circle since $R > 1+ 2|\\D|$. In particular, we don't cross this pole when we deform from the contour $\\mathcal{C}_R$ to the contour $\\widehat{\\Gamma}$, since the contour $\\widehat{\\Gamma}$ lies outside the unit circle. If $\\xi_k \\in \\widehat{\\Gamma}$, we note that the location of the pole $(2 \\D - \\xi_k)^{-1}$ also lies inside the unit circle except for the region with the small half-circle bump of radius $\\epsilon_2$. We then consider $\\xi_k$ lying on the small half-circle bump of $\\widehat{\\Gamma}$ and we write $\\xi_k = \\textrm{i} + 2 \\D + \\epsilon_2\\, e^{\\textrm{i} \\phi}$. Then, the location of the pole is given by\n\\begin{equation}\n (2 \\D - \\xi_k)^{-1} = (- \\textrm{i} - \\epsilon_2\\, e^{\\textrm{i} \\phi})^{-1} = \\textrm{i} - \\epsilon_2\\, e^{\\textrm{i} \\phi} + \\mathcal{O}(\\epsilon_2^2),\n\\end{equation}\nwhere the last equality follows from $0 < \\epsilon_2 \\ll 1$. Moreover, since $\\epsilon_2 \\ll \\epsilon_1$, we have that the location of the pole $(2 \\D - \\xi_k)^{-1}$ lies inside the large bump of the contour $\\widehat{\\Gamma}$, when $\\xi_k$ lies on the small bump. Then, we have that the pole $(2 \\D - \\xi_k)^{-1}$ lies inside the unit circle if $\\xi_k$ doesn't lie on the small bump, and the pole lies inside the large bump if $\\xi_k$ lies on the small bump. In particular, if $\\xi_k \\in \\mathcal{C}_R \\cup \\widehat{\\Gamma}$, the location of the pole lies inside the contour $\\widehat{\\Gamma}$ and we don't cross any poles, given by the first case of (\\ref{e:poles1}), when we deform form the contour $\\mathcal{C}_R$ to the contour $\\widehat{\\Gamma}$.\n\nIn the second case of (\\ref{e:poles1}), the location of the pole is given by $2 \\D - \\zeta_k$. Additionally, we have that $\\zeta_k \\in \\mathcal{C}_{R'}$. Given the conditions on the radii $R, R'>0$, it follows that $R < R' -2 |\\D|$. Then, the pole given by $2 \\D - \\zeta_k$ lies outside the contour $\\mathcal{C}_R$. In particular, we don't cross the pole when we deform the contour form $\\mathcal{C}_R$ to $\\widehat{\\Gamma}$. Thus, we don't cross any poles, given by the second case of (\\ref{e:poles1}), when we deform the contours from $\\mathcal{C}_R$ to $\\widehat{\\Gamma}$.\n\nConsider now the integral with respect to $\\xi_{\\ell} \\in \\mathcal{C}_{R}$ with $\\ell \\in J_1$. We deform the contour $\\mathcal{C}_{R}$ to the contour $\\Gamma_{+}$ with $\\xi_j \\in \\widehat{\\Gamma}$ for $j \\in J_2$. The location of the possible poles are given by\n\\begin{equation}\\label{e:poles3}\n (2\\D - \\xi_j)^{-1}, \\quad 2\\D - \\xi_j^{-1}, \\quad \\zeta_k^{-1}.\n\\end{equation}\n\nIn the first case of (\\ref{e:poles3}), the variable $\\xi_j$ may lie on the the contours $\\Gamma_{+}$ or $\\mathcal{C}_R$, depending on the index. In particular, if $\\xi_j \\in \\widehat{\\Gamma}$, then $j = \\tau_{k}$ for some $k$, see (\\ref{e:index_sets}). Moreover, $I_N(\\xi, \\zeta; \\tau)$ is independent of $\\xi_{j} = \\xi_{\\tau_{k}}$ and the pole due to the $f(\\xi, \\zeta; \\tau)$ function is of the form $(2\\D- \\xi_{\\tau_{k}})^{-1}$; see (\\ref{e:integrands}). Thus, for the first case, $\\xi_j$ will never lie on the contour $\\widehat{\\Gamma}$ and only lie on the contours $\\mathcal{C}_{R}$ or $\\Gamma_{+}$. If $\\xi_j \\in \\mathcal{C}_R$, the location of the pole $(2\\D - \\xi_j)^{-1}$ clearly lies inside the unit circle since $R- 2|\\D| >1$. In particular, we don't cross this pole when we deform from the contour $\\mathcal{C}_R$ to the contour $\\Gamma_{+}$, since the contour $\\Gamma_{+}$ lies outside the unit circle. If $\\xi_j \\in \\Gamma_{+}$, the location of the pole $(2\\D - \\xi_j)^{-1}$ will also lie outside the unit circle. This due to the fact the $\\D$ is a real number and $R-2|\\D| >1$. In particular, if $\\xi_j \\in \\mathcal{C}_R \\cup \\Gamma_{+} $, we don't cross a pole, given by the first case of (\\ref{e:poles3}) when we deform from the contour $\\mathcal{C}_R$ to the contour $\\Gamma_{+}$.\n\nIn the second case of (\\ref{e:poles3}), the variable $\\xi_j$ may lie on the the contours $\\Gamma_{+}$, $\\mathcal{C}_R$, or $\\widehat{\\Gamma}$, depending on the index. In all three cases, we have that the $- \\xi_j^{-1}$ point lies inside the unit circle since the contours lie outside the unit circle. Then, the pole $2\\D - \\xi_j^{-1}$ will lie inside $\\Gamma_{+}$ since $\\D$ is a real number and $2(1 + 2|\\D|) < R$. In particular, if $\\xi_j \\in \\mathcal{C}_R \\cup \\Gamma_{+} \\cup \\widehat{\\Gamma}$, we don't cross a pole, given by the second case of (\\ref{e:poles3}), when we deform from the contour $\\mathcal{C}_R$ to the contour $\\Gamma_{+}$.\n\nIn the third case of (\\ref{e:poles3}), we have $\\zeta_k \\in \\mathcal{C}_{R'}$. Then, the location of the pole $\\zeta_k^{-1}$ lies completely inside the unit circle. Then, since $\\Gamma_{+}$ lies outside the unit circle, we don't cross a pole when we deform the contour $\\mathcal{C}_R$ to the contour $\\Gamma_{+}$.\n\n\nLastly, consider the integral with respect to $\\zeta_{\\ell} \\in \\mathcal{C}_{R'}$ with $\\ell \\in K_1$. We deform the contour $\\mathcal{C}_{R'}$ to the contour $\\Gamma_{-}$. The location of the possible poles is given by\n\\begin{equation}\\label{e:poles2}\n (2\\D - \\zeta_j)^{-1}, \\quad 2\\D - \\zeta_j^{-1}, \\quad 2\\D- \\xi_k, \\quad \\xi_k^{-1}\n\\end{equation}\nwhere the variables may lie on different contours depending on the indexes.\n\nIn the first case of (\\ref{e:poles2}), the variable $\\zeta_j$ may lie on the contour $\\mathcal{C}_{R'}$ or on the contour $\\Gamma_{-}$. In either case, the location of the pole lies completely inside the unit circle. When $\\zeta_j \\in \\mathcal{C}_{R'}$, this follows from the bound $R>2(1 +2 |\\D|)$. When $\\zeta_j \\in \\Gamma_{-}$, in addition the bound $R>2(1 +2 |\\D|)$, we also need the fact that $\\D$ is a real number, which means that $(2\\D - \\zeta_j)$ lies outside the unit circle for $\\zeta_j \\in \\Gamma_{-}$. Then, we have that the location of the pole $(2 \\D - \\zeta_j)^{-1}$ lies completely inside the unit circle and we don't cross any poles when we deform the contour $\\mathcal{C}_{R'}$ to the contour $\\Gamma_{-}$.\n\nIn the second case of (\\ref{e:poles2}), the variable $\\zeta_j$ may lie on the contour $\\mathcal{C}_{R'}$ or on the contour $\\Gamma_{-}$. In either case, we know that $\\zeta_j^{-1}$ lies inside the unit circle since $\\mathcal{C}_{R'}$ and $\\Gamma_{-}$ lie outside the unit circle. Then, since $\\D$ is a real number and $4(1+ 2|\\D|) < R'$ , we have that the location of the pole $2\\D -\\zeta_j^{-1}$ lies completely inside the contour $\\Gamma_{-}$. Thus, we don't cross any poles when we deform the contour $\\mathcal{C}_{R'}$ to the contour $\\Gamma_{-}$.\n\nIn the third case of (\\ref{e:poles2}), the variable $\\xi_k$ may lie on $\\widehat{\\Gamma}$ since this pole is due to the $f(\\xi, \\zeta; \\tau)$ factor in the integrand; see (\\ref{e:integrands}). In this case, the location of the pole $2\\D - \\xi_k$ lies completely inside the contour $\\Gamma_{-}$ due to the bumps of the contour $\\widehat{\\Gamma}$. Since $0 < \\epsilon \\ll 1$, the large bump of the contour $\\widehat{\\Gamma}$ lies completely above the horizontal section of the contour $\\Gamma_{-}$. Since the small bump in the contour $\\widehat{\\Gamma}$ lies inside the rectangle, the small bump will also lie completely above the V-section of the $\\Gamma_{-}$ contour. Additionally, since $R + 2|\\D|< R'$, the rest of the contour $\\widehat{\\Gamma}$ will lie completely inside the contour $\\Gamma_{-}$. Then, we don't cross any poles when we deform the contour $\\mathcal{C}_{R'}$ to the contour $\\Gamma_{-}$.\n\nIn the fourth case of (\\ref{e:poles2}), we have $\\xi_k \\in \\Gamma_{+}$. Then, the location of the pole $\\xi_k^{-1}$ lies completely inside the unit circle, since the contour $\\Gamma_{+}$ lies outside the unit circle. Then, since $\\Gamma_{-}$ lies outside the unit circle, we don't cross a pole when we deform the contour $\\mathcal{C}_{R'}$ to the contour $\\Gamma_{-}$.\n\nWe have now shown that we don't cross any poles in any case when we deform the contours. Thus, the result follows.\n\n\\end{proof}\n\n\\textsc{Proposition 7.6} For $\\D \\neq 0$, $\\mathcal{F}_N(x,t)=\\mathbb{P}_Y(X_1(t)\\ge x)$ equals\n\\begin{equation}\n\\sum_{n=0}^N \\sum_{\\tau \\in \\mathcal{T}_n}\\oint_{\\Gamma_{+}} \\cdots \\oint_{\\Gamma_{-}} I_{N}(\\xi, \\zeta; \\tau)\\,\\left( \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi \\right) d^{K_1}\\zeta\\, d^{J_1}\\xi\\,\\hspace{5ex}\\label{Fprob5}\n\\end{equation}\nwhere the integrand is given by (\\ref{e:integrands}), the sets $J_1, J_2, K_1, K_2$ are given by (\\ref{e:index_sets}), the summation is take over the set of maps $\\mathcal{T}_n$ given by (\\ref{e:maps_cont}), and the contours $\\Gamma_{\\pm}$ and $\\widehat{\\Gamma}$ are given by (\\ref{e:deformation_contour}) and Figure \\ref{f:rec_contour} with $R_{+}=R, R_{-}= R'$ so that $\\max\\{2 |\\D|^{-1}, 2(1 + 2|\\D|) \\} < R < \\max\\{4 |\\D|^{-1}, 4(1 + 2|\\D|) \\} < R'\/2$.\n\n\\begin{proof}\nThe result is a direct consequence of \\textsc{Proposition 7.4} and \\textsc{Lemma 7.5}.\n\\end{proof}\n\n\n\n\\section{Asymptotic Analysis, a Conjecture}\\label{s:conjecture}\n\nWe believe that the formula for the probability of the left-most particle given by (\\ref{Fprob4}) in \\textsc{Theorem 7.6} may be suitable for asymptotic analysis when $t \\ll N \\rightarrow \\infty$. Note that we have decomposed the integrand into two factors, $I_N(\\xi, \\zeta; \\tau)$ and $f(\\xi, \\zeta; \\tau)$. In particular, note that that the factor $f(\\xi, \\zeta; \\tau)$ is independent of time $t$. Additionally, for the variables of the term $I_N(\\xi,\\zeta; \\tau)$, we have deformed the contours to steepest descent paths. Thus, in the asymptotic limit, we expect the main contribution for the $I_N(\\xi,\\zeta;\\tau)$ term to come from the saddle point $(\\xi_0,\\zeta_0) =(i,-i)$. Moreover, we expect the asymptotic limit of $I_N(\\xi, \\zeta ; \\tau)$ to be given by the Airy kernel. We give some details of the computation below but, unfortunately, we don't give all the technical details here. The arguments below need more careful consideration.\n\nFix $\\tau \\in \\mathcal{T}_n$ and let's consider the contribution of the contour integrals near the saddle point. We use the following notation for the index sets:\n\\begin{equation}\n\\begin{split}\n K_1 := \\tau^{-1}(0) , \\quad K_2 := (K_1)^c, \\quad \n J_1 := \\tau(K_2)^c , \\quad J_2 := \\tau(K_2)\n\\end{split}\n\\end{equation}\nThe sets $K_1$ and $K_2$ will be used to index the $\\zeta$-variables and the sets $J_1$ and $J_2$ will be used to index the $\\xi$-variables. In particular, variables with index from the sets $K_1$ and $J_1$ will lie on the contours $\\Gamma_{\\pm}$, respectively, and the variables with index from the set $J_2$ will lie on the contour $\\widehat{\\Gamma}$. There are no variables with index from the set $K_2$ because these variable have been integrated out, but nonetheless, this index set will appear in our formulas. Note $K_1 \\cup K_2 = J_1 \\cup J_2 = \\{1, \\dots, N \\}$.\n\n\nRecall that the spectral function $G$ and $H$, given in (\\ref{e:spectral_function}), have a double critical point at $\\xi = i$ and $\\zeta=-i$, respectively, when $x =-2t$. Let $\\mathcal{B}(z,r)$ be an open ball centered at $z \\in \\mathbb{C}$ of radius $r>0$ and $\\mathcal{B}(z,r)^c$ be its complement. Then, we take the following scaling\n\\begin{equation}\\label{e:scaling}\n x= -2t - st^{1\/3}, \\quad \\xi = \\textrm{i} + \\textrm{i}\\, \\tilde{\\xi}\\, t^{-1\/3}, \\quad \\zeta = -\\textrm{i} + \\textrm{i}\\, \\tilde{\\zeta}\\, t^{-1\/3},\\quad y_j+1 = v_j \\, t^{1\/3}\n\\end{equation}\nif $\\xi \\in \\mathcal{B}(i, t^{- \\alpha})$ and $\\zeta \\in \\mathcal{B}(-i, t^{-\\alpha})$ with $1\/3 < \\alpha < 1\/4$.\n\nWe also have that the integrand $I_N(\\xi, \\zeta; \\tau)$ is exponentially small if $\\xi_j \\in \\mathcal{B}(i, t^{-\\alpha})^c$, for $j \\in J_1$, or $\\zeta_j \\in \\mathcal{B}(-i, t^{- \\alpha})^c$, for $j \\in K_1$. This follows from \\textsc{Lemma 6.1}. Additionally, we may uniformly bound the factor $f(\\xi, \\zeta; \\tau)$, independently of $t$, on all the $\\xi$ and $\\zeta$ variables. Then, we may restrict the contours $\\Gamma_{\\pm}$ to the a neighborhood around the saddle points and only lose an exponentially small term. That is,\n\\begin{equation}\\label{e:approx1}\n \\begin{split}\n &\\oint_{\\Gamma_{+}} \\cdots \\oint_{\\Gamma_{-}} I_{N}(\\xi, \\zeta; \\tau)\\, \\left( \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi \\right) d^{K_1}\\zeta\\, d^{J_1}\\xi\\\\\n &= \\oint_{\\Gamma_{+} \\cap \\mathcal{B}(\\textrm{i}, t^{-\\alpha})} \\cdots \\oint_{\\Gamma_{-} \\cap \\mathcal{B}(-\\textrm{i} , t^{-\\alpha})} I_{N}(\\xi, \\zeta; \\tau)\\, \\left( \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi\\right) d^{K_1}\\zeta\\, d^{J_1}\\xi\\, + \\mathcal{O}(e^{- C t^{1-3\\alpha}})\n \\end{split}\n\\end{equation}\nfor some positive constant $C>0$, based on \\textsc{Lemma 6.1}, and $1\/3 < \\alpha <1\/4$.\n\nLet us now approximate the integrands $I_N(\\xi, \\zeta;\\tau)$ and $f(\\xi, \\zeta;\\tau)$ when $\\xi_j \\in \\Gamma_{+} \\cap \\mathcal{B}(i, t^{- \\alpha})$, for $j \\in J_1$, and $\\zeta_k \\in \\Gamma_{-} \\cap \\mathcal{B}(-i, t^{-\\alpha})$, for $k \\in K_1$. In particular, we take the scaling (\\ref{e:scaling}) for the variables with indexes in the sets $J_1$ and $K_1$, for the $\\xi$-variables and $\\zeta$-variables respectively. \n\nNote that $I_N(\\xi, \\zeta;\\tau)$ only depends on the variables with indexes from the sets $K_1$ and $J_1$. Then, we have\n\\begin{equation}\\label{e:approx2}\n \\begin{split}\n I_N(\\xi, \\zeta; \\tau)&= (-1)^{|\\mathcal{X}_1|+\\sum_{j \\in \\mathcal{Z}_2}\\tau(j)-j}\\sum_{\\gamma : \\mathcal{Z}_1 \\rightarrow \\mathcal{X}_1} \\prod_{k \\in \\mathcal{Z}_1}(-1)^{\\gamma(k) - k} (\\textrm{i})^{y_{\\gamma(k)}- y_k} g(\\tilde{\\xi}_{\\gamma(k)}, \\tilde{\\zeta}_k; v_{\\gamma(k)}, v_k)\\, t^{n\/3} + \\mathcal{O}(t^{(n-1)\/3})\\\\\n g(\\xi, \\zeta; x, z)&= \\frac{\\exp\\left( \\frac{1}{3}\\xi^3 - \\frac{1}{3}\\zeta^3 - (s + x)\\,\\xi +(s + z)\\,\\zeta\\right)}{(\\xi - \\zeta)},\n \\end{split}\n\\end{equation}\nwhere the sum is taken over all bijections $\\gamma : K_1\\rightarrow J_1$. This approximation is obtained by expanding the determinant in the term $I_{N}(\\xi, \\zeta;\\tau)$, given by (\\ref{e:integrands}) and taking the scaling (\\ref{e:scaling}). More details regarding this approximation are given in Appendix \\ref{s:I_approx}.\n\nNow, consider the approximation of the term $f(\\xi, \\zeta;\\tau)$ when $\\xi_j \\in \\Gamma_{+} \\cap \\mathcal{B}(i, t^{- \\alpha})$, for $j \\in J_1$, and $\\zeta_k \\in \\Gamma_{-} \\cap \\mathcal{B}(-i, t^{-\\alpha})$, for $k \\in K_1$. We introduce the following function\n\\begin{equation}\\label{e:B_fun}\n B(\\xi;\\tau) = \\prod_{\\substack{j < k, j,k\\in K_2 \\\\ \\tau(k) < \\tau(j)}} \\left(\\frac{1 + \\xi_{\\tau(k)} \\xi_{\\tau(j)}- 2\\D \\xi_{\\tau(j)}}{1 + \\xi_{\\tau(k)} \\xi_{\\tau(j)}- 2\\D \\xi_{\\tau(k)}} \\right),\n\\end{equation}\nwith the indexes $j,k \\in K_2$ and $\\tau(k), \\tau(j) \\in J_2$. Also, let us denote the number of inversions of the $\\tau$ map as follows,\n\\begin{equation}\\label{e:nu1}\n \\begin{split}\n \\nu_1(j; \\tau) &: = \\# \\{j' \\in K_2 \\mid j' < j , \\quad \\tau(j')>\\tau(j)\\ \\}\\\\\n \\nu_2(j; \\tau) &: = \\# \\{j' \\in K_2 \\mid j < j' , \\quad \\tau(j)>\\tau(j')\\ \\}\\\\\n \\nu(j;\\tau) &:= j-\\tau(j) +\\nu_2(j; \\tau) - \\nu_1(j; \\tau)\n \\end{split}.\n\\end{equation}\nNote that, in the case $K_2 = \\{1, 2, \\dots, N\\}$, we have $\\nu(j;\\tau) = 0$ for $j=1, \\dots, N$. Then, by taking the scaling (\\ref{e:scaling}), we obtain\n\\begin{equation}\\label{e:approx3}\n f(\\xi, \\zeta; \\tau) = B(\\xi;\\tau) \\prod_{j \\in K_2}\\left( \\frac{\\xi_{\\tau(j)} - (2\\D +\\textrm{i})}{(2\\textrm{i} \\D +1) \\xi_{\\tau(j)}- \\textrm{i}}\\right)^{\\nu(j;\\tau)} \\prod_{j\\in K_2} \\xi_{\\tau(j)}^{y_{j}- y_{\\tau(j)}-1} + \\mathcal{O}(t^{-1\/3}).\n\\end{equation}\nThis approximation is obtained by applying the scaling (\\ref{e:scaling}) and taking the leading term in the $t^{-1\/3}$ expansion of the $f(\\xi,\\zeta; \\tau)$ function. More details regarding this approximation are given in Appendix \\ref{s:f_approx}. \n\nWe now combine the approximations (\\ref{e:approx1}), (\\ref{e:approx2}) and (\\ref{e:approx3}), given above. Note that the leading term of the approximation (\\ref{e:approx3}) is independent of the $\\tilde{\\xi}$ and $\\tilde{\\zeta}$ variables. We then introduce the term\n\\begin{equation}\\label{e:F}\n F(\\tau) = (\\textrm{i})^{|K_2|}\\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}} B(\\xi;\\tau) \\prod_{j \\in K_2}\\left( \\frac{\\xi_{\\tau(j)} - (2\\D +\\textrm{i})}{(2\\textrm{i} \\D +1) \\xi_{\\tau(j)}- \\textrm{i}}\\right)^{\\nu(j; \\tau)} \\prod_{j\\in K_2} (\\textrm{i}\\,\\xi_{\\tau(j)})^{y_{j}- y_{\\tau(j)}-1}d^{J_2} \\xi,\n\\end{equation}\nwhere we have taken the leading term of the $f(\\xi,\\zeta;\\tau)$ function and also incorporated the $(\\textrm{i})^{y_{\\gamma(k)} - y_k}$ term from the approximation of $I_N(\\xi, \\zeta;\\tau)$ given by (\\ref{e:approx2}), noting that $\\sum_{k\\in K_1}y_{\\gamma(k)} - y_k +\\sum_{k\\in K_2}y_{\\tau(k)} - y_k =0$. Then, for fixed $\\tau \\in \\mathcal{T}_n$, we obtain the following approximation near the saddle point\n\\begin{equation}\\label{e:approx4}\n \\begin{split}\n &\\oint_{\\Gamma_{+}} \\cdots \\oint_{\\Gamma_{-}} I_{N}(\\xi, \\zeta; \\tau)\\,\\left( \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi\\right) \\, d^{K_1}\\zeta\\, d^{J_1}\\xi\\\\\n &=t^{-n\/3}(-1)^{|J_1|+\\sum_{k \\in K_2}\\tau(k)-k} \\sum_{\\gamma : K_1 \\rightarrow J_1} F(\\tau) \\prod_{k \\in K_1}(-1)^{\\gamma(k) - k} \\mathbf{K}_{Ai}\\left(s + v_{\\gamma(k)}, s + v_k \\right)\\\\\n &\\quad \\quad + \\mathcal{O}(t^{(1-n)\/3}) + \\mathcal{O}(e^{-Ct^{1- 3 \\alpha}}).\n \\end{split}\n\\end{equation}\nThe $t^{-n\/3}$ term and the Airy kernel $\\mathbf{K}_{Ai}$ are obtained by taking the change of variables (\\ref{e:scaling}) and the following expression for the Airy kernel\n\\begin{equation}\\label{e:Airy}\n \\mathbf{K}_{Ai}(x,z) = \\int_{\\infty\\, e^{- 2\\pi \\textrm{i} \/3}}^{\\infty\\, e^{2\\pi \\textrm{i} \/3}} \\int_{\\infty\\, e^{- \\pi \\textrm{i} \/3}}^{\\infty\\,e^{ \\pi \\textrm{i} \/3}} \\frac{\\exp\\left(\\frac{1}{3}\\xi^3 - \\frac{1}{3}\\zeta^3 - x\\, \\xi + z\\, \\zeta\\right)}{\\xi - \\zeta}\\, d\\xi\\, d\\zeta,\n\\end{equation}\nwhere the contours for the $\\xi$ (resp.~$\\zeta$) variable starts at $\\infty\\, e^{-\\pi i \/3}$ (resp.~$\\infty\\, e^{-2\\pi i \/3}$) goes through the origin and ends at $\\infty\\, e^{\\pi i \/3}$ (resp.~$\\infty\\, e^{2\\pi i \/3}$).\n\nLet's now consider the formula (\\ref{Fprob5}) and, in particular, the summation over $\\mathcal{T}_n$ and $n$. We substitute the term in the summation by the right side of the approximation (\\ref{e:approx4}). The result is a summation over $\\mathcal{T}_n$, $n$, and injective maps $\\gamma: K_1\\rightarrow J_1$. More precisely, the summation is over a pair of bijective maps\n\\begin{equation}\n \\tau: K_2 \\rightarrow J_2, \\quad \\gamma: K_1 \\rightarrow J_1,\n\\end{equation}\nwhere $K_1 \\cup K_2 = J_1 \\cup J_2 = \\{1, 2, \\dots, N\\}$. This means that we may write the summation, over $\\mathcal{T}_n$, $n$ and the injective maps $\\gamma: K_1 \\rightarrow J_1$ and $\\tau: K_2 \\rightarrow J_2$, as the summation over permutations of the set $[N] =\\{1, 2, \\dots, N\\}$. In particular, we may uniquely identify a pair of bijective maps $(\\tau, \\gamma)$ with a permutation $\\sigma \\in \\mathcal{S}_N$ and a subset $S \\subset [N]$ so that $(\\tau, \\gamma) = (\\sigma|_{S^c}, \\sigma|_{S})$, where the right side are restrictions of the permutation to the indicated sets. Then, under this identification, we rewrite some the notation introduced earlier. For $(\\sigma, S)$ with $\\sigma|_{S^c} = \\tau$, we have\n\\begin{equation}\n B(\\xi;\\sigma, S) = B(\\xi; \\tau) = \\prod_{\\substack{j, k \\in S^c, j \\sigma(k)}} \\left(\\frac{1 + \\xi_{\\sigma(k)} \\xi_{\\sigma(j)} - 2\\D \\xi_{\\sigma(j)}}{1 + \\xi_{\\sigma(k)} \\xi_{\\sigma(j)} - 2\\D \\xi_{\\sigma(k)}} \\right).\n\\end{equation}\nAdditionally, for $(\\sigma, S)$ with $\\sigma|_{S^c} = \\tau$, we write the inversion sets as follows,\n\\begin{equation}\n \\begin{split}\n \\nu_1(j; \\sigma, S) &= \\nu_1(j; \\tau) = \\# \\{j' \\in S^c \\mid j' < j , \\quad \\sigma(j')>\\sigma(j)\\ \\}\\\\\n \\nu_2(j; \\sigma, S) &= \\nu_2(j; \\tau) = \\# \\{j' \\in S^c \\mid j < j' , \\quad \\sigma(j)>\\sigma(j')\\ \\}\\\\\n \\nu(j; \\sigma, S) &= \\nu(j;\\tau) = j-\\sigma(j) +\\nu_2(j; \\sigma, S) - \\nu_1(j; \\sigma, S).\n \\end{split}.\n\\end{equation}\nLastly, for $(\\sigma, S)$ with $\\sigma|_{S^c} = \\tau$, we write\n\\begin{equation}\n F(\\sigma, S) = F(\\tau) = (\\textrm{i})^{|S^c|} \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}} B(\\xi;\\sigma, S) \\prod_{j \\in S^c}\\left( \\frac{\\xi_{\\sigma(j)} - (2\\D +\\textrm{i})}{(2\\textrm{i} \\D +1) \\xi_{\\sigma(j)}- \\textrm{i}}\\right)^{\\nu(\\sigma, S)} \\prod_{j\\in S^c} (\\textrm{i} \\, \\xi_{\\sigma(j)})^{y_{j}- y_{\\tau(j)}-1}d^{\\sigma(S^c)} \\xi.\n\\end{equation}\nThen, under the identification of the pair of injective maps and the permutations, we have\n\\begin{equation}\n \\begin{split}\n &\\sum_{n=0}^N \\sum_{\\tau \\in \\mathcal{T}_n}\\oint_{\\Gamma_{+}} \\cdots \\oint_{\\Gamma_{-}} I_{N}(\\xi, \\zeta; \\tau)\\,\\left( \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}f(\\xi, \\zeta; \\tau) d^{J_2} \\xi\\right)\\, d^{K_1}\\zeta\\, d^{J_1}\\xi\\\\\n &=\\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\sum_{S \\subset [N]} (-1)^{|S|} t^{- |S|\/3} \\left(F(\\sigma, S)\\prod_{k \\in S} \\mathbf{K}_{Ai}\\left( s + \\frac{y_{\\sigma(k)}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}}\\right) + \\mathcal{O}(t^{-1\/3}) + \\mathcal{O}(e^{-C t^{1- 3\\alpha}})\\right)\n \\end{split}.\n\\end{equation}\n\nAssuming that the error terms don't contribute in the limit, we have the following conjecture.\n\n\\textsc{Conjecture 8.1} As $t\\ll N \\rightarrow \\infty$, $\\mathcal{F}_N(x,t)=\\mathbb{P}_Y(X_1(t)\\ge x)$, with $x = -2t -s \\, t^{-1\/3}$ and $y_j +1 = v_j\\, t^{1\/3}$, equals to the limit of\n\\begin{equation}\\label{Fprob6}\n \\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\sum_{S \\subset [N]} (-1)^{|S|} t^{- |S|\/3} F(\\sigma, S)\\prod_{k \\in S} \\mathbf{K}_{Ai}\\left( s + v_{\\sigma(k)}, s +v_k\\right) \n\\end{equation}\nwhere $F$ is given by (\\ref{e:F}) and the Airy kernel $\\mathbf{K}_{Ai}$ is given by (\\ref{e:Airy}).\n\nAt the moment, we are not able to control the limit of (\\ref{Fprob6}) when $t \\ll N \\rightarrow \\infty$. The main obstacle is the term $F(\\sigma, S)$ on (\\ref{Fprob6}). However, under some assumptions, we may simplify (\\ref{Fprob6}) as a determinant of the difference of two kernels. For instance, assume\n\\begin{equation}\\label{e:assump}\n F(\\sigma,S) = \\prod_{j \\in S^c} \\mathbf{Q}(\\sigma(j), j)\n\\end{equation}\nfor some kernel $\\mathbf{Q}$ on the set $\\{1, \\dots, N\\}$. Then, we have\n\\begin{equation}\n \\begin{split}\n &\\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\sum_{S \\subset [N]} (-1)^{|S|} t^{- |S|\/3} F(\\sigma,S)\\prod_{k \\in S} \\mathbf{K}_{Ai}\\left( s + \\frac{y_{\\sigma(k)}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}}\\right) \\\\\n &=\\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\sum_{S \\subset [N]} (-1)^{|S|} t^{- |S|\/3} \\prod_{j \\in S^c} \\mathbf{Q}(\\sigma(j), j) \\prod_{k \\in S} \\mathbf{K}_{Ai}\\left( s + \\frac{y_{\\sigma(k)}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}}\\right)\\\\\n &= \\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\prod_{k=1}^N\\left(\\mathbf{Q}(\\sigma(k), k) - t^{-1\/3}\\, \\mathbf{K}_{Ai}\\left(s + \\frac{y_{\\sigma(k)}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}} \\right) \\right)\\\\\n & = \\det \\left( \\mathbf{Q}(j, k) - t^{-1\/3}\\mathbf{K}_{Ai}\\left(s + \\frac{y_{j}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}} \\right)\\right)_{j,k=1}^N,\n \\end{split}\n\\end{equation}\ngiven the assumption (\\ref{e:assump}). In fact, when $\\Delta = 0$, one may check the assumption to be true and we have\n\\begin{equation}\n F(\\sigma,S) = \\mathds{1}\\left(\\sigma|_{S^c} = \\mathrm{Id}_{S^c} \\right) = \\prod_{j \\in S^c} \\mathds{1}(\\sigma(j) = j),\n\\end{equation}\nwhere the functions with $\\mathds{1}$ are indicator functions. This identity is easy to check since the first two terms in the intergand for $F(\\sigma,S)$, given by (\\ref{e:F}), are identically equal to one when $\\Delta =0$. Then, we have\n\\begin{equation}\n \\begin{split}\n &\\det \\left( \\mathbf{Id}(j, k) - t^{-1\/3}\\mathbf{K}_{Ai}\\left(s + \\frac{y_{j}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}} \\right)\\right)_{j, k=1}^N\\\\\n &=\\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma} \\sum_{S \\subset [N]} (-1)^{|S|}t^{- |S|\/3} F(\\sigma, S)\\prod_{k \\in S} \\mathbf{K}_{Ai}\\left( s + \\frac{y_{\\sigma(k)}+1}{t^{1\/3}}, s + \\frac{y_k+1}{t^{1\/3}}\\right)\n \\end{split}\n\\end{equation}\nwhen $\\Delta = 0$. This means that \\textsc{Conjecture 8.1} is true when $\\Delta =0$. Moreover, if $\\Delta=0$ and $y_j=j$, we may take the limit $t \\ll N \\rightarrow \\infty$. The right side becomes a sum of Riemann integrals, corresponding to the series expansion of a Fredholm determinant. Then, we have\n\\begin{equation}\n \\begin{split}\n \\lim_{N \\rightarrow \\infty}\\mathbb{P}_Y\\left(\\frac{X_1(t)+2t}{t^{1\/3}}\\ge -s\\right) &= \\lim_{t \\ll N \\rightarrow \\infty} \\sum_{\\sigma \\in \\mathcal{S}_N} \\sum_{S \\subset [N]} t^{- |S|\/3} \\det \\left( \\mathbf{K}_{Ai}\\left( s + \\frac{j+1}{t^{1\/3}}, s + \\frac{k+1}{t^{1\/3}}\\right) \\right)_{j, k \\in S}\\\\\n &= \\det \\left(\\mathbf{Id} - \\mathbf{K}_{Ai} \\right)_{L^2(s, \\infty)}\\\\\n &=F_2(s).\n \\end{split}\n\\end{equation}\nThis matches the earlier result (\\ref{TWF2}) for $\\Delta =0$.\n\nWe also may compute the terms in (\\ref{Fprob6}) when $S=\\emptyset$ and $S^c = \\{1, \\dots, N\\} =[N]$. In that case, the formula for $F(\\sigma, \\emptyset)$ simplifies as follows\n\\begin{equation}\n F(\\sigma, \\emptyset) = \\oint_{\\widehat{\\Gamma}} \\cdots \\oint_{\\widehat{\\Gamma}}\\prod_{\\substack{j < k \\\\ \\sigma(k) < \\sigma(j)}} \\left(\\frac{1 + \\xi_{\\tau(k)} \\xi_{\\tau(j)}- 2\\D \\xi_{\\tau(j)}}{1 + \\xi_{\\tau(k)} \\xi_{\\tau(j)}- 2\\D \\xi_{\\tau(k)}} \\right) \\prod_{j=1}^N \\xi_{\\tau(j)}^{y_{j}- y_{\\tau(j)}-1}d^N \\xi,\n\\end{equation}\nwhere $i,j= 1, 2, \\dots, N$ on the first product of the integrand. Additionally, we may deform the contours $\\widehat{\\Gamma}$ to arbitrarily large circles centered at the origin. Note that $(-1)^{\\sigma}F(\\sigma,\\emptyset)$ is equal to the integral inside the sum of (\\ref{psi_N_large}) with $x_i =y_i$, for $i=1, \\dots, N$, and $t=0$. Then, by \\textsc{Theorem 1a}, we have\n\\begin{equation}\n \\sum_{\\sigma \\in \\mathcal{S}_N} (-1)^{\\sigma}F(\\sigma, \\emptyset) = 1\n\\end{equation}\nfor any $N>0$.\n\n\n\n\\section*{Acknowledgement}\n\nThe authors thank F.~Colomo, B.~Nachtergaele, and L.~Petrov for their helpful communications. This work was supported by the National Science Foundation under the grant DMS--1809311 (second author). This work was also supported by the National Science Foundation under Grant No. DMS--1928930 while the first author participated in the program \\emph{``Universality and Integrability in Random Matrix Theory and Interacting Particle Systems\"} hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2021 semester. Additionally, the first author was partially supported by the \\emph{Engineering and Physical Sciences Research Council (EPSRC)} through grant EP\/R024456\/1.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\n\\subsection{Motivations}\nThe recent trend of using a large number of parameters to model large datasets in machine learning and statistics has created a strong demand for optimization algorithms that have low computational cost per iteration and exploit model structure. Empirical risk minimization is a class of important optimization problems in this area. Such problems can be written as\n\\begin{align}\n\\min_{\\mb{x} \\in \\mc{P}} F(\\mb{x}) \\equiv \\frac{1}{n}\\sum_{i = 1}^n f_i(\\mb{x}), \\label{erm_problem}\n\\end{align}\nwhere $\\mc{P}$ is a compact polyhedral set in $\\mr{R}^p$ and each $f_i(\\cdot)$ is a convex function. A popular approach for solving (\\ref{erm_problem}) is the proximal gradient method which solves a projection sub-problem in each iteration. The major drawback of this method is that the projection step can be expensive in many situations. As an alternative, the Frank-Wolfe (FW) algorithm \\cite{FW56}, also known as the conditional gradient method, solves a linear optimization sub-problem in each iteration which is much faster than the standard projection technique when the feasible set is a simple polytope \\cite{N15}. On the one hand, when the number of observations $N$ in problem (\\ref{erm_problem}) is large, calculating the gradient of $F(\\cdot)$ in every FW iteration becomes a computationally intensive task. The question of whether `cheap' stochastic gradient can be used as a surrogate in FW immediately arises. On the other hand, when the dimension of the parameter space $d$ in problem (\\ref{erm_problem}) is large, taking advantage of any underlying structure in the problem can further accelerate the algorithm. In this paper, we present a linearly convergent semi-stochastic Frank-Wolfe algorithm with away-steps for empirical risk minimization and extend it to problems with block-coordinate structure. Specifically, we apply the proposed algorithm to structural support vector machine and graph-guided fused lasso problems.\n\\subsection{Contributions}\nThe contribution of this paper has two aspects. On the theoretical side, it presents the first stochastic conditional gradient algorithm that converges linearly in expectation. In addition, this algorithm uses an adaptive step size rather than one determined by exact line search. On the application side, the algorithm can be applied to various statistical and machine learning problems including support vector machine, multi-task learning, regularized generalized linear models and many others. Furthermore, when the problem has a block-coordinate structure, performance of this algorithm can be enhanced by extending it to a randomized coordinate version which solves several small sub-problems instead of a large problem.\n\\subsection{Related Work}\nThe Frank-Wolfe algorithm was proposed sixty years ago \\cite{FW56} for minimizing a convex function over a polytope and is known to converge at an $O(1\/k)$ rate. In \\cite{LP66} the same convergence rate was proved for compact convex constraints. When both objective function and the constraint set are strongly convex, \\cite{GH15} proved that the Frank-Wolfe algorithm has an $O(1\/k^2)$ rate of convergence with a properly chosen step size. Motivated by removing the influence of ``bad\" visited vertices, the away-steps variant of the Frank-Wolfe algorithm was proposed in \\cite{Wol70}. Later, \\cite{GM86} showed that this variant converges linearly under the assumption that the objective function is strongly convex and the optimum lies in the interior of the constraint polytope. Recently, \\citep{GH13} and \\cite{LJJ13} extended the linear convergence result by removing the assumption of the location of the optimum and \\cite{BS15} extended it further by relaxing the strongly convex objective function assumption. Stochastic Frank-Wolfe algorithms have been considered by \\cite{LAN13} and \\cite{LWM15} in which an $O(1\/k)$ rate of convergence in expectation is proved. In addition, the Frank-Wolfe algorithm has been applied to solve several different classes of problems, including non-linear SVM \\cite{OG10}, structural SVM \\cite{LJJSP13}, and comprehensive principal component pursuit \\cite{MZWG15} among many others.\n\\subsection{Notation}\nBold letters are used to denote vectors and matrices, normal fonts are used to denote scalers and sets. Subscripts represent elements of a vector, while superscripts with parentheses represent iterates of the vector, i.e. $\\mathbf{x}^{(k)}$ is a vector at iteration $k$, and $x_i^{(k)}$ is the $i$-th element of $\\mathbf{x}^{(k)}$. The cardinality of the set $I$ is denoted by $\\vert I \\vert$. Given two vectors $\\mathbf{x}, \\mathbf{y} \\in \\mathbb{R}^n$, their inner product is denoted by $\\langle \\mathbf{x}, \\mathbf{y} \\rangle$. Given a matrix $\\mathbf{A}\\in \\mathbf{R}^{m \\times n}$ and vector $\\mathbf{x} \\in \\mathbb{R}^n$, $\\norm{\\mathbf{A}}$ denotes the spectral norm of $\\mathbf{A}$, and $\\norm{\\mathbf{x}}$ denotes the $l_2$ norm of $\\mathbf{x}$. $\\mathbf{A}^\\top$ represent the transpose of $\\mathbf{A}$. We denote the $i$th row of a given matrix $\\mathbf{A}$ by $\\mathbf{A}_i$, and for a given set $I \\subset \\{1, \\ldots, m\\}$, $\\mathbf{A}_I \\in \\mathbf{R}^{\\vert I \\vert \\times n}$is the submatrix of $\\mathbf{A}$ such that $(\\mathbf{A}_I)_j = \\mathbf{A}_{I_j}$ for any $j = 1, \\ldots, \\vert I \\vert$. Given matrices $\\mathbf{A} \\in \\mathbb{R}^{n \\times m}$ and $\\mathbf{B} \\in \\mathbb{R}^{n \\times k}$, the matrix $[\\mathbf{A}, \\mathbf{B}] \\in \\mathbb{R}^{n \\times (m + k)}$ is their horizontal concatenation. Let $\\lceil x \\rceil$ be the ceiling function that rounds $x$ to the smallest integer larger than $x$.\n\n\\section{Stochastic Condtional Gradient Method for Empirical Risk Minimization}\n\\subsection{Problem description, notation and assumptions}\nConsider the minimization problem\n\\begin{align}\n\\min_{\\mb{x} \\in \\mc{P}} \\Big\\{F(\\mb{x}) \\equiv \\frac{1}{n}\\sum_{i = 1}^n f_i(\\mb{a}^\\top_i \\mb{x}) + \\langle \\mb{b}, \\mb{x} \\rangle \\Big\\}, \\label{general_problem} \\tag{P1}\n\\end{align}\nwhere $\\mathcal{P}$ is a non-empty compact polyhedron given by $\\mathcal{P} = \\{x \\in \\mathbb{R}^p : \\mathbf{Cx} \\leq \\mb{d}\\}$ for some $\\mathbf{C} \\in \\mathbb{R}^{m \\times p}$, $\\mathbf{d} \\in \\mathbb{R}^m$. For every $i = 1, \\ldots, n$, $\\mb{a}_i \\in \\mr{R}^p$, and $f_i : \\mr{R} \\rightarrow \\mr{R}$ is a strongly convex function with parameter $\\sigma_i$ and has a Lipschitz continuous gradient with constant $L_i$. $\\mb{b}$ in the linear term of (\\ref{general_problem}) is a vector in $\\mr{R}^p$. Note that the gradient of $F(\\cdot)$ is also Lipschitz continuous with constant $L \\leq (\\sum_{i=1}^nL_i \\norm{a_i}) \/ n$. However, because of the affine transformation in the argument of each $f_i(\\cdot)$, $F(\\cdot)$ may not be a strongly convex function.\n\n\\noindent\\textbf{Remark:}\n\\begin{description}\n\\item[1] Many statistics and machine learning problems can be modeled as problem (\\ref{general_problem}). For example:\n\\begin{description}\n\\item[i] The LASSO problem : $\\min_{\\boldsymbol\\beta \\in \\mathbb{R}^p} \\sum_{i=1}^n (y_i - \\mb{x}_i^\\top\\boldsymbol\\beta)^2 + \\lambda \\vert\\vert \\boldsymbol\\beta \\vert\\vert_1$, where $n$ is the sample size and for $i = 1, 2,\\ldots, n$, $y_i$'s are responses and $\\mb{x}_i$'s are the covariates. $\\boldsymbol\\beta$ is the regression coefficient to be estimated by solving the minimization problem and $\\lambda$ is a regularization parameter for sparsity.\n \n\\item[ii]$l_1$-Regularized Poisson Regression: $\\min_{\\boldsymbol\\beta \\in \\mathbb{R}^p} \\sum_{i = 1}^n \\exp\\{\\mathbf{x}_i^\\top\\boldsymbol\\beta\\} - y_i\\mathbf{x}_i^\\top\\boldsymbol\\beta +\\log(y_i!) + \\lambda \\vert\\vert \\boldsymbol\\beta \\vert\\vert_1 $, where $n$ is the sample size and for $i = 1, 2, \\ldots, n$, $\\mb{x}_i$'s are the covariates and $y_i \\in \\mr{N}$ are the responses; that is, $y_i$'s are obtained from event counting. $\\boldsymbol\\beta$ is the regression coefficient to be estimated by solving the minimization problem and $\\lambda$ is a regularization parameter for sparsity.\n\n\\item[iii] $l_1$-Regularized Logistic Regression: $\\min_{\\boldsymbol\\beta \\in \\mathbb{R}^p}\\sum_{i = 1}^n \\log(1 + \\exp\\{-y_i\\mathbf{x}_i^\\top\\boldsymbol\\beta\\}) + \\lambda \\vert\\vert \\boldsymbol\\beta \\vert\\vert_1$, where $n$ is the sample size and for $i = 1, 2, \\ldots, n$, $\\mb{x}_i$'s are the covariates and $y_i \\in \\{0, 1\\}$ are the responses; that is, $y_i$'s are binary labels obtained from classification. $\\boldsymbol\\beta$ is the regression coefficient to be estimated by solving the minimization problem and $\\lambda$ is a regularization parameter for sparsity.\n\n\\item[iv] Dual Problem of $l_1$-loss Support Vector Machine: given training label-instance pairs $(y_i, \\mb{z}_i)$ for $i = 1, \\ldots, l$, the problem is formulated as \n\\begin{align*}\n&\\text{minimize}_{\\boldsymbol\\alpha}\\quad \\frac{1}{2}\\boldsymbol\\omega^\\top\\boldsymbol\\omega - \\boldsymbol1^\\top\\boldsymbol\\alpha\\\\\n&\\text{subject to}\\quad \\boldsymbol\\omega = \\mathbf{A}\\boldsymbol\\alpha, \\\\\n&\\quad \\quad \\quad \\quad \\quad \\; 0 \\leq \\alpha_i \\leq C, i = 1, \\ldots, l.\n\\end{align*}\nwhere $\\mb{A} = [y_1\\mb{z}_1, \\ldots, y_l\\mb{z}_l]$, $\\boldsymbol1$ is the vector of ones, and $C$ is a given upper bound. This problem can be transformed to the form of (\\ref{general_problem}) by replacing the $\\boldsymbol\\omega$ is in the objective function by $\\mb{A}\\boldsymbol\\alpha$.\n\\end{description}\n\\item[2] The objective functions in the unconstrained problems (i),(ii) and (iii) all involve a non-smooth regularization term. They however, can be modeled as (\\ref{general_problem}). For example, in the LASSO problem, we can always take $\\boldsymbol\\beta = \\boldsymbol0$ to get an upper bound on the function value and add the constraint $\\norm{\\boldsymbol\\beta}_1 \\leq \\sum_{i=1}^n y_i^2$ to the original problem. Then by introducing new variables $u_i$, $i = 1, \\ldots, p$, we can express $\\norm{\\boldsymbol\\beta}_1$ as $\\sum_{i=1}^p u_i$ after adding the constraints $\\beta_i \\leq u_i$ and $-\\beta_i \\leq u_i$ for all $i = 1, \\ldots, p$ to the problem. We can apply the same method to $l_1$-regularized logistic regression and $l_1$-regularized Poisson regression problems.\n\\end{description}\n\n\\subsection{The Main Result}\nLet $\\mathcal{O}: \\mathbb{R}^p \\rightarrow \\mathcal{P}$ be a linear oracle that given $\\mathbf{c} \\in \\mathbb{R}^p$, returns $\\mathbf{z} = \\mathcal{O}(\\mathbf{c}) \\in \\mathcal{P}$ that $\\mathbf{c}^\\top\\mb{z} \\leq \\mathbf{c}^\\top \\mb{x}$ for every $\\mb{x} \\in \\mc{P}$; i.e. $\\mb{z} = \\arg\\min\\{\\mb{c}^\\top \\mb{x} \\; \\vert \\; \\mb{x} \\in \\mc{P}\\}$. Let $V$ be the set of vertices of polytope $\\mc{P}$.\n\n\\begin{algorithm}\n\\caption{Semi-Stochastic Frank-Wolfe algorithm with Away-Steps}\n\\label{cond_grad_1}\n\\begin{algorithmic}[1]\n\\STATE {\\bfseries Input:} $\\mathbf{x}^{(1)} \\in V$, $f_i$, $\\mb{a}_i$, $\\mb{b}$ and $L$\n\\STATE Set $\\mu^{(1)}_{\\mathbf{x}^{(1)}} = 1$, $\\mu_\\mathbf{v}^{(1)} = 0$ for any $\\mathbf{v} \\in \\mathcal{V} \/ \\{\\mathbf{x}^{(1)}\\}$ and $U^{(1)} = \\{\\mathbf{x}^{(1)}\\}$.\n\\FOR{$k = 1, 2, \\ldots$}\n\\STATE Uniformly sample $\\mc{J} =\\{j_1, \\ldots, j_{m_k}\\}$ from $\\{1, \\ldots, n\\}$ without replacement, and denote $\\mathbf{g}^{(k)} = \\frac{1}{m_k}\\sum_{i=1}^{m_k}f'_{j_i}(\\mb{a}_{j_i}^\\top\\mb{x}^{(k)})\\mb{a}_{j_i} + \\mb{b}$.\n\\STATE Compute $\\mathbf{p}^{(k)} = \\mathcal{O}(\\mathbf{g}^{(k)})$.\n\\STATE Compute $\\mathbf{u}^{(k)} \\in {\\arg\\!\\max}_{\\mathbf{v} \\in U^{(k)}}\\langle \\mathbf{g}^{(k)}, \\mathbf{v} \\rangle$.\n\\IF{$\\langle \\mathbf{g}^{(k)}, \\mathbf{p}^{(k)} + \\mathbf{u}^{(k)}- 2\\mathbf{x}^{(k)} \\rangle \\leq 0$}\n\\STATE Set $\\mathbf{d}^{(k)} = \\mathbf{p}^{(k)} - \\mathbf{x}^{(k)}$ and $\\gamma^{(k)}_{\\text{max}} = 1$. \n\\ELSE\n\\STATE Set $\\mathbf{d}^{(k)} = \\mathbf{x}^{(k)} - \\mathbf{u}^{(k)}$ and $\\gamma^{(k)}_{\\text{max}} = \\frac{\\mu^{(k)}_{\\mathbf{u}^{(k)}}}{1 - \\mu^{(k)}_{\\mathbf{u}^{(k)}}}$.\n\\ENDIF\n\\STATE Set $\\gamma^{(k)} = \\min\\{-\\frac{\\langle \\mb{g}^{(k)}, \\mb{d}^{(k)}\\rangle}{L\\norm{\\mb{d}^{(k)}}^2}, \\gamma^{(k)}_\\text{max}\\}$\n\\STATE Set $\\mathbf{x}^{(k+1)} = \\mathbf{x}^{(k)} + \\gamma^{(k)}\\mathbf{d}^{(k)}$.\n\\STATE Update $U^{(k + 1)}$ and $\\mathbf{\\mu}^{(k+1)}$ by Procedure VRU.\n\\ENDFOR\n\\STATE{\\bfseries Return:} $\\mathbf{x}^{(k+1)}$.\n\\end{algorithmic}\n\\end{algorithm}\n\\noindent The following algorithm updates a vertex representation of the current iterate and is called in Algorithm \\ref{cond_grad_1}.\n\\begin{center}\n\\begin{algorithm}[h]\n\\caption{Procedure Vertex Representation Update (VRU)}\n\\begin{algorithmic}[1]\n\\STATE {\\bfseries Input:} $\\mb{x}^{(k)}$, $(U^{(k)}, \\boldsymbol\\mu^{(k)})$, $\\mb{d}^{(k)}$, $\\gamma^{(k)}$, $\\mb{p}^{(k)}$ and $\\mb{v}^{(k)}$.\n\\IF{$\\mb{d}^{(k)} = \\mb{x}^{(k)} - \\mb{u}^{(k)}$}\n\\STATE Update $\\mu^{(k)}_\\mb{v} = \\mu_\\mb{v}^{(k)}(1 + \\gamma^{(k)})$ for any $\\mb{v} \\in U^{(k)} \/ \\{\\mb{u}^{(k)}\\}$.\n\\STATE Update $\\mu^{(k+1)}_{\\mb{u}^{(k)}} = \\mu^{(k)}_{\\mb{u}^{(k)}}(1 + \\gamma^{(k)}) - \\gamma^{(k)}$.\n\\IF{$\\mu^{(k+1)}_{\\mb{u}^{(k)}} = 0$}\n\\STATE Update $U^{(k+1)} = U^{(k)} \/\\{\\mb{u}^{(k)}\\}$\n\\ELSE\n\\STATE Update $U^{(k+1)} = U^{(k)}$\n\\ENDIF\n\\ENDIF\n\\STATE Update $\\mu_\\mb{v}^{(k+1)} = \\mu^{(k)}_\\mb{v}(1 - \\gamma^{(k)})$ for any $\\mb{v} \\in U^{(k)} \/ \\{\\mb{p}^{(k)}\\}$.\n\\STATE Update $\\mu^{(k+1)}_{\\mb{p}^{(k)}} = \\mu^{(k)}_{\\mb{p}^{(k)}}(1 - \\gamma^{(k)}) + \\gamma^{(k)}$.\n\\IF{$\\mu_{\\mb{p}^{(k)}}^{(k+1)} = 1$}\n\\STATE Update $U^{(k+1)} = \\{\\mb{p}^{(k)}\\}$.\n\\ELSE\n\\STATE Update $U^{(k+1)} = U^{(k)} \\cup \\{\\mb{p}^{(k)}\\}$.\n\\ENDIF\n\\STATE (Optional) Carath\\'eodory's theorem can be applied for the vertex representation of $\\mb{x}^{(k+1)}$ so that $\\vert U^{(k+1)}\\vert = p+1$ and $\\boldsymbol\\mu^{(k+1)} \\in \\mr{R}^{p+1}$.\n\\STATE {\\bfseries Return:} $(U^{(k+1)}, \\boldsymbol\\mu^{(k+1)})$\n\\end{algorithmic}\n\\end{algorithm}\n\\end{center}\n\n\\noindent We need to introduce some definitions before presenting the theorems in this paper. Write $\\mb{A} = [\\mb{a}_1, \\mb{a}_2, \\ldots, \\mb{a}_n]^\\top$ and $\\mb{G}(\\mb{x})= (1\/n)[f_1'(\\mb{a}_1^\\top\\mb{x}), \\ldots, f_n'(\\mb{a}_n^\\top\\mb{x})]^\\top $. Define $D = \\sup\\{\\norm{\\mb{x} - \\mb{y}}\\, \\vert \\, \\mb{x}, \\mb{y} \\in \\mc{P}\\}$ and $G = \\sup\\{\\norm{\\mb{G}(\\mb{x})}\\, \\vert\\, \\mb{x} \\in \\mc{P}\\}$. It follows from compactness of $\\mc{P}$ and continuity of $F(\\cdot)$ that $D < \\infty$ and $G < \\infty$. Write $\\kappa =\\theta^2 \\{D\\norm{\\mb{b}} + 3GD\\norm{\\mb{A}} + \\frac{2n}{\\sigma_F}(G^2+1)\\}$ where $\\sigma_F = \\min\\{\\sigma_1, \\ldots, \\sigma_n\\} > 0$ and $\\theta$ is the Hoffman constant associated with matrix $[\\mb{C}^\\top,\\mb{A}^\\top, \\mb{b}^\\top]$ that is $\\theta = \\max\\{1 \/ \\lambda_{\\min}(\\mb{BB}^\\top) \\;\\vert \\; \\mb{B} \\in B\\}$, $\\lambda_{\\min}(\\mb{BB}^\\top)$ is the smallest eigenvalue of $\\mb{BB}^\\top$, where $B$ is the set of all matrices constructed by taking linearly independent rows from the matrix $[\\mb{C}^\\top,\\mb{A}^\\top, \\mb{b}^\\top]$. We denote by $I(\\mb{x})$, the index set of active constraints at $\\mb{x}$; that is, $I(\\mb{x}) = \\{i \\in {1, \\ldots, m} \\; \\vert \\; \\mb{C}_i\\mb{x} = d_i\\}$. In a similar way, we define the set of active constraints for a set $U$ by $I(U) = \\{i \\in \\{1, \\ldots, m\\} \\; \\vert \\; \\mb{C}_i\\mb{v} = d_i, \\forall \\mb{v} \\in U\\} = \\cap_{\\mb{v} \\in U}I(\\mb{v})$. Let $V$ be the set of vertices of $\\mc{P}$, then define $\\Omega_\\mathcal{P} = \\frac{\\zeta}{\\phi}$ where \n\\begin{align*}\n\\zeta &= \\min_{\\mathbf{v} \\in V, i \\in \\{1, \\ldots, m\\}: d_i > \\mathbf{C}_i \\mathbf{v}} (d_i - \\mathbf{C}_i\\mathbf{v}), \\\\\n\\phi &= \\max_{i \\in \\{1, \\ldots, m\\} \/ I(V)}\\norm{\\mathbf{C}_i}.\n\\end{align*}\n\\noindent\\textbf{Remark:} The vertex representation update procedure can also be implemented by using Carath\\'eodory's theorem so that each $\\mb{x}^{(k)}$ can be written as a convex combination of at most $N = p+1$ vertices of the polytope $\\mc{P}$. Then the set of vertices $U^{(k)}$ and their corresponding weights $\\boldsymbol\\mu^{(k)}$ can be updated according to the convex combination.\n\\begin{Theorem}\nLet $\\{\\mathbf{x}^{(k)}\\}_{k \\geq 1}$ be the sequence generated by Algorithm \\ref{cond_grad_1} for solving Problem $(\\ref{general_problem})$, $N$ be the number of vertices used to represent $\\mb{x}^{(k)}$ (if VRU is implemented by using Carath\\'eodory's theorem, $ N = p + 1$, otherwise $N = \\vert V \\vert$) and $F^*$ be the optimal value of the problem. Let $\\rho = \\Omega_{\\mathcal{P}}^2 \/\\{8N^2 \\kappa D\\max (G, LD)\\}$ and set $m_k = \\lceil n \/ (1 + n(1 - \\rho)^{2\\alpha k})\\rceil$ for some $0 < \\alpha < 1$. Let $D^{(k)}$ be the event that the algorithm removes a vertex from the current convex combination at iteration $k$, that is, the algorithm performs a `drop step' at iteration $k$. Assume $\\mr{P}(D^{(k)}) \\leq (1 - \\rho)^{\\beta k}$ for some $ 0 < \\beta < 1$.\nThen for every $k \\geq 1$\n\\begin{align}\\label{rate_of_convergence}\n\\mathbb{E}\\{F(\\mathbf{x}^{(k+1)}) - F^*\\} \\leq C_3(1 - \\rho)^{\\min (\\alpha, \\beta)k}\n\\end{align}\nwhere\n\\begin{align*}\nC_3 = F(\\mb{x}^{(1)}) - F^* + \\frac{G\\sqrt{C_2}}{L\\{1 - (1 - \\rho)^{1 -\\alpha}\\}} + \\frac{G^2}{2L\\{1 - (1 - \\rho)^{1 - \\beta}\\}}\n\\end{align*}\nand \n\\begin{align*}\nC_2 &= \\sup_{\\mb{x} \\in \\mc{P}}[ \\frac{1}{n}\\sum_{i=1}^n \\{f_i'(\\mb{a}_i^\\top\\mb{x})\\}^2\\mb{a}_i^\\top\\mb{a}_i \\\\\n&\\quad - \\frac{1}{n(n-1)}\\sum_{i \\neq j}f_i'(\\mb{a}_i^\\top\\mb{x})f_j'(\\mb{a}_i^\\top\\mb{x})\\mb{a}^\\top_i\\mb{a}_j ] < \\infty.\n\\end{align*}\n\\end{Theorem}\n\\noindent Proof of the Theorem 1 is presented in the appendix. It is worth noting that by dividing both sides of (\\ref{rate_of_convergence}) by $F^{(1)} - F^*$, this linear convergence result for Algorithm 1 depends on relative function values. Therefore, the linear convergence of the proposed algorithm is invariant to scaling of the function.\n\n\\noindent\\textbf{Remarks}: The constant $\\rho$ depends on the \"vertex-face\" distance of a polytope as discussed in \\cite{BS15}. This also gives some intuition as to why constraint set $\\mc{P}$ has to be a polytope for linear convergence of the Frank-Wolfe algorithm with away-steps. When the boundary of the constraint set is `curved' as is the case for the $l_2$-ball, every point on the boundary is an extreme point. Then, faces can always be constructed so that the vertex-face distance is infinitesimal. Thus, we cannot get linear convergence when we apply the proof to a general convex constraint.\n\n\\section{Block Coordinate Semi-Stochastic Frank-Wolfe Algorithm with Away-Steps}\nIn this section, we assume the domain $\\mc{P}$ takes the form $\\mc{P} = \\mc{P}_{[1]} \\times \\mc{P}_{[2]} \\times \\cdots \\times \\mc{P}_{[q]}$, where each $\\mc{P}_{[i]}$ is a compact polytope that can be expressed as $\\mc{P}_{[i]} = \\{\\mb{x} \\in \\mr{R}^{p_i} \\, \\vert \\, \\mb{C}_{[i]}\\mb{x} \\leq \\mb{d}_{[i]}\\}$ where $ \\mb{C}_{[i]} \\in \\mr{R}^{m_i \\times p_i}$, $\\mb{d}_{[i]} \\in \\mr{R}^{m_i}$, $\\sum_{i = 1}^q m_i = m$ and $\\sum_{i=1}^q p_i = p$. Hence, $\\mc{P}$ still has the polytopic representation $\\mc{P} = \\{\\mb{x} \\in \\mr{R}^p \\: \\vert \\: \\mb{C}\\mb{x} \\leq \\mb{d}\\}$ where\n\\begin{align*}\n\\mb{C}= \\begin{bmatrix}\n\\mb{C}_{[1]} & & &\\\\\n& \\mb{C}_{[2]} & & \\\\\n& &\\ddots & \\\\\n& & & \\mb{C}_{[q]}\n\\end{bmatrix} \\quad \\quad \\text{ and }\\quad \\quad \\mb{d} = \\begin{bmatrix}\n\\mb{d}_{[1]}\\\\\n\\mb{d}_{[2]}\\\\\n\\vdots\\\\\n\\mb{d}_{[q]}\n\\end{bmatrix}.\n\\end{align*}\nExamples of such Cartesian product constraints include the dual problem of structural SVMs, fitting marginal models in multivariate regression and multi-task learning. Let $V_{[i]}$ be the set of vertices of polytope $\\mc{P}_{[i]}$. We use subscripts $[i]$ to denote vectors, matrices, sets and other quantities correspond to the $i$-th block of constraints. Specifically,\nlet $\\mb{x}_{[i]} \\in \\mr{R}^{p_i}$ and $\\mb{x} = [\\mb{x}_{[1]}^\\top, \\mb{x}_{[2]}^\\top, \\cdots, \\mb{x}_{[q]}^\\top]^\\top \\in \\mr{R}^p$. Define $\\mb{U}_{[i]} \\in \\mr{R}^{p\\times p_i}$ as the sub-matrices of $p \\times p$ identity matrix that corresponds to the $i$-th block of the product domain, that is $[\\mb{U}_{[1]},\\mb{U}_{[2]}, \\ldots, \\mb{U}_{[q]}] = \\mb{I}_{p \\times p}$. Hence $\\mb{x}_{[i]} = \\mb{U}_{[i]}^\\top\\mb{x}$ and $\\mb{x} = \\sum_{i = 1}^q \\mb{U}_{[i]}\\mb{x}_{[i]}$. Let $\\mc{O}_{[i]}$ be the linear oracle corresponding to the $i$-th block polytope in a lower dimension space. With the above notation, we are ready to present the block-coordinate version of the semi-stochastic Frank-Wolfe algorithm with away-steps and prove its linear convergence.\n\n\\begin{algorithm}[t]\n\\caption{Block Coordinate Semi-Stochastic Frank-Wolfe Algorithm with Away-Steps}\n\\label{block_cond_grad_1}\n\\begin{algorithmic}[1]\n\\STATE {\\bfseries Input:} $\\mathbf{x}^{(1)} \\in V_{[1]} \\times \\cdots \\times V_{[q]}$, $f_i$, $\\mb{a}_i$, $\\mb{b}$ and $L$\n\\STATE Let $\\mu^{(1)}_{\\mathbf{x}^{(1)}_{[i]}} = 1$, $\\mu_\\mathbf{v}^{(1)} = 0$ for any $\\mathbf{v} \\in V_{[i]} \/ \\{\\mathbf{x}^{(1)}_{[i]}\\}$ and $U^{(1)}_{[i]} = \\{\\mathbf{x}^{(1)}_{[i]}\\}$.\n\\FOR{$k = 1, 2, \\ldots$}\n\\STATE Uniformly sample $\\mc{J} =\\{j_1, \\ldots, j_{m_k}\\}$ from $\\{1, \\ldots, n\\}$ without replacement, and denote $\\mathbf{g}^{(k)} = \\frac{1}{m_k}\\sum_{i=1}^{m_k}f'_{j_i}(\\mb{a}_{j_i}^\\top\\mb{x}^{(k)})\\mb{a}_{j_i} + \\mb{b}$.\n\\STATE Uniformly sample $\\mc{L} = \\{l_1, \\ldots, l_r \\}$ from $\\{1, \\ldots, q\\}$ without replacement\n\\FOR{$i = 1, 2, \\ldots, r$, in parallel}\n\t\\STATE Compute $\\mathbf{p}^{(k)}_{[l_i]} = \\mathcal{O}_{[l_i]}(\\mb{U}_{[l_i]}^\\top\\mathbf{g}^{(k)})$\n\t\\STATE Compute $\\mathbf{u}^{(k)}_{[l_i]} \\in {\\arg\\!\\max}_{\\mathbf{v} \\in U^{(k)}_{[l_i]}}\\langle\\mb{U}_{[l_i]}^\\top \t\t\\mathbf{g}^{(k)}, \\mathbf{v} \\rangle $.\n\\IF{$\\langle \\mb{U}_{[l_i]}^\\top\\mathbf{g}^{(k)}, \\mathbf{p}^{(k)}_{[l_i]} + \\mathbf{u}^{(k)}_{[l_i]}- 2\\mathbf{x}^{(k)}_{[l_i]}\\rangle \\leq 0$}\n\\STATE Set $\\mathbf{d}^{(k)}_{[l_i]} = \\mathbf{p}^{(k)}_{[l_i]} - \\mathbf{x}^{(k)}_{[l_i]}$ and $\\bar{\\gamma}^{(k)}_{[l_i]} = 1$. \n\\ELSE\n\\STATE Set $\\mathbf{d}^{(k)}_{[l_i]} = \\mathbf{x}^{(k)}_{[l_i]} - \\mathbf{u}^{(k)}_{[l_i]}$ and $\\bar{\\gamma}^{(k)}_{[l_i]} = \\frac{\\mu^{(k)}_{\\mathbf{u}^{(k)}_{[l_i]}}}{1 - \\mu^{(k)}_{\\mathbf{u}^{(k)}_{[l_i]}}}$.\n\\ENDIF\n\\STATE Set $\\gamma^{(k)}_{[l_i]} = \\min\\{-\\frac{\\langle \\mb{U}_{[l_i]}^\\top \\mb{g}^{(k)}, \\mb{d}^{(k)}_{[l_i]} \\rangle}{L\\norm{\\mb{d}^{(k)}_{[l_i]}}^2}, \\bar{\\gamma}^{(k)}_{[l_i]} \\}$.\n\t\\STATE Update $U^{(k + 1)}_{[i]}$ and $\\mathbf{\\mu}^{(k+1)}_{[i]}$ by Procedure VRU.\n\\ENDFOR\n\\STATE Update $\\mathbf{x}^{(k+1)} = \\mathbf{x}^{(k)} + \\sum_{i= 1}^r \\gamma^{(k)}_{[l_i]}\\mb{U}_{[l_i]}\\mathbf{d}^{(k)}_{[l_i]}$.\n\\ENDFOR\n\\STATE {\\bfseries Return:} $\\mathbf{x}^{(k+1)}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{Theorem}\nLet $\\{\\mathbf{x}^{(k)}\\}_{k \\geq 1}$ be the sequence generated by Algorithm \\ref{block_cond_grad_1} for solving Problem $(\\ref{general_problem})$ with block-coordinate structure constraints ,$N$ be the number of vertices used to represent $\\mb{x}^{(k)}$ (if VRU is implemented by using Carath\\'eodory's theorem, $ N = p + 1$, otherwise $N = \\vert V \\vert$) and $F^*$ be the optimal value of the problem. Let $\\hat{\\rho} = r\\Omega_{\\mathcal{P}}^2 \/\\{8N^2 \\kappa q^2D \\max (G, LD)\\}$ and set $m_k = \\lceil n \/ (1 + n(1 - \\hat{\\rho})^{2\\mu k})\\rceil$ for some $0 < \\mu < 1$. Let $D^{(k)}_{[i]}$ be the event that the algorithm removes a vertex from the current convex combination in block $i$ at iteration $k$, that is, the algorithm performs a `drop step' at iteration $k$. Assume $\\max_{i}\\{\\mr{P}(D^{(k)}_{[i]})\\} \\leq (1 - \\hat{\\rho})^{\\lambda k}$ for some $ 0 < \\lambda < 1$ and all $k \\geq 1$. Then for any $k \\geq 1$\n\\begin{align*}\n\\mathbb{E}\\{F(\\mathbf{x}^{(k+1)}) - F^*\\}\n&\\leq C_4(1 - \\hat\\rho)^{\\min(\\mu, \\lambda)}\n\\end{align*}\nwhere\n\\begin{align*}\nC_4 = \\{F(\\mb{x}^{(1)}) - F^* + \\frac{rG\\sqrt{C_2}}{qL\\{1 - (1 - \\hat{\\rho})^{1 -\\mu}\\}} + \\frac{rG^2}{2L\\{1 - (1 - \\hat{\\rho})^{1 - \\lambda}\\}}\\}.\n\\end{align*}\n\\end{Theorem}\n\\noindent Proof of the Theorem 2 can be found in the appendix.\n\n\\noindent\\textbf{Remark:} All of the sub-problems in the inner loop of the proposed algorithm can be solved in parallel. This feature can further boost the performance of the algorithm. It seems that the rate of convergence of the algorithm for block-coordinate problems is worse since $\\hat{\\rho} < \\rho$. One reason for this is the algorithm only uses sampled blocks instead of a full data pass in every iteration. Another reason is that the theoretical result is based on the worst case scenario happening at the same time in every sampled block (which won't happen in real implementations) for the ease of proving linear convergence. The worst case scenario is the so called `drop-step' and the detailed analysis can be found in the supplementary material.\n\\section{Numerical Studies}\nIn this section, we apply the semi-stochastic Frank-Wolfe algorithm with away-steps to two popular problems in machine learning. The first one is a graph guided fused LASSO problem for multi-task learning using a simulated data set. The second one is a structural support vector machine (SVM) problem using a real data set in speech recognition.\n\\subsection{Multi-task Learning via Graph Guided Fused LASSO}\n\\begin{figure*}\n\\centering\n\\begin{tabular}{rrl}\n\\includegraphics[scale = 0.28]{true_coeff.eps}&\n\\includegraphics[scale = 0.28]{proxgrad.eps} &\n\\multirow{2}*[2.7cm]{\\includegraphics[scale=0.5]{function_value.eps}}\\\\\n\\includegraphics[scale = 0.28]{bcfwas.eps} &\n\\includegraphics[scale = 0.28]{Correlation_Block.eps}&\n\\end{tabular}\n\\caption{(a) is the true regression coefficient matrix in the simulated data for GFLASSO problem, (b) is the estimated regression coefficient matrix by using proximal gradient method (c) is the estimated regression coefficient matrix by using block-coordinate Frank-Wolfe algorithm with away-steps (BCFWAS), (d) is the truncated correlation matrix of the outputs $\\mb{Y}$ in the simulated data set base on which the graph in the GFLASSO problem is constructed and (e) is the plot of the logarithmic objective function values of both methods. Median of the 10 sample-paths when running both algorithms are plotted in the lines. The shaded areas shows the upper and lower bounds at each iteration in the 10 replications. The detailed plot in the middle shows that the BCFWAS beats the Prox-Grad after the 20th iteration in this experiment.}\n\\label{gflasso}\n\\end{figure*}\n\nConsider the following multivariate linear model:\n\\begin{align*}\n\\mathbf{y}_i = \\mathbf{X}\\boldsymbol\\beta_i + \\boldsymbol\\epsilon_i \\quad\\quad i = 1,\\ldots, N\n\\end{align*}\nwhere $\\mathbf{X} \\in \\mathbb{R}^{N \\times J}$ denotes the matrix of input data for $J$ covariates over $N$ samples, $\\mathbf{Y}=[\\mathbf{y}_1, \\ldots, \\mathbf{y}_K] \\in \\mathbb{R}^{N \\times K}$ denotes the matrix of output data for $K$ tasks, $\\mathbf{B} = [\\boldsymbol\\beta_1, \\ldots, \\boldsymbol\\beta_K] \\in \\mathbb{R}^{J \\times K}$ denotes the matrix of regression coefficients for the $K$ tasks and the $\\boldsymbol\\epsilon_i$'s denote the noise terms that are independent and identically distributed. Let $G = (V, E)$ be a graph where $V$ is the set of vertices and $E$ is the set of edges. In \\citep{CLKCX11}, the graph is constructed by taking each task as a vertex and each non-zero off-diagonal entry of the correlation matrix of $\\mb{Y}$, as an edge. When the correlation between two tasks is small, it will be truncated to $0$ when calculating the correlation matrix and hence there won't be edges between such tasks. The graph guided fused LASSO (GFLASSO) problem for multi-task sparse regression problem is formulated as \n\\begin{align*}\n\\min_{\\mathbf{B}\\in \\mathbb{R}^{J \\times K}} \\frac{1}{2}\\norm{\\mathbf{Y} - \\mathbf{XB}}_F^2 + \\Omega(\\mathbf{B}) + \\lambda\\norm{\\mathbf{B}}_1,\n\\end{align*}\nwhere $$\\Omega(\\mathbf{B}) = \\gamma \\sum_{e = (m, l) \\in E} \\vert r_{ml}\\vert\\sum_{j=1}^J \\vert \\beta_{jm} - \\text{sign}(r_{ml})\\beta_{jl}\\vert, $$ $r_{ml}$ is the $(m, l)$-th entry of \\textbf{cor}($\\mathbf{Y}$), the truncated correlation matrix of ${\\mathbf{Y}}$ at a pre-determined level, $\\norm{\\cdot}_F$ denotes the Frobenius norm, and $\\gamma$ and $\\lambda$ are the regularization parameters.\nThe GFLASSO was first proposed by \\cite{KSX09} for problems in quantitative trait network in genomic association studies. To solve GFLASSO, \\cite{KSX09} formulated it as a quadratic programming problem and proposed using an active-set method. Later \\cite{CLKCX11} proposed a smoothing proximal gradient method which is more saclable than the active-set method. We propose to use the block-coordinate semi-stochastic Frank-Wolfe algorithm with away-steps to solve this problem. When the underlying graph is a union of several connected components (as it is the case in most genetic studies) instead of being fully connected, the proposed algorithm can be applied by considering each connected component as a block. Such a structure effectively transforms a large original problem into several small subproblems which are easier to solve. When the regression coefficient matrix $\\mb{B}$ is sparse, which is also common in most genetic studies, the sparse update in each step of the Frank-Wolfe type algorithms will also have the advantage of being able to extract information from the data sets efficiently. In our numerical example, we follow the data generation procedure in \\cite{CLKCX11} which simulates genetic association mapping data. We set $N = 200$, $J = 50$, $K = 20$, entries of $\\mb{X}$ and $\\boldsymbol\\epsilon_k$'s are generated as standard normal random variables, $\\boldsymbol\\beta_k$'s are generated such that the correlation matrix of outputs of $\\mb{y}_k$'s has a block structure and all non-zero elements of the $\\boldsymbol\\beta_k$'s are equal to 1. The entries in the correlation matrix of $\\mb{Y}$ are truncated to $0$ when their absolute value is smaller than $0.7$. In the illustration, fitting GFLASSO using the proposed algorithm reveal the structures of the coefficients very quickly and the proposed algorithm converges much faster than the Prox-Grad method that was proposed in \\cite{CLKCX11}.\n\n\\noindent\\textbf{Remark:} Initial values for Algorithm \\ref{block_cond_grad_1} must be vertices of each `block' polytope and even with a relatively bad initialization, the proposed algorithm has a better performance than the Prox-Grad algorithm in this problem. To obtain the same initialization as in the Prox-Grad algorithm, we can write the initial value as a convex combination of the vertices in each block and set $U_{[i]}^{(1)}$ and $\\boldsymbol\\mu_{[i]}^{(1)}$ accordingly. It is worth noting that due to the numerical error, the points returned by the oracles may not be the vertices of the constraint polytopes. As a result, we may obtain a smaller function value than the optimal value due to the tiny in-feasibilities. Such problems do not occur when the vertices are integral, which is the case of our next example.\n\n\\subsection{Structural Support Vector Machines (SVM)} \nThe idea of using the Frank-Wolfe algorithm to solve the dual of structural SVM problems was introduced by \\cite{LJJSP13}. Briefly speaking, structural SVM solves multi-label classification problems through a linear classifier (with parameter $\\mb{w}$) $h_\\mb{w}(\\mb{x}) = \\arg\\!\\max_{\\mb{y} \\in \\mc{Y}(\\mb{x})}\\langle \\mb{w}, \\phi(\\mb{x}, \\mb{y}) \\rangle$, where $\\mb{y}$ is a structured output given an input $\\mb{x}$, $\\mc{Y}(\\mb{x})$ denotes the set of all possible labels for an input $\\mb{x}$, $\\phi: \\mc{X} \\times \\mc{Y} \\rightarrow \\mr{R}^d$ is a given feature or basis map, and $\\mc{X}$ denotes the set of all possible inputs. To learn $\\mb{w}$ from training samples $\\{(\\mb{x}_i, \\mb{y}_i), i = 1, \\ldots, n\\} $, we solve \n\\begin{align*}\n\\min_{\\mb{w}, \\boldsymbol\\xi} \\quad &\\frac{\\lambda}{2}\\norm{\\mb{w}} + \\frac{1}{n}\\sum_{i=1}^n \\xi_i \\\\\n\\text{s.t.} \\quad & \\langle \\mb{w}, \\psi_i(\\mb{y}) \\rangle \\geq L_i(y) -\\xi_i \\\\\n & \\mb{y} \\in \\mc{Y}_i \\\\\n & i = 1, 2, \\ldots, n,\n\\end{align*}\nwhere $\\psi_i(\\mb{y}) = \\phi(\\mb{x}_i, \\mb{y}_i) -\\phi(\\mb{x}_i, \\mb{y})$ is the potential function, $L_i(\\mb{y})$ denotes loss incurred by predicting $\\mb{y}$ instead of the observed label $\\mb{y}_i$, $\\mc{Y}_i = \\mc{Y}(\\mb{x}_i)$, $\\xi$'s are the surrogate losses for the data points and $\\lambda$ is a regularization parameter. Now consider the Lagrange dual of the above problem. Let $\\alpha_i(\\mb{y}) \\in \\mr{R}$ be the dual variable associated with training sample $i$ and possible output $\\mb{y} \\in \\mc{Y}_i$, $\\boldsymbol\\alpha_i \\in \\mr{R}^{m_i}$ be the concatenation of all $\\alpha_i(\\mb{y})$ over all $ \\mb{y} \\in\\mc{Y}_i$, and $\\boldsymbol\\alpha \\in R^m$ be the column concatenation of all $\\alpha_i(\\mb{y})$, where $m_i = \\vert \\mc{Y}_i \\vert$ and $m = \\sum_{i=1}^n m_i$. Then the dual problem can be written as\n\\begin{align*}\n\\min_{\\boldsymbol\\alpha} \\quad &F(\\boldsymbol\\alpha) := \\frac{\\lambda}{2}\\norm{\\mb{A}\\boldsymbol\\alpha}^2 -\\mb{b}^\\top \\boldsymbol\\alpha \\\\\n\\text{s. t.} \\quad & \\boldsymbol\\alpha \\geq \\mb{0} \\\\\n\t\t\t\t\t\t\t\t\t & \\sum_{\\mb{y} \\in \\mc{Y}_i}\\alpha_i(\\mb{y}) = 1\\\\\n\t\t\t\t\t\t\t\t\t & i = 1, \\ldots, n,\n\\end{align*}\nwhere $A \\in \\mr{R}^{d \\times m}$ whose columns are given by $\\psi_i(\\mb{y}) \/(\\lambda n), \\mb{y} \\in \\mc{Y}_i, i = 1, 2, \\ldots, n$ and $b \\in \\mr{R}^m$ whose entries are $L_i(\\mb{y})\/n, \\mb{y} \\in \\mc{Y}_i, i = 1, 2, \\ldots, n$. Given a dual solution $\\hat{\\boldsymbol\\alpha}$, the corresponding primal solution can be retrieved from the relation $\\hat{\\mb{w}} = \\mb{A}\\hat{\\boldsymbol\\alpha}$ which is implied by KKT conditions. Observe that $\\norm{\\mb{A}\\boldsymbol\\alpha}^2 = \\sum_{i=1}^d (\\mb{A}_i \\boldsymbol\\alpha)^2$ and the constraint set for the dual problem is a Cartesian product of polytopes that can be written as $\\mc{M} := \\Delta_{\\vert \\mc{Y}_1 \\vert} \\times \\cdots \\times \\Delta_{\\vert \\mc{Y}_n \\vert}$ where $\\Delta_{\\vert\\mc{Y}_i\\vert}$ is the simplex generated by the elements in $\\mc{Y}_i$. This suggests the block coordinate stochastic Frank-Wolfe algorithm with away-steps can be applied to this problem if we have linear oracles for each block which solves\n\\begin{align*}\n\\min_{\\boldsymbol\\alpha_i \\in \\Delta_{\\vert \\mc{Y}_i \\vert}} \\langle \\nabla_{[i]} F(\\cdot), \\boldsymbol\\alpha_i \\rangle\n\\end{align*}\nwhere $\\nabla_{[i]} F(\\cdot)$ denotes the partial derivative of $F(\\cdot)$ with respect to $i$-th block of variables.\nSince the gradient of $F$, $\\nabla F = \\lambda \\mb{A}^\\top \\mb{A}\\boldsymbol\\alpha - \\mb{b} = \\lambda\\mb{A}^\\top\\mb{w} - \\mb{b}$ whose $(i, \\mb{y})$-th entry is $\\{\\langle \\mb{w}, \\psi_i(\\mb{y}) \\rangle - L_i(\\mb{y})\\} \/ n := -H_i(\\mb{y}; \\mb{w}) \/ n$, the above linear oracle is equivalent to the following maximization oracle\n\\begin{align*}\n\\max_{\\mb{y} \\in \\mc{Y}_i} H_i(\\mb{y}; \\mb{w}).\n\\end{align*}\n\\begin{algorithm}[tb]\n\\caption{Block Coordinate Semi-Stochastic Frank-Wolfe Algorithm with Away-Steps for Structural SVM}\n\\label{block_cond_grad_2}\n\\begin{algorithmic}[1]\n\\STATE Let $\\mb{y}^{(1)} \\in \\mc{Y}_1 \\times \\cdots \\times \\mc{Y}_n$ where $\\mu^{(1)}_{\\mb{y}^{(1)}_{[i]}} = 1$, $\\mu_\\mathbf{v}^{(1)} = 0$ for any $\\mathbf{v} \\in \\mathcal{Y}_i \/ \\{\\mb{y}^{(1)}_{[i]}\\}$ and $U^{(1)}_{[i]} = \\{\\mathbf{y}^{(1)}_{[i]}\\}$.\n\\STATE Set $\\mb{w}^{(1)}_i = \\frac{1}{\\lambda n}\\psi_i(\\mb{y}^{(1)}_{[i]})$, $\\ell^{(1)}_i = \\frac{1}{n}L_i(\\mb{y}^{(1)}_{[i]})$, $\\mb{w}^{(1)} = \\sum_{i=1}^n \\mb{w}^{(1)}_i$, and $\\ell^{(1)}= \\sum_{i=1}^n \\ell^{(1)}_i$.\n\\FOR{$k = 1, 2, \\ldots$}\n\\STATE Uniformly sample $\\mc{J} =\\{j_1, \\ldots, j_{m_k}\\}$ from $\\{1, \\ldots, d\\}$ without replacement.\n\\STATE Randomly pick $i$ from $\\{1, \\ldots, n\\}$\n\\STATE Compute $\\mathbf{p}^{(k)}_{[i]} = \\arg\\!\\max_{\\mb{y} \\in \\mc{Y}_i}H_i(y; I_\\mc{J}\\mb{w}^{(k)})$\n\\STATE Compute $\\mathbf{u}^{(k)}_{[i]} \\in {\\arg\\!\\max}_{\\mathbf{v} \\in U^{(k)}_{[i]}}\\langle I_\\mc{J}\\mb{w}^{(k)}, \\psi_i(\\mathbf{v}) \\rangle - L_i(\\mb{v})$. \n\t\\IF{$\\langle I_\\mc{J}\\mb{w}^{(k)}, \\frac{1}{\\lambda n}\\psi_i(\\mb{p}^{(k)}_{[i]}) - \\mb{w}^{(k)}_i \\rangle + \\ell^{(k)}_i - \\frac{1}{n}L_i(\\mb{p}_{[i]}^{(k)}) \\leq \\langle I_\\mc{J}\\mb{w}^{(k)}, \\mb{w}^{(k)}_i - \\frac{1}{\\lambda n}\\psi_i(\\mb{u}^{(k)}_{[i]}) \\rangle + \\frac{1}{n}L_i(\\mb{u}_{[i]}^{(k)}) - \\ell^{(k)}_i $}\n\\STATE Set $\\mb{d}^{(k)}_{i} = \\frac{1}{\\lambda n}\\psi_i(\\mb{p}^{(k)}_{[i]}) - \\mb{w}^{(k)}_i$, ${e}^{(k)}_{i} = \\frac{1}{n}L_i(\\mb{p}_{[i]}^{(k)}) - \\ell_i^{(k)}$ and $\\bar{\\gamma}^{(k)}_{i} = 1$.\n\t\\ELSE\n\t\\STATE Set $\\mathbf{d}^{(k)}_{i} = \\mathbf{w}^{(k)}_{i} - \\frac{1}{\\lambda n}\\psi_i(\\mathbf{u}^{(k)}_{[i]})$, ${e}^{(k)}_{i} = \\ell_i^{(k)} - \\frac{1}{n}L_i(\\mb{u}_{[i]}^{(k)}) $, and $\\bar{\\gamma}^{(k)}_{i} = \\frac{\\mu^{(k)}_{\\mathbf{u}^{(k)}_{[i]}}}{1 - \\mu^{(k)}_{\\mathbf{u}^{(k)}_{[i]}}}$.\n\t\\ENDIF\n\t\\STATE Set $\\gamma^{(k)} = \\max\\{0, \\min\\{\\frac{-\\lambda\\langle \\mb{w}^{(k)}, \\mb{d}^{(k)}_{i}\\rangle + e^{(k)}_{i}}{\\lambda\\norm{\\mb{d}^k_{i}}^2}, \\bar{\\gamma}^{(k)}_{i}\\} \\}$\n\t\\STATE Update $\\mb{w}^{(k+1)}_i = \\mb{w}^{(k)} + \\gamma^{(k)} \\mb{d}^{(k)}_{i}$\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\nAccording to \\cite{LJJSP13}, this maximization problem is known as the \\textit{loss-augmented decoding} subproblem which can be solved efficiently. The last thing which may prohibit us from applying the proposed algorithm is the potentially exponential number $\\vert \\mc{Y}_i \\vert$ of dual variables due to its combinatorial nature, which makes maintaining a dense $\\boldsymbol\\alpha$ impossible. This is also the reason why other algorithms, for example QP, become intractable for this problem. However, the simplex structure of the constraint sets enables us to keep the only non-zero coordinate of the solution to the linear oracle which corresponds to the solution of the loss-augmented decoding sub-problem in each iteration. Thus, we only need to keep track of the initial value $\\boldsymbol\\alpha^{(0)}$ and a list of previously seen solutions to the maximum oracle. Another way to keep track of the vertices is to maintain a list of primal variables from the linear transformation $\\mb{w} = \\mb{A}\\boldsymbol\\alpha$ when the dimension of $\\mb{w}$ is moderate. Denote the vertex set of $\\Delta_{\\vert \\mc{Y}_i \\vert}$ as $\\mc{V}_i$ and for an set $\\mc{J} \\subset \\{1, 2, \\ldots, d\\}$, let $I_\\mc{J}$ be a $d \\times d$ matrix whose $j$-th diagonal entry equals $1$ for all $j \\in \\mc{J}$ and all other entries are zero. With above notation, we can apply the stochastic block coordinate Frank-Wolfe algorithm with away-steps the the structural SVM problem.\nWe apply the above algorithm on the OCR dataset ($n = 6251, d = 4028$) from \\cite{TGK03} and compare it with the block-coordinate Frank-Wolfe algorithm. To make a fair comparison, the initial vertex $\\mb{y}^{(1)}_i$ in $i$-th coordinate block is chosen to be the observed tag $\\mb{y}_i$ corresponding to the choice of primal variables $\\mb{w}_i^{(1)} = 0$ as in the implementation of \\cite{LJJSP13}. It is worth noting that in the experiments, the performance of both algorithms was worse when the initializations were changed. Computational results for OCR dataset with regularization parameters $\\lambda = 0.05, 0.01 $ and $0.001$ are presented here. From the figures, we can see that the stochastic Block-Coordinate Frank-Wolfe Algorithm with Away-Steps (BCFWAS) dominates the Block-Coordinate Frank Wolfe algorithm (BCFW) for every $\\lambda$ when an accurate solution is required.\n\\begin{figure*}\n\\vskip 0.2in\n\\begin{center}\n\\subfigure[]{\\includegraphics[scale = 0.55]{lambda_05.eps}}\n\\subfigure[]{\\includegraphics[scale = 0.55]{lambda_01.eps}}\n\\subfigure[]{\\includegraphics[scale = 0.55]{lambda_005.eps}}\n\\subfigure[]{\\includegraphics[scale = 0.55]{lambda_001.eps}}\n\\caption{Each figure is plotted the number of effective passes of data versus the relative duality gap. In the implementation, we used the weighted average technique introduced in \\cite{LJJSP13} in both algorithms which outputs the series $\\bar{\\mb{w}}^{(k+1)} = k\/(k+2)\\bar{\\mb{w}}^{(k)} + 2\/(k+2)\\mb{w}^{(k+1)}$ and $\\bar{\\mb{w}^{(0)}} = \\mb{w}^{(0)}$. The embedded figures are the corresponding (by frame color) details in the original plots.}\n\\label{pics:ssvm}\n\\end{center}\n\\vskip -0.2in\n\\end{figure*}\n\\section{Conclusion, Discussion and Future Work}\nIn this paper, we established the linear convergence of a semi-stochastic Frank-Wolfe algorithm with away-steps for empirical risk minimization problems and extended it to problems with block-coordinate structure. We applied the algorithms to solve the graph-guided fused LASSO problem and the structural SVM problem. Numerical results indicate the proposed algorithms outperform competing algorithms for these two problems in terms of both iteration cost and number of effective data passes. In addition, the stochastic nature of the proposed algorithms can use an approximate solution to the sub-problems, that is, an inexact oracle which can further reduce the computational cost. Possible extensions of this work include:\n\\begin{description}\n\\item[1] In the algorithms, we assume that the Lipschitz constants of the gradient of $F(\\cdot)$ are known. This assumption might be avoided by performing back-tracking. \n\\item[2] The algorithms are semi-stochastic since we need to use all data to calculate gradients at the final steps. Variance reduced gradient techniques such as the one proposed by \\cite{JZ13} might be applicable in stochastic versions of the Frank-Wofe algorithm with away-steps to make it fully stochastic and linearly convergent in high probability.\n\\item[3] The compact polytope constraint is crucial in the theoretical analysis of the proposed algorithms. Finding ways to relax this assumption is another direction for future work.\n\\end{description}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Abstract}\\label{ch1}\n\nWe present a new efficient method to compute Boltzmann collision integral for all two-particle interactions in relativistic plasma. Plasma is assumed to be homogeneous in coordinate space and isotropic in momentum space. The set of reactions consists of: Moeller and Bhabha scattering, Compton scattering, two-photon pair annihilation, and two-photon pair production, which are described by QED matrix elements. In our method exact energy and particle number conservation laws are fulfilled. Drastic improvement in computation time with respect to existing methods is achieved. Reaction rates are compared, where possible, with the corresponding analytical expressions and convergence of numerical rates is demonstrated.\n\\section{Introduction}\\label{ch2}\n\nRelativistic plasma, for which $kT\\geq mc^2$, where $k$ is Boltzmann constant, $c$ is speed of light, $m$ is electron mass, $T$ is temperature, is relevant in different branches of astrophysics. In the early universe ultrarelativistic electron-positron pairs contribute to the matter contents of the Universe \\cite{weinberg2008cosmology}. X-ray and gamma-ray radiation from numerous astrophysical sources such as gamma-ray bursts \\cite{1999PhR...314..575P,2010PhR...487....1R,2015PhR...561....1K}, active galactic nuclei \\cite{2012AAT...27..557A, blandford2013active}, and X-ray binaries \\cite{2006ARA&A..44..323F} points out to existence of relativistic electron-positron plasma in these objects.\nThe upcoming high-energy laser facilities aiming at generation of femtosecond laser pulses with intensity more than $10^{21}W\/cm^2$ aim generation of relativistic plasma by interacting laser pulses. At present relativistic electron-positron jets are generated by interaction of laser pulses with condensed matter \\cite{2015NatCo...6E6747S,0741-3335-53-1-015009,1742-6596-454-1-012016,PhysRevLett.102.105001}. \n\n\nSolving the Boltzmann equations with collision integral containing a quantum cross-section represents the most general and complete method to describe a behavior of relativistic plasma \\cite{vereshchagin2017relativistic,cercignani2012relativistic,groot1980relativistic}. \nThe one-particle distribution function (DF) is defined on a seven dimensional space, three dimensions for the physical space and three dimensions for the momentum space, and one dimension for the time. Thus one has a multidimensional problem which is a real challenge from the computational point of view. Beside the dimensionality problem, there are other difficulties which are related to kinetic equations in general \\cite{1990JMP....31..245B,1991RvMaP...3..137B}. \nOur main goal in this paper is to tackle the challenge associated with the calculation of the collision integral, dealing with two key issues. First, the computational cost related to the evaluation of the collision operator involving multidimensional integrals which should be solved in each point of the coordinate space. Second, the presence of multiple scales requires the development of adapted numerical schemes capable of solving stiff dynamics. Different deterministic approaches are used to tackle collision integral from a numerical point of view: finite volume, semi-Lagrangian and spectral schemes \\cite{1991RvMaP...3..137B,dimarco_pareschi_2014,2006MaCom..75.1833M,DIMARCO2017,WU201327}. While the deterministic methods could normally reach high order of accuracy, the probabilistic ones, such as Monte-Carlo (MC) method, are often faster.\n\nMC methods are traditionally used to model Coulomb interactions in non-relativistic plasma\\cite{SHERLOCK20082286,HUTHMACHER2016535,TURRELL2015144,BOBYLEV2013123}. As a rule MC techniques are based on the random pairing of particles in close vicinity and the calculation of a scattering angle due to the interaction. Small-angle Coulomb collisions which allow small energy and momentum transfer are often described in diffusion approximation by the Fokker-Planck equation \\cite{1981phki.book.....L}. The principal feature of relativistic plasma is a presence of pair creation\nand pair annihilation processes, which are often included in MC based models \\cite{0741-3335-53-1-015009,1742-6596-454-1-012016}. However Fokker-Planck approximation is no longer valid in relativistic plasma \\cite{2009PhRvD..79d3008A}.\n\n\nClassical Boltzmann equation does not take into account quantum statistics of particles. The generalization of classical Boltzmann equation including quantum corrections is Uehling-Uhlenbeck (U-U) equation, which contains additional Pauli blocking and Bose enhancement multipliers that give rise to equilibrium solution with Bose-Einstein and Fermi-Dirac distributions \\cite{1934PhRv...46..917U,1933PhRv...43..552U}. The main problem of the MC methods in application to U-U equations is that total reaction rate is unknown as distribution function is unknown too. Compensation methods include smoothing of the delta-function distribution of MC-particles over cells in the phase space, but it suffers from a large number of simulation particles and cells needed to reproduce Bose-Einstein steady state distribution. Spectral methods based on the Fourier transformation of the velocity distribution function require very dense computational grid to reach high accuracy \\cite{2017JCoPh.330.1010Y,Hu2015,huying12,PhysRevE.68.056703,2010arXiv1009.3352F}.\nProcess-oriented approach to the U-U collision integral presented in this work allows one to get high accuracy results with low computational cost.\n\nIn this paper we further develop the method first used in the work \\cite{2004ApJ...609..363A}. This method was successfully applied to follow the thermalization of relativistic plasma \\cite{2007PhRvL..99l5003A,2009AIPC.1111..344A,2009PhRvD..79d3008A,2010AIPC.1205...11A} and to investigate thermalization timescales for an electron-positron plasma \\cite{2010PhRvE..81d6401A}. In section \\ref{ch3} we recall Boltzmann and UU equations and present usual scheme of their analytic treatment. Section \\ref{ch4} is devoted to the description of our numerical scheme, while section \\ref{ch5} shows comparison between our code results and known analytic formulae for non-degenerate case. Conclusion follows.\n\n\\section{Formulation}\\label{ch3}\n\nThe Boltzmann equation governs an evolution of one-particle distribution function $f(\\mathbf x,\\mathbf p,t)$. We assume that plasma is homogeneous and isotropic in coordinate space and isotropic in momentum space, thus distribution function depends on absolute value of momentum (energy) and time. DF is normalized on particles concentration, so that $n=\\int f(\\mathbf p,t)d^3p$.\\\\\nConsider an interaction of two initial particles of type I and II which are in states 1 \u0438 2, correspondingly, and creation of two final particles of type III \u0438 IV which are in states 3 \u0438 4, correspondingly. Let us image the process by the following scheme:\n\\begin{equation}\\label{2pdir}\nI_1 + II_2 \\rightarrow III_3 + IV_4.\n\\end{equation}\nThe corresponding inverse process is:\n\\begin{equation}\\label{2pinv}\nIII_3 + IV_4\\rightarrow I_1 + II_2.\n\\end{equation}\nIf every particle has momentum $p_i$, which lies in interval $d^3p_i$, then a number of interactions in unit time and unit space volume is:\n\\begin{equation}\nw(p_1,p_2;p_3,p_4)f_{I} f_{II} d^3p_1d^3p_2d^3p_3d^3p_4,\n\\end{equation}\nfunction $w$ is called a transition rate for a given reaction. \\\\\nAn effective cross-section is defined by the formula:\n\\begin{equation}\nd\\sigma=\\frac{w}{v}d^3p_3 d^3p_4,\n\\end{equation}\nwhere $v=c\\epsilon_1^{-1}\\epsilon_2^{-1}\\sqrt[]{\\left( \\epsilon_1\\epsilon_2-(\\mathbf{p_1}\\mathbf{p_2})c^2 \\right)^2-(m_1 m_2 c^4)^2}$ is a relative velocity of particles. \\\\\nIn quantum field theory an expression for interaction cross-section is:\n\\begin{equation}\\label{dsigma}\nd\\sigma=\\frac{\\hbar^2 c^6}{(2\\pi)^2}\\frac{1}{v}\\frac{|M_{if}|^2}{16\\epsilon_1\\epsilon_2\\epsilon_3\\epsilon_4}\n\\delta(\\epsilon_1+\\epsilon_2-\\epsilon_3-\\epsilon_4)\\delta(\\mathbf p_1+\\mathbf p_2-\\mathbf p_3-\\mathbf p_4)d^3p_3d^3p_4,\n\\end{equation}\nwhere $|M_{if}|$ are a matrix elements calculated with a methods of quantum field theory. \\\\\nComparing two last formulas one can derive the following expression for transition rate:\n\\begin{equation}\\label{transw}\nw(p_3,p_4;p_1,p_2)=\\frac{\\hbar^2 c^6}{(2\\pi)^2}\\frac{|M_{if}|^2}{16\\epsilon_1\\epsilon_2\\epsilon_3\\epsilon_4}\\delta(\\epsilon_1+\\epsilon_2-\\epsilon_3-\\epsilon_4)\n\\delta(\\mathbf p_1+\\mathbf p_2-\\mathbf p_3-\\mathbf p_4),\n\\end{equation}\nNow let us write the Boltzmann equation for DF of particle I for a given process:\n\\begin{multline}\\label{StfI}\n\\dot{f_I}=\\int d^3p_2 d^3p_3 d^3p_4 [w(p_3,p_4;p_1,p_2)f_{III}(\\mathbf p_3,t)f_{IV}(\\mathbf p_4,t)\\\\\n-w(p_1,p_2;p_3,p_4)f_{I}(\\mathbf p_1,t)f_{II}(\\mathbf p_2,t)],\n\\end{multline}\nwhere a dot denotes time derivative. Equations for particle DFs of remaining types can be derived by the corresponding replacement of indices.\\\\\nSpecifically, for a scattering with $I=III$ and $II=IV$ the inverse process is the same as the direct one since pairs of indices $(1,2)$ and $(3,4)$ can be interchanged. The relation $w(p_3,p_4;p_1,p_2)=w(p_1,p_2;p_3,p_4)$ holds for all processes listed in Table \\ref{2pptable}.\nThe right hand side of the Boltzmann equation is a collision integral denoted as St$f_I$. The first term in collision integral describes particle outcome and the second term describes particle income, we will denote it as $\\text{St}^-f$ and $\\text{St}^+f$, respectively.\\\\\n\n\\begin{table}\n\\caption{Two-particle processes in electron-positron-photon plasma\\label{2pptable}}\n\\center\n\\begin{tabular}[c]{|l|l|l|l|l|}\n\\hline\nProcess (q) & I & II & III & IV \\\\\n\\hline\nCompton Scattering (CS) & $e^{\\pm}$ & $\\gamma$ & $e^{\\pm}$ & $\\gamma$\\\\\n\\hline\nBhabha Scattering (BS) & $e^{\\pm}$ & $e^{\\mp}$ & $e^{\\pm}$ & $e^{\\mp}$\\\\\n\\hline\nM{\\o}ller Scattering (MS) & $e^{\\pm}$ & $e^{\\pm}$ & $e^{\\pm}$ & $e^{\\pm}$\\\\\n\\hline\nPair Annihilation (PA) & $e^{-}$ & $e^{+}$ & $\\gamma$ & $\\gamma$\\\\\n\\hline\nPair Creation (PC) & $\\gamma$ & $\\gamma$ & $e^{-}$ & $e^{+}$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe generalization of Boltzmann equation for the case of particles obeying quantum statistics is U-U equation. For the particle $I$ in the state $1$ U-U equation has the following form:\n\\begin{multline} \\label{uustfI}\n\\dot{f_I} =\\int d^{3} \\mathbf{p}_{2} d^{3} \\mathbf{p}_{3} d^{3} \\mathbf{p}_{4} \\\\\n\\shoveleft{\\ \\times\\biggl[w(p_3,p_4;p_1,p_2)\nf_{III}(\\mathbf{p}_{3},t) f_{IV}(\\mathbf{p}_{4},t)\n\\left( 1+\\eta \\frac{f_{I}(\\mathbf{p}_{1},t)}{2 h^{-3}}\\right)\\left( 1+\\eta \\frac{f_{II}(\\mathbf{p}_{2},t)}{2 h^{-3}}\\right)}\\\\\n\\shoveleft{\\quad -w(p_1,p_2;p_3,p_4)\nf_{I}(\\mathbf{p}_{1},t)f_{II}(\\mathbf{p}_{2},t)}\n\\left(1+\\eta \\frac{f_{III}(\\mathbf{p}_{3},t)}{2 h^{-3}}\\right) \\left(1+\\eta \\frac{f_{IV}(\\mathbf{p}_{4},t)}{2 h^{-3}}\\right) \\biggr],\n\\end{multline}\nwhere $\\eta$ is defined through\n\\begin{equation}\n \\eta=\n \\begin{cases}\n +1, & \\text{for Bose-Einstein statistics,}\\\\\n \t -1, & \\text{for Fermi-Dirac statistics,}\\\\\n 0, & \\text{for Maxwell-Boltzmann statistics.}\n \\end{cases}\n\\end{equation}\nWhen incoming or outgoing particles coincide ($I=II$ and\/or $III=IV$) quantum indistinguishability gives the term $\\frac12$ in front of the corresponding\noutcome and income terms, see e.g. \\cite{1973rela.conf....1E}, \\cite{groot1980relativistic}.\n\nFor numerical evaluation phase space is divided into zones, in calculations we approximate continuous DF by its averaging over each zone (see Eq.~\\eqref{Ydef}). For this purpose we add an integral over $\\mathbf{p}_{1}$ in UU equation \\ref{uustfI}, the RHS of resulting equation will have the same form for each particle type differing only by sign and its integration limits:\n\\begin{multline}\\label{uustfIint}\n\\pm \\int d^{3} \\mathbf{p}_{1} d^{3} \\mathbf{p}_{2} d^{3} \\mathbf{p}_{3} d^{3} \\mathbf{p}_{4} \\\\\n\\shoveleft{\\ \\times\\biggl[w(p_3,p_4;p_1,p_2)\nf_{III}(\\mathbf{p}_{3},t) f_{IV}(\\mathbf{p}_{4},t)\n\\left( 1+\\eta \\frac{f_{I}(\\mathbf{p}_{1},t)}{2 h^{-3}}\\right)\\left( 1+\\eta \\frac{f_{II}(\\mathbf{p}_{2},t)}{2 h^{-3}}\\right)}\\\\\n\\shoveleft{\\quad -w(p_1,p_2;p_3,p_4)\nf_{I}(\\mathbf{p}_{1},t)f_{II}(\\mathbf{p}_{2},t)}\n\\left(1+\\eta \\frac{f_{III}(\\mathbf{p}_{3},t)}{2 h^{-3}}\\right) \\left(1+\\eta \\frac{f_{IV}(\\mathbf{p}_{4},t)}{2 h^{-3}}\\right) \\biggr],\n\\end{multline}\nwhere the upper sign corresponds to particle type I and II and the lower sign corresponds to particle type III and IV.\n\nEvaluating collision integral in the framework of reaction-oriented approach we use one expression \\eqref{uustfIint} and distribute the result to each particle type according to integration limits in \\eqref{uustfIint}.\n\nIn this paper we deal with all two-particle QED processes in relativistic plasma, which are collected in Table \\ref{2pptable}. The exact QED matrix elements for these processes can be found in the standard textbooks, e.g. \\cite{2003spr..book.....G,1982els..book.....B}.\\\\\n\nLet us make a notice connected with a conservation laws for interacting particles. Energy and momentum conservations read\n\\begin{gather}\\label{2pconslaws}\n\\hat\\varepsilon = \\varepsilon_{1} + \\varepsilon_{2}=\\varepsilon_{3} +\n\\varepsilon_{4},\\qquad \\hat{\\mathbf p}=\\mathbf p_{1} + \\mathbf p_{2}=\\mathbf p_{3} + \\mathbf\np_{4}.\n\\end{gather}\nThere are 4 delta-functions in Eq.~\\eqref{transw} representing conservation of energy and momentum \\eqref{2pconslaws}. Three integrations over momentum of particle $III$ can be performed immediately\n\\begin{equation}\n \\int d\\mathbf p_3 \\delta^3(\\mathbf{p}_{1}+\\mathbf{p}_{2}-\\mathbf{p}_{3}-\\mathbf{p}_{4})\\longrightarrow 1.\n\\end{equation}\nIn the integration over energy $\\varepsilon_{4}$ of particle $IV$ it is necessary to take into account that $\\varepsilon_3$ is now a function of energy and angles of particles $I$ and $II$, as well as angles of particle $IV$, so we have\n\\begin{equation}\n \\int d\\varepsilon_4 \\delta(\\varepsilon_{1}+\\varepsilon_{2} -\\varepsilon_{3}-\\varepsilon_{4}) \\longrightarrow\n \\frac{1}{1-(\\beta_{3}\/\\beta_{4})\\mathbf n_{3}\\cdot\\mathbf n_{4}},\n\\end{equation}\nwhere $\\mathbf n=\\mathbf p\/p$ is the unit vector in the direction of particle momentum, $p=|\\mathbf p|=\\sqrt{(\\varepsilon\/c)^2 -m^2c^2}$ is the absolute value of particle momentum, $\\beta=pc\/\\varepsilon$, and a dot denotes scalar product of 3-vectors.\\\\\nWe use spherical coordinates in momentum space: $\\{\\varepsilon, \\mu ,\\phi \\}$, $\\mu=\\cos\\vartheta$, where $\\varepsilon$ is the particle energy, and $\\vartheta$ and $\\phi$ are polar and azimuthal angles, respectively. Then energy and angles of particle $III$ and energy of particle $IV$ follow from energy and momentum conservation \\eqref{2pconslaws} and relativistic energy-momentum relation, namely\n\\begin{gather}\\label{KinematicsStart}\n \\varepsilon_4=c\\sqrt{p_4^2+m_{IV}^2c^2},\\qquad \\mathbf p_4=p_4 \\mathbf n_4,\\\\\n \\varepsilon_3=\\hat\\varepsilon-\\varepsilon_4,\\qquad\n \\mathbf p_3=\\hat{\\mathbf p}-\\mathbf p_4,\\nonumber\\\\\n \\mathbf n_3=\\frac{\\mathbf p_3}{p_3},\\qquad \\mathbf n_4=\\frac{\\mathbf p_4}{p_4},\\\\\n \\mathbf{n}_3=\\left(\\sqrt{1-{\\mu_3}^2}\\cos\\phi_3, \\sqrt{1-{\\mu_3}^2}\\sin\\phi_3,\n \\mu_3\\right),\\\\\n \\mathbf{n}_4=\\left(\\sqrt{1-\\mu_4^2}\\cos\\phi_4, \\sqrt{1-\\mu_4^2}\\sin\\phi_4,\n \\mu_4\\right),\\\\\n p_4=\\frac{AB\\pm\\sqrt{A^2+4m_{IV}^2c^2(B^2-1)}}{2(B^2-1)},\\label{Kinematics}\\\\\n A=\\frac{c}{\\hat\\varepsilon}[\\hat{p}^2+(m_{III}^2-m_{IV}^2)c^2]\n -\\frac{\\hat\\varepsilon}{c},\\qquad\n B=\\frac{c}{\\hat\\varepsilon}\\mathbf n_4\\cdot\\hat{\\mathbf p}.\\nonumber\n\\\\\n \\mathbf n_{3}\\cdot\\mathbf n_{4}=\\mu_3\\mu_4+\\sqrt{(1-\\mu_3^2)(1-\\mu_4^2)}\\cos(\\phi_3-\\phi_4).\n \\label{KinematicsEnd}\n\\end{gather}\n\nThen we introduce these relations into collision integral \\eqref{uustfIint}. We also use spherical symmetry in momentum space to fix angles of the particle $I$: $\\mu_1=1,\\phi_1=0$, and to perform the integration over azimuthal angle of particle $II$: $\\int d\\phi_2\\longrightarrow2\\pi$, setting $\\phi_2=0$ in the remaining integrals. Then final expression for collision integral is\n\\begin{multline}\\label{stfI1}\n\\text{St}f_I=\\frac{\\hbar^2}{32\\pi}\\int d\\varepsilon_2 d\\mu_2 \\ d\\mu_4 d\\phi_4\n\\frac{p_2 p_4|M_{fi}|^2}{\\varepsilon_{1}\\varepsilon_{3}[1-(\\beta_{3}\/\\beta_{4})\\mathbf n_{3}\\cdot\\mathbf n_{4}]}\\\\\n\\shoveleft{\\ \\times\\biggl[f_{III}(\\varepsilon_{3},t) f_{IV}(\\varepsilon_{4},t) \\left( 1+\\eta \\frac{f_{I}(\\varepsilon_{1},t)}{2 h^{-3}}\\right)\n\\left( 1+\\eta \\frac{f_{II}(\\varepsilon_{2},t)}{2 h^{-3}}\\right)}\\\\\n\\shoveleft{\\quad -f_{I}(\\varepsilon_{1},t)f_{II}(\\varepsilon_{2},t)}\\left(1+\\eta\\frac{f_{III}(\\varepsilon_{3},t)}{2 h^{-3}}\\right)\n\\left(1+\\eta\\frac{f_{IV}(\\varepsilon_{4},t)}{2 h^{-3}}\\right) \\biggr].\n\\end{multline}\n\nFor numerical integration, however, another expression is proved useful\n\\begin{multline}\\label{stfI2}\n\\int d\\varepsilon_1 \\text{St}f_I =\\frac{\\hbar^2}{32\\pi}\\Biggl[ \\int d\\varepsilon_3\\ d\\varepsilon_4 d\\mu_4 \\ d\\mu_2 d\\phi_2\n\\frac{p_2 p_4|M_{fi}|^2}{\\varepsilon_{1}\\varepsilon_{3}\\,[1-(\\beta_{1}\/\\beta_{2})\\mathbf n_{1}\\cdot\\mathbf n_{2}]}\\\\\n \\times f_{III}(\\varepsilon_{3},t) f_{IV}(\\varepsilon_{4},t)\\left(1+\\eta\\frac{f_{I}(\\varepsilon_{1},t)}{2 h^{-3}}\\right) \\left(1+\\eta \\frac{f_{II}(\\varepsilon_{2},t)}{2 h^{-3}}\\right)\\\\\n-\\int d\\varepsilon_1 \\ d\\varepsilon_2 d\\mu_2 \\ d\\mu_4 d\\phi_4\n\\frac{p_2 p_4|M_{fi}|^2}{\\varepsilon_1 \\varepsilon_{3}\\,[1-(\\beta_{3}\/\\beta_{4})\\mathbf n_{3}\\cdot\\mathbf n_{4}]}\\\\\n\\shoveleft{\\ \\times f_{I}(\\varepsilon_1,t)f_{II}(\\varepsilon_{2},t)}\\left(1+\\eta\\frac{f_{III}(\\varepsilon_{3},t)}{2 h^{-3}}\\right) \\left(1+\\eta \\frac{f_{IV}(\\varepsilon_{4},t)}{2 h^{-3}}\\right)\\Biggr],\n\\end{multline}\nwhere the first term is expressed in the form ready for replacement by the sum over incoming particles $III$ and $IV$. In this term $\\varepsilon_1,\\mu_1,\\phi_1,\\varepsilon_2$ are given by relations \\eqref{Kinematics} with indices exchange $1\\leftrightarrow3$, $2\\leftrightarrow4$, $I\\leftrightarrow III$, $II\\leftrightarrow IV$.\n\nThis collision integral of any of two-particle processes is a four-dimensional integral in momentum space. In Sec.~\\ref{ch4} we show how such integral is computed numerically on finite grid.\n\nHere we note that in the case of homogeneous and isotropic pair plasma one has to satisfy only two conservation laws, namely of energy and particle number. Momentum conservation should be added for nonisotropic in momentum space DF, see e.g.~\\cite{BENEDETTI2013206}. In our method electric charge is conserved due to conservation of particles because we use between cell interpolation for the same kind of particles described in the next Section. \n\\section{Numerical Scheme}\\label{ch4}\n\n\nThe phase space is divided in zones. The zone $\\Omega^\\alpha_{a,j,k}$ for particle specie $\\alpha$ corresponds to energy $\\varepsilon_a$, cosine of polar angle $\\mu_j$ and azimuthal angle $\\phi_k$, where indices run in the following ranges $1\\leq a\\leq a_{\\mathrm{max}}$, $1\\leq j\\leq j_{\\mathrm{max}}$, and $1\\leq k\\leq k_{\\mathrm{max}}$. The zone boundaries are $\\varepsilon_{a\\mp1\/2}$, $\\mu_{j\\mp1\/2}$, $\\phi_{k\\mp1\/2}$. The length of the $a$-th energy zone $\\Omega^\\alpha_{a}$ is $\\Delta\\varepsilon_{a} \\equiv \\varepsilon_{a+1\/2} - \\varepsilon_{a-1\/2}$. On finite grid $f_\\alpha$ does not depend on $\\mu$ and $\\phi$, and number density of particle $\\alpha$ in zone $a$ is\n\\begin{multline}\\label{Ydef}\nY^\\alpha_{a}(t)=4\\pi\\int_{\\varepsilon_{a-1\/2}}^{\\varepsilon_{a+1\/2}}c^{-3}\\varepsilon\\sqrt{\\varepsilon^2-m_\\alpha^2c^4}\nf_\\alpha(\\varepsilon,t)d\\varepsilon\\\\= 4\\pi c^{-3}\\varepsilon_a\\sqrt{\\varepsilon_a^2-m_\\alpha^2c^4} f_\\alpha(\\varepsilon_a,t)\\Delta\\varepsilon_a.\n\\end{multline}\nIn this variables discretized U-U equation for particle $I$ and energy zone $a$ reads\n\\begin{equation}\\label{discrBoltzmann}\n \\frac{d Y^\\alpha_a(t)}{dt}=\\sum \\left[\\text{St}^+Y^{I}_a -\\text{St}^-Y^{I}_a \\right],\n\\end{equation}\nwhere the sum is taken over all processes involving particle $I$. Coefficients of particles income and outcome on the grid are obtained by integration of \\eqref{stfI2} for two-particle processes over the zone. The corresponding integrals are replaced by sums on the grid.\nFor instance, coefficient of particle $I$ outcome in two-particle process \\eqref{2pdir} is\n\\begin{multline}\\label{YrateI}\n \\text{St}^-Y^{I}_a=\\frac{\\hbar^2c^{4}}{8(4\\pi)^2} \\sum_{b,j,s,k}\\Delta\\mu^{II}_{j}\\Delta\\mu^{IV}_{s} \\Delta\\phi^{IV}_{k} |M_{fi}|^2 \\frac{p_4}{\\varepsilon_{3}[1-(\\beta_{3}\/\\beta_{4})\\mathbf n_{3}\\cdot\\mathbf n_{4}]}\\times \\\\\n\\times \\frac{Y^{I}_{a}(t)}{\\varepsilon^{I}_{a}}\\frac{Y^{II}_{b}(t)}{\\varepsilon^{II}_{b}}\n \\times\\left[1+\\eta \\frac{Y^{III}_c(t)}{\\bar Y^{III}_c}\\right]\\left[1+\\eta \\frac{Y^{IV}_d(t)}{\\bar Y^{IV}_d}\\right],\n\\end{multline}\nand coefficient of particle $I$ income in process (\\ref{2pinv}) from integration of \\eqref{stfI2} is\n\\begin{multline}\\label{YrateII}\n \\text{St}^+Y^{I}_a=\\frac{\\hbar^2c^{4}}{8(4\\pi)^2} \\sum_{c,d,j,s,k} C_a(\\varepsilon_1) \\Delta\\mu^{IV}_{j}\\Delta\\mu^{II}_{s} \\Delta\\phi^{II}_{k} |M_{fi}|^2 \\frac{p_2}{\\varepsilon_{1}[1-(\\beta_{1}\/\\beta_{2})\\mathbf n_{1}\\cdot\\mathbf n_{2}]} \\times \\\\\n\\times \\frac{Y^{III}_{c}(t)}{\\varepsilon^{III}_{c}}\\frac{Y^{IV}_{d}(t)}{\\varepsilon^{IV}_{d}}\n \\times\\left[1+\\eta \\frac{Y^{I}_a(t)}{\\bar Y^{I}_a}\\right]\\left[1+\\eta \\frac{Y^{II}_b(t)}{\\bar Y^{II}_b}\\right],\n\\end{multline}\nwhere $\\bar Y^\\alpha_{a}=4\\pi\\int_{\\varepsilon_{a-1\/2}}^{\\varepsilon_{a+1\/2}}\n c^{-3}\\varepsilon\\sqrt{\\varepsilon^2-m_\\alpha^2c^4}\\ (2 h^{-3})d\\varepsilon\n = 8\\pi(hc)^{-3}\\varepsilon_a\\sqrt{\\varepsilon_a^2-m_\\alpha^2c^4} \\Delta\\varepsilon_a,$\nand\n\\begin{equation}\\label{csol}\n C_a(\\varepsilon_1)=\n \\begin{cases}\n \\dfrac{\\varepsilon_{a}-\\varepsilon_1}{\n \\varepsilon_{a}-\\varepsilon_{a-1}}, &\n \\varepsilon_{a-1} < \\varepsilon_1 < \\varepsilon_{a},\\\\[2.5ex]\n \\dfrac{\\varepsilon_{a+1}-\\varepsilon_1}{\n \\varepsilon_{a+1}-\\varepsilon_{a}}, &\n \\varepsilon_{a} < \\varepsilon_1 < \\varepsilon_{a+1},\\\\[2.5ex]\n 0, & \\text{otherwise.}\n \\end{cases}\n\\end{equation}\nIn integration of (\\ref{stfI2}) over the zone one can integrate out the $\\delta$-function $\\int \\delta(\\varepsilon_1-\\varepsilon)d\\varepsilon_1\n\\longrightarrow 1$. However, when energies of incoming particles are fixed on the grid, the energies of outgoing particles are not on the grid. Hence an interpolation \\eqref{csol} is adopted, which enforces the exact number of particles and energy conservation in each two-particle process due to redistribution of outgoing particle $\\alpha$ with energy $\\varepsilon$ over two energy zones $\\Omega^{\\alpha}_{n}, \\Omega^{\\alpha}_{n+1}$ with $\\varepsilon_{n} < \\varepsilon < \\varepsilon_{n+1}$. Further we denote this technique as particle splitting.\n\nThe redistribution of final particles should also satisfy requirements of quantum statistics. Therefore if a process occurs, when fermionic final particle should be distributed over the quantum states which are fully occupied, such process should be forbidden. Thus we introduce the Bose enhancement\/Pauli blocking coefficients in (\\ref{YrateI}) and (\\ref{YrateII}) as\n\\begin{gather}\n \\left[1+\\eta \\frac{Y^{\\alpha}_a(t)}{\\bar Y^{\\alpha}_a}\\right]=\n \\min\\left(1+\\eta \\frac{Y^{\\alpha}_{n}(t)}{\\bar Y^{\\alpha}_{n}},\n 1+\\eta \\frac{Y^{\\alpha}_{n+1}(t)}{\\bar Y^{\\alpha}_{n+1}}\\right).\n\\end{gather}\nThe sum over angles $\\mu_j, \\mu_s, \\phi_k$ can be found once and for all at the\nbeginning of the calculations. We then store in the program for each set of the\nincoming and outgoing particles the corresponding terms and redistribution\ncoefficients given by Eq.~(\\ref{csol}).\n\nRepresentation of discretized collisional integral for particle $I$ and energy zone $a$ in processes \\eqref{2pdir},\n\\eqref{2pinv} is\n\\begin{multline}\\label{BoltzmannFinal}\n \\dot{Y}^I_a=\n \\sum P_{abcd}\\times Y^{III}_{c}(t) Y^{IV}_{d}(t)\n \\times \\left[1+\\eta \\frac{Y^{I}_{a}(t)}{\\bar Y^{I}_{a}}\\right]\n \\left[1+\\eta \\frac{Y^{II}_{b}(t)}{\\bar Y^{II}_{b}}\\right]\\\\\n-\\sum R_{abcd}\\times Y^{I}_{a}(t) Y^{II}_{b}(t)\n \\times \\left[1+\\eta \\frac{Y^{III}_{c}(t)}{\\bar Y^{III}_{c}}\\right]\n \\left[1+\\eta \\frac{Y^{IV}_{d}(t)}{\\bar Y^{IV}_{d}}\\right],\n\\end{multline}\nwhere constant coefficients $P,R$ are obtained from the summation over angles in the sums \\eqref{YrateI}, \\eqref{YrateII}. In the nondegenerate case of Boltzmann equation the indices $b$ in the first sum and $c,d$ in the second sum become dummy, equation \\eqref{BoltzmannFinal} can be partially summed and takes the following form:\n\\begin{gather}\\label{BoltzmannFinal2}\n \\dot{Y}^I_a=\n \\sum P_{acd}\\times Y^{III}_{c}(t) Y^{IV}_{d}(t)\n -\\sum R_{ab}\\times Y^{I}_{a}(t) Y^{II}_{b}(t),\n\\end{gather}\nwhere $P_{acd}=\\sum_{b} P_{abcd},\\ R_{ab}=\\sum_{c,d} R_{abcd}$. The last quantity is essentially \\emph{reaction rate} usually used for description of binary processes and simply connected to the total cross section.\n\nThe full U-U equation \\eqref{BoltzmannFinal} contains similar sums for all processes from Table \\ref{2pptable}. Each individual term in these sums appears in the system of discretized equations four times in emission and absorption coefficients for each particle entering a given process. Then each term can be computed only once and added to all\ncorresponding sums, that is the essence of our \\emph{\"reaction-oriented\" approach} \\cite{ivanphd,2015AIPC.1693g0007S}.\n\nWe point out that unlike classical Boltzmann equation for binary interactions\nsuch as scattering, more general interactions are typically described by four\ncollision integrals for each particle that appears both among incoming and\noutgoing particles. \n\\section{Numerical results}\\label{ch5}\n\nThe results of numerical calculations are presented below. As all known analytical expressions for reaction rates in relativistic plasma concern nondegenerate case, here we compare our results for collision integral to that of nondegenerate plasma. Notice that for Coulomb scattering we have implemented a cutoff scheme based on minimal scattering angle \\cite{2009PhRvD..79d3008A,1988A&A...191..181H}.\n\nWe consider mildly relativistic plasma with\n\\begin{equation}\n0.01\\lesssim e \\lesssim100,\n\\end{equation}\nwhere $e$ is particle kinetic energy divided by electron rest energy, this range contains both relativistic and non-relativistic domains. The upper limit is chosen to avoid thermal production of other particles such as neutrinos and muons, while the lower limit is required to have sufficient pair density.\n\nWe introduce logarithmic energy grid with $a_{max}=40$ nodes for all calculations and different homogeneous grids for angular variables, $\\phi$-grid is 2 time denser then $\\mu$-grid (typically $\\mu$-grid contains $j_{max}=64$ nodes). To compare results with known analytical expressions we use definition of angle-averaged reaction rate per pair of particles \n\\begin{equation}\\label{sv1}\n\\overline{v\\sigma}(e_1,e_2)=\\int_{-1}^{+1} \\frac{d\\mu_{2}}2 \\int_{\\vec{p}_3,\\vec{p}_4} v d\\sigma,\n\\end{equation}\nand angle-averaged emissivity per pair of particles \n\\begin{equation}\\label{sv2}\n\\overline{v\\frac{d\\sigma}{de_3}}(e_1,e_2,e_3)=\\int_{-1}^{+1} \\frac{d\\mu_{2}}2 \\int_{\\vec{p}_3,\\vec{p}_4} v \\frac{d\\sigma}{de_3},\n\\end{equation} introduced by Svensson~\\cite{1982ApJ...258..321S}, where $d\\sigma$ is given by standard definition \\eqref{dsigma} and we have used spherical symmetry as described before Eq.~\\eqref{stfI1}. We use Coppi \\& Blandford \\cite{1990MNRAS.245..453C} analytical expressions (2.3), (3.2), (4.3) for $\\overline{v\\sigma}$, which corresponds to $R_{ab}$. Svensson \\cite{1982ApJ...258..321S} formula (55), Peer \\& Waxman \\cite{2005ApJ...628..857P} formulae (19, 28) are used for quantity $\\overline{v\\frac{d\\sigma}{de}}$, which corresponds to $P_{abc}\/\\Delta e_a$.\n\nTo compare numerical results with analytical ones, we introduce the following quantity for each process\n\\begin{equation}\nQ=\\frac{1}{a_\\text{max}^2}\\sum_{a,b} |R_{ab}\/\\overline{v\\sigma}(e_a,e_b)-1|,\n\\end{equation}\nexpressing average relative deviation of numerical results from analytical ones for all energy grid nodes. Table \\ref{qtab} presents values of $Q$ for selected number of angular grid nodes. It is evident that the relative error decreases with increasing of number of angular grid nodes reaching about 1~\\% with 128 nodes. This demonstrates convergence of numerical results to the corresponding analytical ones.\n\n\\begin{table}[tbp]\n\\caption{Values of $Q$ for selected number of angular grid nodes ($a_{max}=40, k_{max}=2j_{max}$). \\label{qtab}} \\center\n\\begin{tabular}[c]{|l|l|l|l|l|}\n\\hline\nProcess\/$j_{max}$ & 16 & 32 & 64 & 128 \\\\\n\\hline\nCS & 0.0855 & 0.0403 & 0.0207 & 0.0145\\\\\n\\hline\nPA & 0.0231 & 0.00693 & 0.00313 & 0.00138\\\\\n\\hline\nPC & 0.146 & 0.0657 & 0.0303 & 0.0116\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nBelow we present some representative plots for the reaction rates of all reactions together with analytical curves (where they are known). Energy is measured in electron rest energy units. Presented results reproduce both nonrelativistic and relativistic energy cases. All computations were carried on Intel Core i3-7100 CPU @3.90 GHz processor using one processor core. The code is written in C and compiled in Windows 7 environment with Microsoft Visual Studio 2015 in fully optimized x64 mode. Computation time of initial angular integration of collision integrals for each reaction from Table~\\ref{2pptable} is shown in Table~\\ref{timing}. It shows even lower than expected $O(j_{max}^3)$ behaviour due to kinematic cuts on the phase space of reactions.\n\n\\begin{table}[tbp]\n\\caption{CPU time (in seconds) of each reaction initial angular integration for selected number of angular grid nodes ($a_{max}=40, k_{max}=2j_{max}$), and its exponent of computational cost $O(j_{max}^n)$. \\label{timing}} \\center\n\\begin{tabular}[c]{|l|l|l|l|l||l|}\n\\hline\nProcess\/$j_{max}$ & 16 & 32 & 64 & 128 & n \\\\\n\\hline\nCS & 2.215 & 14.48 & 113.2 & 590.1 & 2.7 \\\\\n\\hline\nPA & 2.106 & 14.73 & 100.2 & 543.1 & 2.7 \\\\\n\\hline\nPC & 0.531 & 3.619 & 28.82 & 223.2 & 2.9 \\\\\n\\hline\nMS & 2.418 & 16.87 & 130.5 & 1030 & 2.9 \\\\\n\\hline\nBS & 3.354 & 22.74 & 178.6 & 1113 & 2.8 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nCompton scattering presents well-known challenge for numerical treatment as all the analytical formulas for scattering rate behave badly numerically in different parameter areas, see e.g. \\cite{2005ApJ...628..857P, 2009A&A...506..589B}. We easily bypass this difficulty as we numerically integrate well-behaved differential cross-section, as one can see for non-relativistic regime in Fig.~\\ref{CS_D} and for relativistic regime in Fig.~\\ref{CS_I}. Figure~\\ref{CS_D} presents analytic photon spectrum for the reaction $\\gamma+e^{\\pm}\\rightarrow\\gamma'+e^{\\pm}{}'$ as solid line and our numerical results shown by dots. Overall there is good agreement between numerical and analytical results. Small deviations in high-energy of the spectrum arise from leakage of the particles to kinematically forbidden area at the boundary of energy zones. Due to particle splitting (between cell interpolation) some final paricles would be placed on a grid node, that is kinematically forbidden, and it is indeed the case of Fig.~\\ref{CS_D}. To show this effect we enlarge the plot range especially on this figure. On the other spectrum figures these points appear to be outside the presented plot range. There the maximum allowed photon energy is $e_{max}=0.291$, but we have particles of energies from 0.275 up to $e_{max}$ that are splitted between energy zones of 0.275 and 0.327 -- the second is kinematically forbidden.\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig1}\n\\caption{Compton scattered-photon distribution $\\overline{v\\frac{d\\sigma}{de_\\gamma'}}(e_\\gamma,e_{\\pm},e_\\gamma ').$}\\label{CS_D}\n\\end{figure}\nFigure~\\ref{CS_I} shows the total reaction rate of the same process. Again there is good agreement between numerical and analytical results. Small discrepancy arises from truncation of reactions where final particles get out of the grid to higher or lower energies. As a result numerical reaction rates are systematically lower than analytic ones.\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig2}\n\\caption{Compton scattering rate $\\overline{v\\sigma}(e_\\pm,e_\\gamma)$.}\\label{CS_I}\n\\end{figure}\n\nAnnihilation photon spectrum for reaction $e^{+}+e^{-}\\rightarrow\\gamma+\\gamma'$ is illustrated in Fig.~\\ref{AN_D} and total reaction rate in this process in Fig.~\\ref{AN_I}. Figure~\\ref{AN_D} shows that the method is able to accurately reproduce the spectrum of annihilation photons in the range of more than two orders of magnitude. Reaction truncation errors, hardly seen at Fig.~\\ref{AN_I}, are much lower for annihilation as low-energy photons are rare in this process.\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig3}\n\\caption{Distribution for pair annihilation $\\overline{v\\frac{d\\sigma}{de_\\gamma}}(e_\\gamma,e_{+},e_{-}).$}\\label{AN_D}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig4}\n\\caption{Pair annihilation rate $\\overline{v\\sigma}(e_{+},e_{-}).$}\\label{AN_I}\n\\end{figure}\n\nBalance between pair creation and annihilation represent an independent test for the numerical scheme, as it is not automatically satisfied due to different numerical treatment of incoming and outgoing particles in the reactions. Pair creation spectra for reaction $\\gamma_1+\\gamma_2\\rightarrow e^{+}+e^{-}$ are reproduced well, see Fig.~\\ref{PC_D}, as well as total reaction rates, see Fig.~\\ref{PC_I}. Numerical balance can be checked by the form of particle distributions in numerical equilibrium, that was verified to be within 5~\\% of corresponding Boltzmann distributions.\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig5}\n\\caption{Distribution for pair production $\\overline{v\\frac{d\\sigma}{de_\\pm}}(e_{\\gamma_1},e_{\\gamma_2},e_{\\pm}).$}\\label{PC_D}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig6}\n\\caption{Pair production rate $\\overline{v\\sigma}(e_{\\gamma_1},e_{\\gamma_2}).$}\\label{PC_I}\n\\end{figure}\n\nFor completeness we present also the results for M{\\o}ller and Bhabha scattering, they show that these processes are indeed dominant for electrons and positrons in relativistic plasma, compare Figs.~\\ref{BS_I}, \\ref{MS_I} with Figs.~\\ref{CS_I}, \\ref{AN_I}, \\ref{PC_I}.\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig7}\n\\caption{Bhabha scattering rate $\\overline{v\\sigma}(e_+,e_-).$}\\label{BS_I}\n\\end{figure}\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=70mm]{fig8}\n\\caption{M{\\o}ller scattering rate $\\overline{v\\sigma}(e_{\\pm},e_{\\pm}).$\\label{MS_I}}\n\\end{figure}\n\n\n\\begin{figure}[ptb!]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=70mm]{figron1}\n\\end{tabular}\n\\caption{Time evolution of energy density in components of electron-positron-photon plasma: photon energy density (blue), electron\/positron energy density (orange), total energy density (green). Solid lines correspond to Boltzmann case, dashed lines correspond to Uehling-Uhlenbeck case.}\\label{evolro}\n\\end{figure}\n\\begin{figure}[ptb!]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=70mm]{figron2}\n\\end{tabular}\n\\caption{Time evolution of number density in components of electron-positron-photon plasma: photon concentration (blue), electron\/positron concentration (orange), total concentration (green). Solid lines correspond to Boltzmann case, dashed lines correspond to Uehling-Uhlenbeck case. Note the difference in the final pair and photon density due to rest mass of elecron\/positron (in both cases) and difference in statistics (for U-U case).}\\label{evoln}\n\\end{figure}\n\n\\begin{figure}[ptb!]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=70mm]{figsp10} &\\includegraphics[width=70mm]{figsp11}\n\\end{tabular}\n\\caption{Final energy spectra at $t_{final}=10^{-14}$~s. Solid lines are equilibrium Boltzmann and Bose-Einstein\/Fermi-Dirac fits of numerical results: photon Boltzmann energy spectrum (blue), photon Bose-Einstein energy spectrum (cyan), pairs Boltzmann energy spectrum (orange), pairs Fermi-Dirac energy spectrum (red).}\\label{evolge}\n\\end{figure}\n{Finally, we present a time evolution of energy density and concentration to demonstrate the difference between the classical Boltzmann and U-U equations. Both systems \\eqref{BoltzmannFinal} and \\eqref{BoltzmannFinal2} were solved numerically with the same initial conditions under $a_{max}=60, j_{max}=64, k_{max}=2j_{max}$ and using Gear's method for resulting stiff ODE system \\cite{1976oup..book.....H}.\n\nThe energy spectrum $d\\rho\/d\\varepsilon$ is shown instead of the distribution function $f$, that are related by $d\\rho\/d\\varepsilon=4\\pi |\\mathbf{p}|\\varepsilon^2 c^{-2}f$. We chose an initial state without electrons and positrons but with photons only, initial spectrum has a power-law shape $d\\rho\/d\\varepsilon=a (\\varepsilon\/\\varepsilon_0)^b$, with $a=3.63\\times10^{28}\\text{ cm}^{-3}$ and $b=-0.438$, $\\varepsilon_0=1$~erg, between $e=0.157$ and $e=157$. The initial spectrum corresponds to a total energy density $\\rho=4.10\\times 10^{26}\\text{ erg cm}^{-3}$ and a total number density of particles $n=8.15\\times 10^{31}\\text{ cm}^{-3}$. In general, initial spectrum can have an arbitrary shape and thermalization process transforms it to an equilibrium form. Fig.~\\ref{evolge} represents energy spectra at final equilibrium state. They attain corresponding shapes of Boltzmann and Bose-Einstein\/Fermi-Dirac with some deviations in high-energy tails that are attributed to reaction truncation errors described before. We note that total energy and number densities do not change in time due to particle splitting applied, this feature does not depend on a form of a system of equations or a type of numerical ODE solver.} \\\\\n\n\\section{Conclusions}\\label{ch6}\n\nIn this paper, we propose a new numerical method to accurately calculate Uehling\u2013Uhlenbeck collision integral for two-particle interactions in relativistic plasma. Exact energy and particle number conservation laws are achieved by using interpolation scheme \\eqref{csol}. After calculation of collision integral discretized Uehling\u2013Uhlenbeck equations transforms into system of ODEs, which can be treated by various methods suitable to solve stiff ODEs. The method admits parallelization on GPU\/CPU. Improvement in computation time with respect to previous work is achieved. Our reaction-oriented approach can be easily applied to any other types of particles and any other binary interactions, for instanse, weak interactions of neutrinos or electromagnetic ones of protons.\nGeneralization of the proposed method for triple interactions is straightforward.\n\nOur results show that reaction rates in relativistic plasma are well reproduced with moderate number of grid nodes in energy and angles (see Figures and Table \\ref{2pptable}) both for non-relativistic and relativistic particle energies. This allows development of an efficient method of solution for relativistic Uehling\u2013Uhlenbeck equation. \n\n\n\\section{Acknowledgements}\nWe thank anonymous referees for their remarks which improved the presentation of our results.\n\\newpage\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nIn this paper we give very concrete applications of \ncontinuous logic in group theory. \nWe consider classes of (locally compact) metric groups \nwhich can be also viewed as (reducts of) axiomatizable \nclasses of continuous structures. \nThen by some standard logical tricks we obtain \nseveral interesting consequences. \nUsually we concentrate on classes which are \ntypical in geometric group theory. \\parskip0pt \n\nThe following notion is one of the main objects of the paper. \nA class of groups $\\mathcal{K}$ is called {\\em bountiful} if \nfor any pair of infinite groups $G\\le H$ with $H\\in \\mathcal{K}$ \nthere is $K\\in \\mathcal{K}$ such that $G\\le K\\le H$ and $|G|=|K|$. \nIt was introduced by Ph.Hall and was studied in \npapers \\cite{KM}, \\cite{phillips}, \\cite{sabbagh} and \\cite{thomas}. \nSome easy logical observations from \\cite{KM} show that if \n$\\mathcal{K}$ is a reduct of a class axiomatizable in \n$L_{\\omega_1 \\omega}$ then $\\mathcal{K}$ is bountiful. \n\\parskip0pt \n\nWhen one considers topological groups, the definition of \nbountiful classes should be modified as follows. \n\\begin{definition} \nA class of topological groups $\\mathcal{K}$ is called bountiful \nif for any pair of infinite groups $G\\le H$ with $H\\in \\mathcal{K}$ \nthere is $K\\in \\mathcal{K}$ such that $G\\le K\\le H$ and \nthe density character of $G$ \n(i.e. the smallest cardinality of a dense subset of the space) \ncoincides with the density character of $K$. \n\\end{definition} \nWe mention paper \\cite{HHM} where similar questions were studied \nin the case of locally compact groups. \nWe will see below that under additional assumptions of metricity \nlogical tools become helpful in this class of groups. \nWe should only replace first-order logic (or $L_{\\omega_1 \\omega}$) \nby continuous one. \nWe concentrate on negations of properties {\\bf (T)}, {\\bf FH}, \n{\\bf F}$\\mathbb{R}$ (\\cite{BHV}, \\cite{HV}) and on negations \nof boundedness properties classified in \\cite{rosendalN}. \n\nIn the final part of the paper we consider separable locally compact groups \nwhich have separably categorical continuous theory, i.e. the group \nis determined uniquely (up to metric isomorphism) by its continuous theory \nand the the density character. \nIt is interesting that some basic properties of the automorphism groups \nof such structures are strongly connected with some classes examined on \nbountifulness below. \n\n\nIn the rest of this introduction we briefly remind \nthe reader some preliminaries of continuous logic. \nThen we finish this section by some remarks on sofic groups. \n\n\\bigskip \n\n\n\\paragraph{Continuous structures.} \n\nWe fix a countable continuous signature \n$$\nL=\\{ d,R_1 ,...,R_k ,..., F_1 ,..., F_l ,...\\}. \n$$ \nLet us recall that a {\\em metric $L$-structure} \nis a complete metric space $(M,d)$ with $d$ bounded by 1, \nalong with a family of uniformly continuous operations on $M$ \nand a family of predicates $R_i$, i.e. uniformly continuous maps \nfrom appropriate $M^{k_i}$ to $[0,1]$. \nIt is usually assumed that to a predicate symbol $R_i$ \na continuity modulus $\\gamma_i$ is assigned so that when \n$d(x_j ,x'_j ) <\\gamma_i (\\varepsilon )$ with $1\\le j\\le k_i$ \nthe corresponding predicate of $M$ satisfies \n$$ \n|R_i (x_1 ,...,x_j ,...,x_{k_i}) - R_i (x_1 ,...,x'_j ,...,x_{k_i})| < \\varepsilon . \n$$ \nIt happens very often that $\\gamma_i$ coincides with $id$. \nIn this case we do not mention the appropriate modulus. \nWe also fix continuity moduli for functional symbols. \nNote that each countable structure can be considered \nas a complete metric structure with the discrete $\\{ 0,1\\}$-metric. \n\nBy completeness continuous substructures of a continuous structure are always closed subsets. \n\nAtomic formulas are the expressions of the form $R_i (t_1 ,...,t_r )$, \n$d(t_1 ,t_2 )$, where $t_i$ are terms (built from functional $L$-symbols). \nIn metric structures they can take any value from $[0,1]$. \n{\\em Statements} concerning metric structures are usually \nformulated in the form \n$$\n\\phi = 0 \n$$ \n(called an $L$-{\\em condition}), where $\\phi$ is a {\\em formula}, \ni.e. an expression built from \n0,1 and atomic formulas by applications of the following functions: \n$$ \nx\/2 \\mbox{ , } x\\dot- y= max (x-y,0) \\mbox{ , } min(x ,y ) \\mbox{ , } max(x ,y )\n\\mbox{ , } |x-y| \\mbox{ , } \n$$ \n$$ \n\\neg (x) =1-x \\mbox{ , } x\\dot+ y= min(x+y, 1) \\mbox{ , } sup_x \\mbox{ and } inf_x . \n$$ \nA {\\em theory} is a set of $L$-conditions without free variables \n(here $sup_x$ and $inf_x$ play the role of quantifiers). \n \nIt is worth noting that any formula is a $\\gamma$-uniformly continuous \nfunction from the appropriate power of $M$ to $[0,1]$, \nwhere $\\gamma$ is the minimum of continuity moduli of $L$-symbols \nappearing in the formula. \n\nThe condition that the metric is bounded by $1$ is not necessary. \nIt is often assumed that $d$ is bounded by some rational number $d_0$. \nIn this case the (dotted) functions above are appropriately modified. \nSometimes predicates of continuous structures map $M^n$ to some \n$[q_1 ,q_2 ]$ where $q_1 ,q_2 \\in \\mathbb{Q}$. \n\nThe following theorem is one of the main tools of this paper. \n\\bigskip \n \n{\\bf L\\\"{o}wenheim-Skolem Theorem.} (\\cite{BYBHU}, Proposition 7.3) \n{\\em Let $\\kappa$ be an infinite cardinal number and assume \n$|L|\\le \\kappa$. \nLet $M$ be an $L$-structure and suppose $A\\subset M$ has \ndensity $\\le\\kappa$. \nThen there exists a substructure $N\\subseteq M$ containing $A$ such that \n$density(N) \\le\\kappa$ and $N$ is an elementary substructure of $M$, i.e. \nfor every $L$-formula $\\phi (x_1 ,...,x_n)$ and $a_1 ,...,a_n \\in N$ \nthe values of $\\phi (a_1 ,...,a_n )$ in $N$ and in $M$ are the same.} \n\n\\bigskip \n\n\\begin{remark} \\label{Hofmann}\n{\\em \nIt is proved in \\cite{HHM} that for any locally compact group $G$, \nthe entire interval of cardinalities between $\\aleph_0$ and $w(G)$, \nthe weight of the group, is occupied by the weights of closed \nsubgroups of $G$. \nWe remind the reader that the weight of a topological space $(X,\\tau )$ \nis the smallest cardinality which can be realized as \nthe cardinality of a basis of $(X,\\tau )$. \nIf the group $G$ is metric, the weight of $G$ coincides \nwith the density character of $G$. \nThis yelds the following version of the L\\\"{o}wenheim-Skolem Theorem. }\n\n Let $G$ be a locally compact group which is a continuous structure. \nThen for any cardinality $\\kappa < density(G)$ there is \na closed subgroup $H0$, every $n$-type $p$ is principal. \nThe latter means that for every model $M\\models T$, the predicate \n$dist(\\bar{x},p(M))$ is definable over $\\emptyset$. \n\nAnother property equivalent to separable categoricity states that \nfor each $n>0$, the metric space $(S_n (T),d)$ is compact. \nIn particular for every $n$ and every $\\varepsilon$ there is \na finite family of principal $n$-types $p_1 ,...,p_m$ so that \ntheir $\\varepsilon$-neighbourhoods cover $S_n(T)$. \n\nIn first order logic a countable structure $M$ is \n$\\omega$-categorical if and only if $Aut(M)$ is \nan {\\em oligomorphic} permutation group, i.e. \nfor every $n$, $Aut(M)$ has finitely many orbits \non $M^n$. \nIn continuous logic we have the following modification. \n\n\\begin{definition} \nAn isometric action of a group $G$ on a metric space $({\\bf X},d)$ \nis said to be approximately oligomorphic if for every $n\\ge 1$ and $\\varepsilon >0$ \nthere is a finite set $F\\subset {\\bf X}^n$ such that \n$$ \nG\\cdot F = \\{ g\\bar{x} : g\\in G \\mbox{ and } \\bar{x}\\in F\\}\n$$\nis $\\varepsilon$-dense in $({\\bf X}^n,d)$. \n\\end{definition} \n\nAssuming that $G$ is the automorphism group of a non-compact \nseparable continuous metric structure $M$, $G$ is approximately \noligomorphic if and only if the structure $M$ is separably categorical \n(C. Ward Henson, see Theorem 4.25 in \\cite{scho}). \nIt is also known that separably categorical structures are \n{\\em approximately homogeneous} in the following sense: \nif $n$-tuples $\\bar{a}$ and $\\bar{c}$ have the same types \n(i.e. the same values $\\phi (\\bar{a})=\\phi (\\bar{b})$ for all $L$-formulas $\\phi$) \nthen for every $c_{n+1}$ and $\\varepsilon >0$ there is \nan tuple $b_1 ,...,b_n ,b_{n+1}$ of the same type with \n$\\bar{c},c_{n+1}$, so that $d(a_i, b_i )\\le \\varepsilon$ for $i\\le n$. \nIn fact for any $n$-tuples $\\bar{a}$ and $\\bar{b}$ there is \nan automorphism $\\alpha$ of $M$ such that \n$$\nd(\\alpha (\\bar{c}),\\bar{a})\\le d(tp(\\bar{a}),tp(\\bar{c})) +\\varepsilon . \n$$ \n(i.e $M$ is {\\em strongly $\\omega$-near-homogeneous} in the sense\nof Corollary 12.11 of \\cite{BYBHU}). \n\n\\begin{definition} \nA topological group $G$ is called Roelcke precompact \nif for every open neighborhood of the identity $U$, \nthere exists a finite subset $F\\subset G$ such that $G=UFU$. \n\\end{definition} \n\nThe following theorem is a combination of the remark above, \nTheorem 6.2 of \\cite{rosendal}, \nTheorem 2.4 of \\cite{tsankov} and Proposition 1.20 of \\cite{rosendalN}. \n\n\\begin{theorem} \nLet $G$ be the automorphism group of a non-compact separable structure $M$. \n\nThen \\\\ \n(i) the group $G$ is approximately oligomorphic if and only if $M$ is separably categorical; \\\\ \n(ii) if $G$ is Roelcke precompact and approximately oligomorphic for 1-orbits, \nthen $M$ is separably categorical;\\\\ \n(iii) if the structure $M$ is separably categorical, then $G$ is Roelcke precompact. \n\\end{theorem} \n\n\n\n\\paragraph{Axiomatizability in continuous logic, topological properties and sofic groups. } \n\nSuppose $\\mathcal{C}$ is a class of metric $L$-structures. \nLet $Th^c (\\mathcal{C})$ be the set of all closed $L$-conditions \nwhich hold in all structures of $\\mathcal{C}$. \nIt is proved in \\cite{BYBHU} (Proposition 5.14 and Remark 5.15) \nthat every model of $Th^c (\\mathcal{C})$ is elementary equivalent \nto some ultraproduct of structures from $\\mathcal{C}$. \nMoreover by Proposition 5.15 of \\cite{BYBHU} we have the following statement. \n\\begin{quote} \nThe class $\\mathcal{C}$ is axiomatizable in continuous logic \nif an only if it is closed under metric isomorphisms and \nultraproducts and its complement is closed under ultrapowers. \n\\end{quote} \nLet $Th^c_{\\sup} (\\mathcal{C})$ be the set of all closed \n$L$-conditions of the form \n$$ \nsup_{x_1} sup_{x_2} ... sup_{x_n} \\varphi =0 \n\\mbox{ ( $\\varphi$ does not contain $inf_{x_i}$, $sup_{x_i}$ ), } \n$$\nwhich hold in all structures of $\\mathcal{C}$.\nSome standard arguments also give the following theorem. \n\n\\begin{theorem} \\label{axiom} \n(1) The class $\\mathcal{C}$ is axiomatizable in continuous logic \nif an only if it is closed under metric isomorphisms, ultraproducts and \ntaking elementary submodels. \\parskip0pt \n\n(2) The class $\\mathcal{C}$ is axiomatizable in continuous logic \nby $Th^c_{sup} (\\mathcal{C})$ if an only if it is closed under \nmetric isomorphisms, ultraproducts and taking substructures. \n\\end{theorem} \n\nIt is worth noting that when one considers classes axiomatizable \nin continuous logic it is obviously assumed that all operations \nand predicates are uniformly continuous. \nThis shows that some topological properties cannot be described \n(axiomatized) in continuous logic. \n\nSome other obstacles arise from the fact that existentional \nquantifiers cannot be expressed in continuous logic. \nFor example consider the class of all \nmetric groups which are discrete in their metrics (with $id$ as continuity moduli). \nThis class is not closed under metric ultraproducts but \nif we replace all metrics by the $\\{ 0,1\\}$-one \nwe just obtain the (axiomatizable) class of all groups. \n\nIt may also happen that when we extend an axiomatizable \nclass of structures with the $\\{ 0,1\\}$-metric \n\\footnote{in this case axiomatizability in continuous logic is equivalent to axiomatizability in first-order logic} \nby (abstract) structures from this class with all possible \n(not only possible discrete) metrics we lose axiomatizability. \nA nice example of this situation is the class of non-abelian groups \nwith $[0,1]$-metrics. \nFor example there is a sequence of non-abelian groups $G_n \\le Sym (2^n +3 )$ \nwith $G_n \\cong \\mathbb{Z}(2)^n \\times S_3$ so that their metric unltraproduct \n with respect to Hamming metrics is abelian (an easy exercise). \n\nContinuous axiomatizability appears in one of the most active areas in group theory \nas follows. \n\\begin{quote} \nAn abstract group is {\\em sofic} if it is embeddable into a metric \nultraproduct of finite symmetric groups with Hamming metrics. \n\\end{quote} \nLet $\\mathcal{S}$ be the class of complete $id$-continuous \nmetric groups of diameter 1, which are embeddable as closed subgroups \nvia isometric morphisms into a metric ultraproduct of finite symmetric \ngroups with Hamming metrics. \nThis class is axiomatizable by Theorem \\ref{axiom}. \nWe call it the class of {\\em metric sofic groups}. \n\n\\begin{corollary} \nThe class of metric sofic groups is $sup$-axiomatizable (i.e. by its theory $Th^c_{sup}$). \n\\end{corollary} \n\nIt is folklore that any abstract sofic group can be \nembedded into a metric ultraproduct of finite symmetric \ngroups as a discrete subgroup \n(see the proof of Theorem 3.5 of \\cite{pestov}). \nThis means that the set of all abstract sofic groups consists \nof all discrete structures of the class $\\mathcal{S}$. \n\n\n\\section{Boundedness properties} \n\nIt is worth noting that many classes from geometric \ngroup theory are just universal. \nFor example if a group has free isometric actions \non real trees (resp. Hilbert spaces) then any its subgroup \nhas the same property. \nSimilarly a closed subgroup of a locally compact amenable \ngroup is amenable. \n\\footnote{the class of discrete initially amenable groups (see \\cite{cornulier}) is universal too} \nThus these classes are bountiful. \n\nOn the other hand if we extend these classes by non-compact \nlocally compact groups without Kazhdan's property ${\\bf (T)}$\nor by groups admitting isometric actions on real trees \nwithout fixed points then we lose universality. \nAre these classes still bountiful? \nWe may further extend our classes by so called \nnon-{\\em boundedness properties} introduced in \\cite{rosendalN}. \nFor example consider metric groups which satisfy non-${\\bf OB}$ \n(in terms of \\cite{rosendalN}): they have isometric strongly \ncontinuous actions (i.e. the map $g\\rightarrow g\\cdot x$ defined on \n$G$ is continuous for each $x$) on metric spaces with unbounded orbits. \nThe first part of this section is devoted to some modifications of this property. \nWe will show how continuous logic can work in these cases. \nIn fact metric groups from these classes can be presented \nas reducts of continuous metric structures \nwhich induce some special actions. \n\nIn the second part of the section we consider non-${\\bf (T)}$ and \nnon-${\\bf F\\mathbb{R}}$ (of fixed points for isometric actions on real trees). \nNote that ${\\bf (T)}$ and property ${\\bf FH}$ \n(that any strongly continuous isometric affine action on a real Hilbert space has a fixed point) \nare equivalent for $\\sigma$-compact locally compact groups \n(see Chapter 2 in \\cite{BHV}). \nSince definitions of these properties require Hilbert spaces (or unbounded trees), \nwe will here apply a many-sorted version of continuous logic \n(as in Section 15 of \\cite{BYBHU}). \nWe will also present our groups as a union of an increasing \nchain of subsets of bounded diameters treating each subset as a sort. \nThis situation is very natural if the group is $\\sigma$-compact \n(i.e. a union of an increasing chain of compact subsets). \n\nIt is worth noting that by Section 1.10 of \\cite{rosendalN} \nin the case of $\\sigma$-locally compact groups \n(=$\\sigma$-compact locally compact) Roelcke precompatness coincides \nwith all boundedness properties studied in \\cite{rosendalN} excluding only ${\\bf FH}$. \nIn particular it coincides with compactness and property ${\\bf OB}$. \nOn the other hand an elementary submodel of a non-compact (resp. compact) continuous \nstructure is also non-compact (resp. compact, see \\cite{BYBHU}, Section 10). \nThus by the L\\\"{o}wenheim-Skolem theorem (in a 1-sorted language) \nnon-compacness is bountiful in the class of locally compact groups. \nWhen the property non-${\\bf OB}$ coincides with non-compacness \n(as in the case of locally compact Polish groups) it is also bountiful. \nThis explains why in the first part of the section we do not \nassume that a group is locally compact or Polish. \n\n\nIt is worth noting that our methods do not work \nfor the classes of (locally compact) groups satisfying \nproperties {\\bf (T)}, {\\bf F}$\\mathbb{R}$ and {\\bf FH} \n(see discussion before Proposition \\ref{discr}). \nThe case of amenable Polish groups is open and looks very intresting. \nA topological group $G$ is called {\\em amenable} \nif every $G$-flow admits an invariant Borel probability measure. \nIn the case of locally compact groups this definition coincides with \nthe classical one. \nIt is noticed in \\cite{kechrisN}, that the group $Sym(\\omega )$ \nof all permutations of $\\omega$ is amenable. \nSince it has closed non-amenable subgroups, the class of amenable \nPolish groups is not universal (with respect to taking closed subgroups). \n\n\n\\subsection{Negations of strong boundedness and OB} \n\nAn abstract group $G$ is {\\em Cayley bounded} if for every generating subset\n$U\\subset G$ there exists $n\\in \\omega$ such that every element\nof $G$ is a product of $n$ elements of $U\\cup U^{-1}\\cup\\{ 1\\}$.\nIf $G$ is a Polish group then $G$ is {\\em topologically Cayley bounded} \nif for every analytic generating subset $U\\subset G$ \nthere exists $n\\in \\omega$ such that every element\nof $G$ is a product of $n$ elements of $U\\cup U^{-1}\\cup\\{ 1\\}$.\nIt is proved in \\cite{rosendal} that for Polish groups property \n{\\bf OB} is equivalent to topological Cayley boundedness together \nwith {\\em uncountable topological cofinality}: $G$ is not the union \nof a chain of proper open subgroups. \n\n\\paragraph{Discrete groups.} \n\nLet us consider the abstract (discrete) case. \nA group is {\\em strongly bounded} if it is Cayley bounded and\ncannot be presented as the union of a strictly increasing chain\n$\\{ H_n :n\\in \\omega\\}$ of proper subgroups\n(has {\\em cofinality} $>\\omega$). \nIt is known that strongly bounded groups have property {\\bf FA}, \ni.e. any action on a simplicial tree fixes a point. \n\nThe class of strongly bounded groups is not bountiful. \nIndeed, by \\cite{dC} for any finite perfect group $F$ and \nan infinite $I$ the power $F^I$ is strongly bounded. \nSince $F^I$ is locally finite, any its countable subgroup \nhas cofinality $\\omega$. \nSimilar arguments can be applied to property ${\\bf FA}$. \n\nIt is shown in \\cite{dC}, that strongly bounded groups\nhave property {\\bf FH}. \nIt can be also deduced from \\cite{dC} that strongly bounded groups \nhave property ${\\bf F}\\mathbb{R}$ that every isometric action of $G$ \non a real tree has a fixed point (since such a group acting on \na real tree has a bounded orbit, all the elements are elliptic \nand it remains to apply cofinality $>\\omega$). \nIt is now clear that the bountiful class of groups having \nfree isometric actions on real trees (or on real Hilbert spaces) \nis disjoint from strong boundedness. \n \n\n\\begin{proposition} \\label{discr} \nThe following classes of groups are reducts of axiomatizable \nclasses in $L_{\\omega_1 \\omega}$: \\\\ \n(1) The complement of the class of strongly bounded groups; \\\\ \n(2) The class of groups of cofinality $\\le \\omega$; \\\\ \n(3) The class of groups which are not Cayley bounded; \\\\ \n(4) The class of groups presented as non-trivial free products with amalgamation \n(or HNN-extensions); \\\\ \n(5) The class of groups having homomorphisms onto $\\mathbb{Z}$. \n\nAll these classes are bountiful. \nThe class of groups which do not have property {\\bf FA} is bountiful too. \n\\end{proposition} \n\n{\\em Proof.} \n(1) We use the following characterization of strongly bounded \ngroups from \\cite{dC}. \n\\begin{quote} \nA group is strongly bounded if and only if for every \npresentation of $G$ as $G=\\bigcup_{n\\in \\omega} X_n$ for \nan increasing sequence $X_n$, $n\\in \\omega$, with \n$\\{ 1\\} \\cup X^{-1}_n \\cup X_n \\cdot X_n \\subset X_{n+1}$ \nthere is a number $n$ such that $X_n =G$. \n\\end{quote} \nLet us consider the class $\\mathcal{K}_{nb}$ of all structures \n$\\langle G, X_n \\rangle_{n\\in\\omega}$ with the axioms \nstating that $G$ is a group, $\\{ X_n \\}$ is a sequence of \nunary predicates on $G$ defining a strictly increasing \nsequence of subsets of $G$ with \n$\\{ 1\\} \\cup X^{-1}_n \\cup X_n \\cdot X_n \\subset X_{n+1}$ \n(these axioms are first-order) and \n$$ \n(\\forall x) (\\bigvee_{n\\in\\omega} x\\in X_n ). \n$$\nBy the L\\\"{o}wenheim-Skolem theorem for countable fragments of \n$L_{\\omega_1 \\omega}$ (\\cite{keisler}, p.69)\nany subset $C$ of such a structure is contained in an elementary \nsubmodel of cardinality $|C|$ (the countable fragment which we consider \nis the minimal fragment containing our axioms). \nThis proves bountifulness in case (1). \\\\ \n(2) The case groups of cofinality $\\le \\omega$ is similar. \\\\ \n(3) The class of groups which are not Cayley bounded is \na class of reducts of all groups expanded by an unary predicate \n$\\langle G,U\\rangle$ with an $L_{\\omega_1 \\omega}$-axiom stating \nthat $U$ generates $G$ and with a system of first-order axioms \nstating that there exists an element of $G$ which is not a \nproduct of $n$ elements of $U\\cup U^{-1} \\cup \\{ 1 \\}$. \nThe rest is clear. \\\\ \n(4) The class of groups which can be presented as \nnon-trivial free products with amalgamation is the class \nof reducts of all groups expanded by two unary predicates \n$\\langle G,U_1 ,U_2 \\rangle$ with first-order axioms that $U_1$ and $U_2$ \nare subgroups and with $L_{\\omega_1 \\omega}$-axioms \nstating tha $U_1 \\cup U_2$ generates $G$ and a word in the \nalphabeth $U_1 \\cup U_2$ is equal to 1 if and only if this \nword follows from the relators of the free product of $U_1$ \nand $U_2$ amalgamated over $U_1 \\cap U_2$. \nThe rest of (4) is clear. \\\\ \n(5) Groups having homomorphisms onto $\\mathbb{Z}$ \ncan be considered as reducts of structures in the language \n$\\langle \\cdot ,...U_{-n},...,U_0 ,...,U_{m},...\\rangle$, \nwhere predicates $U_t$ denote preimages of the corresponding integer numbers.\n\nTo see that the class of groups without {\\bf FA} is \nbountiful, take any infinite $G\\models {\\bf notFA}$. \nIt is well-known (\\cite{serre}, Section 6.1) \nthat such a group belongs to the union of \nthe classes from statements (2),(4) and (5). \nThus $G$ has an expansion as in one of the cases (2),(4) or (5). \nNow applying the L\\\"{o}wenheim-Skolem theorem, for any \n$C\\subset G$ we find a subgroup of $G$ of cardinality $|C|$ \nwhich contains $C$ and does not satisfy {\\bf FA}. \n$\\Box$\n\n\\bigskip \n\n\n\\paragraph{Topological groups.} \n\nAs we already mentioned in Introduction separably categorical structures have \nRoelcke precompact automorphism groups. \nIn the following definition we consider several versions of this property. \n\n\\begin{definition} \nLet $G$ be a topological group. \\\\ \n(1) The group $G$ is called bounded if for any open $V$ containing $1$ there is \na finite set $F\\subseteq G$ and a natural number $k>0$ such that $G=FV^k$. \\\\ \n(2) The group $G$ is Roelcke bounded if for any open $V$ containing $1$ there is \na finite set $F\\subseteq G$ and a natural number $k>0$ such that $G=V^k FV^k$. \\\\ \n(3) The group $G$ is Roelcke precompact if for any open $V$ containing \n$1$ there is a finite set $F\\subseteq G$ such that $G=VFV$. \\\\ \n(4) The group $G$ has property ${\\bf (OB)_k}$ if for any open symmetric \n$V\\not=\\emptyset$ there is a finite set $F\\subseteq G$ such that $G=(FV)^k$. \n\\end{definition} \nIt is known that for Polish groups property ${\\bf OB}$ is equivalent \nto the property that for any open symmetric $V\\not=\\emptyset$ there is \na finite set $F\\subseteq G$ and a natural number $k$ such that $G=(FV)^k$. \nThus when $G$ is non-{\\bf OB}, there is an non-empty open $V$ such that \nfor any finite $F$ and a natural number $k$, $G\\not=(FV)^k$. \nNote that for such $F$ and $k$ there is a real number $\\varepsilon$ \nsuch that some $g\\in G$ is $\\varepsilon$-distant from $(FV)^k$. \nIndeed, otherwise $(FV)^k V$ would cover $G$. \n \nThis explains why in order to define a suitable class which is \ncomplementary to {\\bf OB} we consider the following property. \n\n\\begin{definition} \nA metric group $G$ is called uniformly non-{\\bf OB} if there is an open \nsymmetric $V\\not=\\emptyset$ so that for any natural numbers $m$ and $k$ \nthere is a real number $\\varepsilon$ such that for any $m$-element subset \n$F\\subset G$ there is $g\\in G$ which is $\\varepsilon$-distant from $(FV)^k$. \n\nUniform non-boundedness, uniform non-Roelcke boundedness, uniform non-Roelcke \nprecompactness and uniform non-${\\bf (OB)_k}$ are defined by the same scheme. \n\\end{definition} \n\nIt is clear that in the case discrete groups if a symmetric \nsubset $V$ has the property that $G\\not= (FV)^k$ for all finite \n$F\\subset G$ and natural numbers $k$, then the corresponding \nuniform version also holds. \n\n\n\\begin{proposition} \\label{uniform} \nThe following classes of metric groups are bountiful: \\\\ \n(1) The class of uniformly non-bounded groups; \\\\ \n(2) The class of uniformly non-Roelcke bounded groups; \\\\ \n(3) The class of uniformly non-Roelcke precompact groups; \\\\ \n(4) The class of uniformly non-${\\bf (OB)_k}$-groups; \\\\ \n(5) The class of uniformly non-${\\bf (OB)}$-groups. \n\\end{proposition} \n \n{\\em Proof.} \nLet us consider the class of uniformly non-${\\bf (OB)}$-groups. \nLet $\\mathcal{K}_{0}$ be the class of all continuous \nmetric structures $\\langle G, P,Q \\rangle$ with the axioms \nstating that $G$ is a group and $P:G\\rightarrow [0,1]$ and \n$Q:G\\rightarrow [0,1]$ are unary predicates on $G$ with $Q(1)=0$ so that \n$$ \nsup_x min (P(x),Q(x))= sup_x |P(x)-P(x^{-1})|=0 \\mbox{ and } inf_x |P(x)-1\/2| =0, \n$$\n$$ \nsup_x |Q(x)-Q(x^{-1})|=0 \\mbox{ and } inf_x |Q(x)-1\/2| =0, \n$$\n$$\n\\mbox{ and for all rational }\\varepsilon \\in [0,1] \n$$ \n$$\nsup_x min ( \\varepsilon \\dot{-} Q(x), inf_y (max(d(x,y)\\dot{-} 2\\varepsilon , \\varepsilon \\dot{-} P(y)))=0. \n$$ \nNote that the last axiom implies that any neighbourhood of an element from the nullset of $Q$ \ncontains an element with non-zero $P$. \n\nFor any natural $m$ and $k$ and any rational $\\varepsilon$ let us consider the following condition (say $\\theta (m,k,\\varepsilon )$): \n$$ \nsup_{x_1 ...x_m } inf _{x} sup_{y_1 ...y_k} min(P(y_1),...,P(y_n), (\\varepsilon \\dot{-} min_{w\\in W_{m,k}}(d(x, w))))=0, \n$$ \n$$ \n\\mbox{ where } W_{m,k} \\mbox{ consists of all words of the form } x_{i_1} y_1 x_{i_2} y_2 ...x_{i_k}y_k . \n$$ \nIf $G$ is a uniformly non-${\\bf (OB)}$-group, then find an open symmetric $V$ \nsuch that for any natural numbers $m$ and $k$ \nthere is a real number $\\varepsilon$ such that for any $m$-element subset \n$F\\subset G$ there is $g\\in G$ which is $\\varepsilon$-distant from $(FV)^k$. \nWe interpret $Q(x)$ by $d(x,V)$ and $P(x)$ by $d(x, G\\setminus V)$ \n(possibly normalizing them to satisfy the axioms of $\\mathcal{K}_0$). \nThen observe that $\\langle G,P,Q\\rangle\\in \\mathcal{K}_0$ and for any \nnatural numbers $m$ and $k$ there is a rational number $\\varepsilon$ \nso that $\\theta(m,k,\\varepsilon )$ holds in $(G,P,Q)$. \nBy the L\\\"{o}wenheim-Skolem theorem for continuous logic \nany infinite subset $C$ of such a structure is contained in an elementary \nsubmodel of the same density character as $C$. \nTo verify uniform {\\bf non-(OB)} in such a submodel take the complement \nof the nullset of $P(x)$ as an open symmetric subset. \nThis proves statement (5). \n\nAll remaining cases are considered in a similar way. \n$\\Box$ \n\\bigskip \n\n\n\n\\subsection{Unbounded actions} \n\\paragraph{{\\bf Negation of (T).}}\n\nLet a topological group $G$ have a strongly continuous unitary \nrepresentation on a Hilbert space ${\\bf H}$. \nA closed subset $Q\\subset G$ \nhas an {\\em almost $\\varepsilon$-invariant unit vector} in ${\\bf H}$ if \n$$ \n\\mbox{ there exists }v\\in {\\bf H} \\mbox{ such that } \nsup_{x\\in Q} \\parallel x\\circ v - v\\parallel < \\varepsilon\n\\mbox{ and } \\parallel v\\parallel =1. \n$$ \nWe call a closed subset $Q$ of the group $G$ a {\\em Kazhdan set} \nif there is $\\varepsilon$ with the following property: \nevery unitary representation of $G$ on a Hilbert space \nwith almost $(Q,\\varepsilon )$-invariant unit vectors also has \na non-zero invariant vector. \nIf the group $G$ has a compact Kahdan subset then it is said that \n$G$ has property ${\\bf (T)}$ of Kazhdan. \n\nIf we want to consider unitary representations in \ncontinuous logic we should fix continuity moduli \nfor the corresponding binary functions \n$G\\times B_n \\rightarrow B_n$ induced by the action, \nwhere $B_n$ is the $n$-ball of the corresponding Hilbert space. \nIn fact if $G$ is $\\sigma$-locally compact, \nthen we can present $G$ as the union of a chain \nof compact subsets $K_1 \\subseteq K_2 \\subseteq ...$ \nand consider continuity moduli for the corresponding \nfunctions $K_m \\times B_n \\rightarrow B_n$. \nNote that each $B_k$ and $K_l$ will be considered as sorts \nof a continuous structure. \nIn this version of continuous logic we do not assume that \nthe diameter of a sort is bounded by 1. \nIt can become any rational number. \n\nWe can now slightly modify the definition of a Kazhdan set \nas follows. \n\n\\begin{definition} \nLet $G$ be the union of a chain of closed subsets \n$K_1 \\subseteq K_2 \\subseteq ...$ of bounded diameters. \nLet $\\mathcal{F} = \\{ F_1 , F_2 , ... \\}$ be a family of continuity \nmoduli for continuous function $K_i \\times B_i \\rightarrow B_i$. \n\nWe call a closed subset $Q$ of the group $G$ \nan $\\mathcal{F} $-Kazhdan set if there is $\\varepsilon$ \nwith the following property: \nevery $\\mathcal{F} $-continuous unitary representation of $G$ \non a Hilbert space with almost $(Q,\\varepsilon )$-invariant unit vectors \nalso has a non-zero invariant vector. \n\\end{definition} \n\nLet us consider such actions in continuous logic. \nWe treat a Hilbert space over $\\mathbb{R}$ \nexactly as in Section 15 of \\cite{BYBHU}. \nWe identify it with a many-sorted metric structure \n$$\n(\\{ B_n\\}_{n\\in \\omega} ,0,\\{ I_{mn} \\}_{m0} B^{n}_{\\rho}(1)$. \nThus there is a number $n$ such that \nthe $\\varepsilon$-neighbourhood of $B^n_{\\rho}(1)$ \ncontains the zeroset of $P(x) = dist (x,G_{\\rho})$.\nIf $(N,Q)$ is an elementary extension of $(G,P)$ then \n$(N,Q)$ satisfies the condition \n$$ \nsup_x inf_{y_1} ...inf_{y_n}max(d(y_1 ,1)\\dot{-}\\rho ,..., d(y_n ,1)\\dot{-}\\rho, |Q(x) -d(x,y_1 \\cdot ...\\cdot y_n )|\\dot{-}\\varepsilon )=0, \n$$ \ni.e. the $\\varepsilon$-neighbourhood of $B^n_{\\rho}(1)$ \ncontains the zeroset of $Q(x)$. \nIn particular the zeroset of $Q$ coincides with the closure of $G_{\\rho}$, \ni.e is $G_{\\rho}$ itself and is a subset of $acl(\\emptyset )$. \nSince $(N,Q)$ is an elementary extension of $(G,P)$,\n$Q(x)$ is the distance from the zeroset of $Q$ (see Theorem 9.12 in \\cite{BYBHU}). \nIn particular any automorphism of $N$ preserves $Q$. \nUsing Corollary 9.11 of \\cite{BYBHU} (cited in Introduction above) \nwe see that $P(x)$ is a definable predicate. \n$\\Box$\n\n\\bigskip \n\n \n\\begin{lemma} \nUnder the circumstances above \nthere is a natural number $n$ so that \n$G_{\\rho}=B^n_{\\rho}(1)$. \nIn particular $G_{\\rho}$ is compact. \n\\end{lemma} \n\n{\\em Proof.} \nIf $G_{\\rho}\\not=B^n_{\\rho}(1)$ for all $n\\in\\omega$, there are \npositive rational numbers $\\varepsilon_1 ,...,\\varepsilon_n ,...$ so that \nthe $\\varepsilon_n$-neighbourhood of $B^n_{\\rho}(1)$ does not cover $G_{\\rho}$. \nThus all statements \n$$\nsup_{x_1 ...x_n}(min(\\varepsilon_n \\dot{-} d(x,x_1 \\cdot ....\\cdot x_n ), \\rho\\dot{-}d(1,x_1 ),..., \\rho\\dot{-}d(1,x_n )))=0\n$$ \nare finitely consistent together with $P(x)=0$. \nBy compactness of continuous logic we obtain a contradiction. \n$\\Box$ \n\n\\bigskip \n\nSince $G_{\\rho}$ is a characteristic subgroup of $G$ with respect to \nthe automorphism group of the metric structure $G$, \nwe see that $Aut(G,d)$ acts correctly on $G\/G_{\\rho}$ \nby permutations of $G\/G_{\\rho}$. \nNote that $G\/G_{\\rho}$ is a discrete space with respect to \nthe topology induced by the topology of $G$. \n\n\\begin{lemma} \nThe action of $Aut(G,d)$ on $G\/G_{\\rho}$ is oligomorphic. \n\\end{lemma} \n\n{\\em Proof.} \nSince $(G,d)$ is separably categorical, $Aut(G,d)$ \nis approximately oligomorphic on $(G,d)$. \nThus for every $n$ there is a finite set $F$ of \n$n$-tuples from $G$ such that the set of orbits \nmeeting $F$ is $\\rho$-dense in $(G,d)$. \nIn particular for any $g_1 ,...,g_n \\in G$ \nthere is a tuple $(h_1 ,...,h_n )\\in F$ and \nan automorphism $\\alpha \\in Aut(G,d)$ \nsuch that $g^{-1}_i \\alpha (h_i )\\in G_{\\rho}$ \nfor all $i\\le n$. \n$\\Box$ \n\n\\bigskip \n\nTo see that Theorem \\ref{catlc} follows from lemmas above \njust take $H$ to be $G_{\\rho}$. \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzaiuk b/data_all_eng_slimpj/shuffled/split2/finalzzaiuk new file mode 100644 index 0000000000000000000000000000000000000000..43c4ffeb95466296c38be63df4612d6879b08a1d --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzaiuk @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{intro}\n\nLet $\\fg$ be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra $\\fh$ and the Weyl group $W$. Let $P_+$ be the set of dominant integral weights. For $\\lambda \\in P_+$, let $L(\\lambda)$ be the irreducible, integrable, highest weight representation of $\\fg$ with highest weight $\\lambda$. For a positive integer $s$, define the {\\em saturated tensor semigroup} as\n\\begin{align*}\n\\Gamma_s:= \\{(\\lambda_1, \\dots, \\lambda_s,\\mu)\\in P_+^{s+1}: \\exists\\,\nN>1 \\,\\,\\text{with}\\,\\, L(N\\mu)\\subset L(N\\lambda_1)\\otimes \\dots \\otimes L(N\\lambda_s)\\}.\n\\end{align*}\nThe aim of this paper is to begin a systematic study of $\\Gamma_s$ in the infinite dimensional symmetrizable Kac-Moody case. \nIn this paper, we produce a set of necessary inequalities satisfied by $\\Gamma_s$, which we describe now. Let $X =G^{\\min}\/B$ be the standard full KM-flag variety\nassociated to $\\fg$, where $G^{\\min}$ is the `minimal' Kac-Moody group with Lie algebra $\\fg$ and $B$ is the standard Borel subgroup of \n$G^{\\min}$. For $w\\in W$, let $X_w = \\overline{BwB\/B}\\subset X$ be the corresponding Schubert variety. Let $\\{\\eps^w\\}_{w\\in W} \\subset H^*(X,\\bz)$ be the \n(Schubert) basis dual (with respect to the standard pairing) to the basis of the singular homology of $X$ given by the fundamental classes of $X_w$. \nThe following result is our first main theorem valid for any symmetrizable $\\fg$ (cf. Theorem \\ref{thm1}).\n\\begin{thm}\nLet\n $(\\lambda_1,\\dots,\\lambda_s,\\mu) \\in \\Gamma_s$. Then, \nfor any $u_1, \\dots , u_s, v\\in W$ such that $n^v_{u_1, \\dots, u_s}\\neq 0$, where\n$$\\eps^{u_1} \\dots \\eps^{u_s} = \\sum_w n^w_{u_1, \\dots, u_s}\\,\\eps^w,$$\nwe have \n\\[\\left(\\sum^s_{j=1} \\lam_j (u_jx_i)\\right) - \\mu (vx_i) \\geq 0, \\,\\,\\,\\text{for any}\\,\\, x_i,\\]\nwhere \n $x_i\\in \\fh$ is dual to the simple roots of $\\fg$.\n\\end{thm}\nThe proof of the theorem relies on the Kac-Moody analogue of the Borel-Weil theorem and the Geometric Invariant Theory (specifically the Hilbert-Mumford index). We conjecture that the above inequalities are sufficient as well to describe $\\Gamma_s$. In fact, we conjecture a much sharper result,\nwhere much fewer inequalities suffice to describe the semigroup $\\Gamma_s$. To explain our conjecture, we need some more notation. \n\nLet $P\\supset B$ be a (standard) parabolic subgroup and let $X_P:=G^{\\min}\/P$ be the corresponding partial flag variety. Let $W_P$ be the Weyl group of $P$ (which is, by definition, the Weyl group of the Levi $L$ of $P$) and let $W^P$ be the set of minimal length coset representatives \nof cosets in $W\/W_P$. \nThe projection map $X\\to X_P$ induces an injective homomorphism $H^*(X_P, \\bz) \\to H^*(X, \\bz)$ and $H^*(X_P, \\bz) $ has the Schubert basis\n$\\{\\eps^w_P\\}_{w\\in W^P}$ such that $\\eps^w_P$ goes to $\\eps^w$ for any $w\\in W^P$. As defined by Belkale-Kumar [BK, $\\S$6] in the finite dimensional case (and extended here in Section 7 for any symmetrizable Kac-Moody case), there is a new deformed product $\\odot_0$ in\n$H^*(X_P, \\bz)$, which is commutative and associative. Now, we are ready to state our conjecture (see Conjecture \\ref{conj1}). \n\\begin{conjecture} \nLet $\\fg$ be any indecomposable symmetrizable Kac-Moody Lie algebra\nand let $(\\lambda_1, \\dots, \\lambda_s, \\mu)\\in P_+^{s+1}$. Assume further that none of $\\lambda_j$ is $W$-invariant and $\\mu-\\sum_{j=1}^s \\lambda_j\\in Q$, where $Q$ is the root lattice of $G$. \nThen, the following are equivalent:\n\n(a) $(\\lambda_1, \\dots, \\lambda_s, \\mu)\\in \\Gamma_s$.\n\n(b) For every standard maximal parabolic subgroup $P$ in $G^{\\min}$ and every choice of\n$s+1$-tuples $(w_1, \\dots, w_s, v)\\in (W^P)^{s+1}$ such that $\\epsilon_P^v$ occurs with coefficient $1$ in \nthe deformed product\n$$\\epsilon_P^{w_1}\\odot_0\\, \\cdots \\,\\odot_0 \\epsilon_P^{w_s}\n\\in \\bigl(H^*(X_P,\\Bbb{Z}), \\odot_0\\bigr),$$\n the following inequality holds:\n \\[\n\\bigl(\\sum_{j=1}^s \\lambda_j(w_jx_{P})\\bigr)-\\mu(vx_P)\\geq 0, \\tag{$I^P_{(w_1,\\dots, w_s,v)}$}\n\\]\nwhere $\\alpha_{i_P}$ is the (unique) simple root not in the Levi of $P$\nand $x_P:=x_{i_P}$.\n\\end{conjecture}\n\nThis conjecture is motivated from its validity in the finite case due to Belkale-Kumar [BK, Theorem 22]. (For a survey of these results in the finite case, see [K$_5$].) So far, the only evidence of its validity in the infinite dimensional case is shown for $s=2$ and $\\fg$ of types $A_1^{(1)}$ and $A_2^{(2)}$ (cf. Theorems \\ref{thm7.5} and \\ref{thm8.6}). In these cases, we explicitly determine $\\Gamma_2$ and thereby show the validity of the conjecture. \n\nA positive integer $d_o$ is called a {\\em saturation factor} for $\\fg$ if for any $\\Lambda$, $\\Lambda'$, $\\Lambda'' \\in P_+$\nsuch that $\\Lambda-\\Lambda'-\\Lambda''\\in Q$ and \n $L(N\\Lambda)$ is a submodule of $L(N\\Lambda')\\otimes L(N\\Lambda'')$, for some $N\\in \\bz_{>0}$, then \n $L(d_o\\Lambda)$ is a submodule of $L(d_o\\Lambda')\\otimes L(d_o\\Lambda'')$.\n\nWe prove the following result on saturation factors (cf. Corollaries \\ref{cor6.4} and \\ref{cor8.7}).\n\\begin{thm} For $A_1^{(1)}$, any integer $d_o>1$ is a saturation factor. For $A_2^{(2)}$, $4$ is a saturation factor.\n\\end{thm}\n\nThe proof in these affine rank-2 cases makes use of basic representation theory of the Virasoro algebra (in particular, Lemma \\ref{virasoro}). \nLet $\\delta$ be the smallest positive imaginary root of $\\fg$. To determine the saturated tensor semigroup, we show that it is enough to know the components of $L(\\lambda_1)\\otimes L(\\lambda_2)$ which are $\\delta$-maximal, i.e., the components $L(\\mu)\\subset L(\\lambda_1)\\otimes L(\\lambda_2)$ such that $L(\\mu+n\\delta) \\nsubseteq L(\\lambda_1) \\otimes L(\\lambda_2)$ for any $n>0$. Let $m^\\mu_{\\lambda_1,\\lambda_2}$ be the multiplicity of $L(\\mu)$ in $L(\\lambda_1) \\otimes L(\\lambda_2)$. If $L(\\mu)$ is a $\\delta$-maximal component of $L(\\lambda_1)\\otimes L(\\lambda_2)$, then $\\sum_{n\\in\\bz_{\\leq 0}} L(\\mu+n\\delta)^{\\oplus m^{\\mu+n\\delta}_{\\lambda_1,\\lambda_2}}$ is a unitarizable coset module for the Virasoro algebra arising from the Sugawara construction for the diagonal embedding $\\fg \\hookrightarrow \\fg \\oplus\\fg$. Proposition \\ref{maximum} for $A_1^{(1)}$ (and the analogous Proposition \\ref{maxS} for $A_2^{(2)}$) determining the maximal $\\delta$-components plays a crucial role in the proofs.\n\n\\vskip2ex\n\\noindent\n{\\bf Acknowledgements.} We thank Evgeny Feigin and Victor Kac for some helpful correspondences. Both the authors were partially supported by the NSF grant number DMS-1201310.\n\\section{Notation}\\label{sec2}\n\nWe take the base field to be the field of complex numbers $\\bc$. By a variety, we\n mean an algebraic variety over $\\bc$, which is reduced but not necessarily irreducible.\n\nLet $G$ be any symmetrizable Kac-Moody group over $\\bc$ completed\nalong\n the negative roots (as opposed to completed along the positive roots as in [K$_3$,\n Chapter 6]) and $G^{\\min}\\subset G$ be the `minimal' Kac-Moody group as in\n [K$_3$, \\S7.4]. Let $B$ be the standard (positive) Borel\nsubgroup, $B^{-}$ the standard negative Borel subgroup, $H=B\\cap B^{-}$ the\nstandard maximal torus and $W$ the Weyl group (cf.\n[K$_3$, Chapter 6]). Let $U$ (resp. $U^-$) be the unipotent radical $[B,B]$ (resp. $[B^-,B^-]$) of\n$B$ (resp. $B^-$). Let\n \\[\n\\bar{X} = G\/B\n \\]\nbe the `thick' flag variety which contains the standard KM-flag\nvariety\n \\[ X = G^{\\min}\/B. \\]\nIf $G$ is not of finite type, $\\bar{X}$ is an infinite\ndimensional non quasi-compact scheme (cf. [Ka, \\S4]) and $X$ is an\nind-projective variety (cf. [K$_3$, \\S7.1]). The group $G^{\\min}$ acts on $\\bar{X}$ and $X$. \n\nMore generally, for any \nstandard parabolic subgroup $P\\supset B$, define the partial flag variety\n \\[ X_P = G^{\\min}\/P, \\]\nand \n\\[\\bar{X}_P=G\/P.\\]\n\nRecall that if $W_P$ is the Weyl group of $P$ (which is, by definition, the Weyl\nGroup $W_L$ of its Levi subgroup $L$), then in each coset of $W\/W_P$ we have a unique member $w$ of minimal length.\n Let $W^P$ be the set of the minimal length representatives\nin the cosets of $W\/W_P$.\n\nFor any $w\\in W^P$, define the Schubert cell:\n\\[\nC_w^P:= BwP\/P \\subset G\/P\n \\]\nendowed with the reduced subscheme structure.\nThen, it is a locally closed subvariety of the ind-variety $G\/P$ isomorphic with the affine\nspace $\\Bbb A^{\\ell(w)}, \\ell(w)$ being the length of $w$ (cf. [K$_3$, $\\S$7.1]). Its closure is denoted by $X^P_w$, \nwhich is an irreducible (projective) subvariety\nof $G\/P$ of dimension $\\ell(w)$. We denote the point $wP\\in C_w^P$ by $\\dot{w}$.\nWe abbreviate $C^B_w, X_w^B$ by $C_w, X_w$ respectively.\n\nSimilarly, define the opposite Schubert cell\n$$\nC^{w}_P:={B^{-}wP\/P}\\subset \\bar{X}_P,\n$$\nand the opposite Schubert variety\n$$\nX^{w}_P:=\\overline{C^w}\\subset \\bar{X}_P,\n$$\nboth endowed with the reduced subscheme structures.\nThen, $X^{w}_P$ is a finite codimensional irreducible subscheme\nof $\\bar{X}_P$ (cf. [K$_3$, Section 7.1] and [Ka, \\S4]). As above, we abbreviate $C_B^w, X^w_B$ by $C^w, X^w$ respectively.\n\nFor any integral\nweight $\\lambda$ (i.e., any character $e^{\\lambda}$ of $H$), we have\na $G^{\\min}$-equivariant line bundle $\\mathcal{L}_B(\\lambda)$ on $X$\nassociated to the character $e^{-\\lambda}$ of $H$. Similarly, we have \na $G$-equivariant line bundle $\\mathcal{L}_{B^-}(\\lambda)$ on $X^-:=G\/B^-$\nassociated to the character $e^{\\lambda}$ of $H$. \n\n By the Bruhat decomposition\n$$X_P=\\sqcup_{w\\in W^P}\\,C_w^P,$$\nthe singular homology $H_*(X_P, \\bz)$ of $X_P$ with integral coefficients\nhas a basis $\\{\\mu(X_w^P)\\}_{w\\in W^P}$, where $\\mu(X_w^P)\\in H_{2\\ell(w)}(X_P, \\bz)$ denotes the \nfundamental class of $X_w^P$. Let $\\{\\epsilon^w_P\\}_{w\\in W^P}$ be the dual \nbasis of the singular cohomology $H^*(X_P, \\bz)$ under the standard pairing of cohomology with homology, i.e., \n$$\\epsilon^u_P(\\mu(X_v^P))=\\delta_{u,v},\\,\\,\\,\\text{for any} \\,\\,u,v\\in W^P.$$\nThus, $\\epsilon^w_P\\in H^{2\\ell(w)}(X_P, \\bz)$. If $P=B$, we abbreviate \n$\\epsilon^u_P$ by $\\epsilon^u$. \n\nLet $\\Delta=\\{\\alpha_1,\\ldots,\\alpha_{r}\\}\\subset \\mathfrak{h}^{*}$ be the\nset of simple roots,\n$\\{\\alpha_1^{\\vee},\\ldots,\\alpha^{\\vee}_{r}\\}\\subset \\mathfrak{h}$\nthe set of simple coroots and $\\{s_1,\\ldots, s_{r}\\}\\subset W$ the\ncorresponding simple reflections, where $\\mathfrak{h}:=\\Lie H$. Let\n$\\rho\\in X(H)$ be any weight satisfying\n$$\n\\rho(\\alpha^{\\vee}_{i})=1,\\quad\\text{for all}\\quad 1\\leq i\\leq r,\n$$\nwhere $X(H)$ is the character group of $H$ (identified as a subgroup of $\\fh^*$ \nvia the derivative). \nWhen $G$ is a finite dimensional semisimple group, $\\rho$ is unique,\nbut for a general Kac-Moody group $G$, it may not be unique.\n\nChoose elements $x_i\\in \\fh$ such that \n\\beqn \\label{eq1} \\alpha_j(x_i)=\\delta_{i,j}, \\,\\,\\,\\text{for any}\\,\\, 1\\leq i,j\\leq r.\n\\eeqn\nObserve that $x_i$ may not be unique.\n\nDefine the set of {\\it dominant integral weights} \n$$P_+:=\\{\\lambda\\in X(H): \\lambda (\\alpha_i^\\vee)\\in \\bz_+ \\,\\forall \\,\n1\\leq i\\leq r\\},$$\n and the set of {\\it dominant integral regular weights} \n$$P_{++}:=\\{\\lambda\\in X(H): \\lambda (\\alpha_i^\\vee)\\in \\bz_{\\geq 1} \\,\\forall \\,\n1\\leq i\\leq r\\},$$\nwhere $\\bz_+$ is the set of non-negative integers. The integrable highest \nweight (irreducible) modules of $G^{\\min}$ are parameterized by $P_+$. For $\\lambda\n\\in P_+$, let $L(\\lambda)$ be the corresponding integrable highest weight (irreducible) $G$-module \nwith highest weight $\\lambda$. \n\n\\section{Necessary Inequalities for the Saturated Tensor Semigroup}\\label{sec3}\nFix a positive integer $s$ and define the {\\em saturated tensor semigroup} $\\Gamma_s=\\Gamma_s(G)$:\n\\beqn\n\\Gamma_s:= \\{(\\lambda_1, \\dots, \\lambda_s,\\mu)\\in P_+^{s+1}: \\exists\\,\nN>1 \\,\\,\\text{with}\\,\\, L(N\\mu)\\subset L(N\\lambda_1)\\otimes \\dots \\otimes L(N\\lambda_s)\\}.\n\\eeqn\nIt is indeed a semigroup by the anlogue of the Borel-Weil theorem for the Kac-Moody case (see the identity \\eqref{ne3.3}\nin the proof of Theorem \\ref{thm1}).\nWe give a certain set of inequalities satisfied by $\\Gamma_s$. But, we first recall some basic results about the Hilbert-Mumford index.\n\n\\begin{definition}\\label{git} Let $S$ be any (not necessarily reductive) algebraic group\nacting on a (not necessarily projective) variety $\\exx$ and let $\\elal$ be\nan $S$-equivariant line bundle on $\\exx$. Let $O(S)$ be the set of all one parameter\nsubgroups (for short OPS) in $S$.\n Take any $x\\in \\exx$ and\n $\\delta \\in O(S)$ such that the limit\n $\\lim_{t\\to 0}\\delta(t)x$\nexists in $\\exx$ (i.e., the morphism ${\\delta}_x:\\Bbb{G}_m\\to \\exx$ given by\n$t\\mapsto \\delta(t)x$ extends to a morphism $\\tilde{\\delta}_x : \\Bbb{A}^1\\to \\exx$).\nThen, following Mumford, define a number $\\mu^{\\elal}(x,\\delta)$ as follows:\nLet $x_o\\in \\exx$ be the point $\\tilde{\\delta}_x(0)$. Since $x_o$ is $\\Bbb{G}_m$-invariant\nvia $\\delta$, the fiber of $\\elal$ over $x_o$ is a\n$\\Bbb{G}_m$-module; in particular, it is given by a character of $\\Bbb{G}_m$. This integer is defined as $\\mu^{\\elal}(x,\\delta)$.\n\\end{definition}\n\nWe record the following standard properties of $\\mu^{\\elal}(x,\\delta)$ (cf.\n [MFK, Chap. 2, $\\S$1]):\n\\begin{proposition}\\label{propn14} For any $x\\in \\exx$ and $\\delta \\in O(S)$ such that $\\lim_{t\\to 0}\\delta(t)x$\nexists in $\\exx$, we have the following (for any $S$-equivariant line bundles\n$\\elal, \\elal_1, \\elal_2$):\n\\begin{enumerate}\n\\item[(a)]\n$\\mu^{\\elal_1\\otimes\\elal_2}(x,\\delta)=\\mu^{\\elal_1}(x,\\delta)+\\mu^{\\elal_2}(x,\\delta).$\n\\item[(b)] If there exists $\\sigma\\in H^0(\\exx,\\elal)^S$ such that $\\sigma(x) \\neq 0$, then $\\mu^{\\elal}(x,\\delta)\\geq 0.$\n\\item[(c)] If $\\mu^{\\elal}(x,\\delta)=0$, then any element of $H^0(\\exx,\\elal)^S$\nwhich does not vanish at $x$ does not vanish at $\\lim_{t\\to 0}\\delta(t)x$ as well.\n\\item[(d)] For any $S$-variety $\\exx'$ together with an $S$-equivariant morphism $f:\\exx'\\to \\exx$ and any $x'\\in \\exx'$ such that $\\lim_{t\\to 0}\\delta(t)x'$\nexists in $\\exx'$, we have\n$\\mu^{f^*\\elal}(x',\\delta)=\\mu^{\\elal}(f(x'),\\delta).$\n\\item[(e)] (Hilbert-Mumford criterion) Assume that $\\exx$ is projective, $S$ is\n connected and reductive\nand $\\elal$ is ample. Then, $x\\in\\exx$ is semistable (with respect to $\\elal$) if\nand only if $\\mu^{\\elal}(x,\\delta)\\geq 0$, for all $\\delta\\in O(S)$.\n\nIn particular, if $x\\in \\exx$ is semistable and $\\delta$-fixed, then\n$\\mu^{\\elal}(x,\\delta)= 0$.\n\\end{enumerate}\n\\end{proposition}\n\nThe following theorem is one of our main results giving a collection of necessary inequalities defining the semigroup\n$\\Gamma_s$.\n \\begin{theorem} \\label{thm1} Let $G$ be any symmetrizable Kac-Moody group and let $(\\lam_1, \\cdots ,\\lam_s, \\mu )\\in \\Gamma_s$.\n Then, for any $u_1, \\dots ,u_s, v\\in W$ such that $n^v_{u_1, \\dots, u_s}\\neq 0$, where\n$$\\eps^{u_1} \\cdots \\eps^{u_s}\n= \\sum_w n^w_{u_1, \\dots, u_s}\\,\\eps^w\\in H^*(X, \\bz),$$ we have\n \\[\n\\bigl(\\sum^s_{j=1} \\lam_j (u_jx_i)\\bigr) - \\mu (vx_i) \\geq 0, \\quad\\text{ for any }x_i,\n \\]\nwhere $x_i$ is defined by the equation \\eqref{eq1}.\n \\end{theorem}\n\n \\begin{proof}\nLet\n \\[\nZ := \\bigl\\{ (\\bar{g}_1, \\dots ,\\bar{g}_s)\\in {(X^-)}^{s}: g_1X^{u_1}\n\\cap \\cdots \\cap g_sX^{u_s}\\cap X_v \\neq \\emptyset\\bigr\\} ,\n \\]\nwhere $X^-:=G\/B^-$ and $\\bar{g}_j= g_jB^-$. \n Then, $Z$ contains a nonempty open set by Proposition \\ref{prop5}.\n(In fact, by Proposition \\ref{prop5}, $Z = (X^-)^{s}$, but we do not need this stronger result.)\n\nTake a nonzero $\\sig\\in H^0 \\bigl( (X^-)^{s}\\times X, \\cl^N\n\\bigr)^{G^{\\min}}$, where\n$$\\cl := \\cl_{B^-}(\\lam_1)\\boxtimes \\cdots\\boxtimes \\cl_{B^-}(\\lam_s)\\boxtimes\n\\cl_B (\\mu ).$$\nSuch a nonzero $\\sigma$ exists, for some $N>0$, since by [K$_3$, Corollary 8.3.12(a) and Lemma 8.3.9], \n\\begin{align} \\label{ne3.3}\nH^0 \\bigl( (X^-)^{s}\\times X, \\cl^N\n\\bigr)^{G^{\\min}}&\\simeq \\Hom_{G^{\\min}}\\bigl( L(N\\lambda_1)^\\vee\\otimes \\dots \\otimes L(N\\lambda_s)^\\vee\\otimes L(N\\mu), \\bc \\bigr)\\notag\\\\\n&\\simeq \\Hom_{G^{\\min}}\\bigl( L(N\\mu), [L(N\\lambda_1)^\\vee\\otimes \\dots \\otimes L(N\\lambda_s)^\\vee]^*\\bigr)\\notag\\\\\n&\\simeq \\Hom_{G^{\\min}}\\bigl( L(N\\mu), [L(N\\lambda_1)^\\vee\\otimes \\dots \\otimes L(N\\lambda_s)^\\vee]^\\vee \\bigr)\\notag\\\\\n&\\simeq \\Hom_{G^{\\min}}\\bigl( L(N\\mu), L(N\\lambda_1)\\otimes \\dots \\otimes L(N\\lambda_s) \\bigr)\\notag\\\\\n&\\neq 0,\n\\end{align}\nsince $(\\lambda_1, \\dots, \\lambda_s,\\mu)\\in \\Gamma_s$, where, for a $G^{\\min}$-module $M$, $M^\\vee$ denotes the direct sum of the $H$-weight spaces of the full dual module $M^*$. \n\nPick $(\\bar{g}_1, \\dots ,\\bar{g}_s)\\in Z$ such that $\\sig (\\bar{g}_1,\n\\dots ,\\bar{g}_s, \\bar{1})\\neq 0$, where \n$\\bar{1} =1\\cdot B$. Since $(\\bar{g}_1, \\dots ,\\bar{g}_s)\\in Z$, there exists $u'_1 \\geq u_1, \\cdots , u'_s \\geq u_s$ and $v' \\leq v$\nsuch that $g_1C^{u'_1} \\cap \\cdots \\cap g_s C^{u'_s}\n \\cap C_{v'}$ is nonempty. Now, pick $g\\in G^{\\min}$ such that\n \\beqn \\label{e101}\ngB \\in g_1C^{u'_1} \\cap \\cdots \\cap g_s C^{u'_s}\n \\cap C_{v'}.\n \\eeqn\nBy Proposition \\ref{propn14}, for any $\\delta \\in O(G^{\\min})$, $\\mu^{\\cl}(\\bar{x}, \\del\n(t))\\geq 0$, where $\\bar{x} = (\\bar{g}_1, \\dots ,\\bar{g}_s, \\bar{1})$\n(since $\\sig (\\bar{x})\\neq 0$). By the following Lemma \\ref{lem2},\napplied to the OPS $\\delta (t)=gt^{x_i}g^{-1}$, we get\n \\beqn \\label{eqn02}\n\\bigl(\\sum^s_{j=1} \\lam_j (u'_jx_i)\\bigr) - \\mu (v' x_i) \\geq 0. \n \\eeqn\nBut, by [K$_3$, Lemma 8.3.3],\n \\[ (u'_j)^{-1}\\lam_j \\leq u_j^{-1} (\\lam_j). \\]\nThus,\n \\[ \\lam_j(u'_j x_i) \\leq \\lam_j(u_jx_i). \\]\nSimilarly,\n \\[ \\mu (v'x_i) \\geq \\mu (v x_i). \\]\nThus, from \\eqref{eqn02}, we get\n \\[ \\bigl(\\sum_{j=1}^s \\lam_j(u_jx_i) \\bigr)- \\mu (vx_i) \\geq 0. \\]\n\n This proves the theorem. \n \\end{proof}\n\n \\begin{lemma} \\label{lem2} Let $g\\in G^{\\min}$ be as in the equation \\eqref{e101}.\nConsider the one parameter subgroup\n $\n\\del (t) = gt^{x_i} g^{-1}\\in O(G^{\\min}).\n $ Then,\n\n(a) $\\mu^{\\cl_{B^-}(\\lam_j)} (g_jB^-, \\del (t)) = \\lam_j(u_j' x_i)$.\n\n(b) $\\mu^{\\cl_B(\\mu )}(1\\cdot B, \\del (t)) = -\\mu (v'x_i)$.\n \\end{lemma}\n\n \\begin{proof} (a) \n $\\mu^{\\cl_{B^-}(\\lam_j)} (g_jB^-, \\del (t))= \\mu^{\\cl_{B^-}\n (\\lam_j)}(g^{-1}g_jB^-, t^{x_i})$.\\\\\nBy assumption, $g^{-1}_jg\\in U^- u'_jB$. Write\n \\[\ng_j^{-1}g = b_j^-u'_jp_j, \\quad\\text{ for some } \\,b_j^-\\in U^-,\\, p_j \\in B.\n \\]\nThus,\n \\[ 1 = g^{-1}g_jb_j^- u_j' p_j. \\]\nLet\n \\[ b_j(t) = b_j^- u'_j t^{-x_i} (u'_j)^{-1} (b_j^-)^{-1} \\in B^- . \\]\nThen,\n \\beqn\\label{eqn01}\nt^{x_i} g^{-1}g_jb_j(t) = t^{x_i}p_j^{-1} t^{-x_i} (u'_j)^{-1} (b_j^-)^{-1}.\n \\eeqn\nConsider the $G_m$-invariant section (via $t^{x_i}$) of $\\cl_{B^-}(\\lam_j):$\n \\begin{align*}\n\\hat{\\sig}(t) &= \\bigl( t^{x_i}\\, g^{-1}g_j, 1\\bigr) \\mod B^-\\\\\n&= \\bigl( t^{x_i}\\, g^{-1}g_jb_j(t), \\lam_j (b_j(t)^{-1})\\bigr) \\mod B^- .\n \\end{align*}\nClearly, $\\lt_{t\\to 0} \\,t^{x_i}\\, g^{-1}g_jb_j(t)$ exists in $G$ by \\eqref{eqn01}.\n\nNow,\n \\begin{align*}\n\\lam_j \\bigl( b_j(t)^{-1}\\bigr) &= \\lam_j\\bigl( b_j^- u'_j t^{x_i} (u'_j)^{-1}\n(b_j^-)^{-1}\\bigr)\\\\\n &= \\lam_j \\bigl( t^{u_j' x_i}\\bigr) .\n \\end{align*}\nThis gives\n \\[\n\\mu^{\\cl_{B^-}(\\lam_j)} (g_jB^-, \\del (t)) = \\lam_j (u'_j(x_i)) .\n \\]\n\nThis proves the (a) part of the lemma.\n\\vskip1ex\n\n (b) $\\mu^{\\cl_B(\\mu )}(1\\cdot B, \\del (t)) = \\mu^{\\cl_B(\\mu )}\n (g^{-1}B, t^{x_i})$. By assumption,\n \\[\n g\\in B v'\\cdot B.\n \\]\nWrite\n \\[\ng = bv'p, \\quad\\text{for }b\\in U, p\\in B.\n \\]\nThus,\n \\[ 1 = g^{-1} b v'p. \\]\nLet\n \\[ b(t) = bv't^{-x_i}(v')^{-1}b^{-1} \\in B. \\]\nNow,\n \\[\nt^{x_i}g^{-1} b(t) = t^{x_i}p^{-1} t^{-x_i}(v' )^{-1}b^{-1}.\n \\]\nThus,\n \\[\n\\lt_{t\\to 0} \\,t^{x_i}g^{-1}b(t) \\text{ exists in } G^{\\min}.\n \\]\n\nConsider the $G_m$-invariant section (via $t^{x_i}$)\n \\begin{align*}\n\\hat{\\sig}(t) &= (t^{x_i}g^{-1}, 1) \\mod B\\\\\n&= \\bigl( t^{x_i}g^{-1}b(t), \\mu (b(t))\\bigr) \\mod B.\n \\end{align*}\nNow,\n \\begin{align*}\n\\mu (b(t)) &= \\mu (bv' t^{-x_i} (v')^{-1}b^{-1})\\\\\n&= \\mu (t^{-v' x_i}).\n \\end{align*}\nThis gives\n \\[\n\\mu^{\\cl_B(\\mu )}(1\\cdot B, \\del (t)) = -\\mu (v' (x_i)).\n \\]\n\n This proves the (b)-part and hence the lemma is proved.\n \\end{proof}\n\n \n \\begin{definition} \\label{n2.1}\n For a quasi-compact scheme $Y$, an $\\co_{Y}$-module $\\cs$ is called {\\it coherent}\n if it is finitely presented as an $\\co_{Y}$-module and any $\\co_{Y}$-submodule of finite\n type admits a\nfinite presentation.\n\n An $\\co_{\\bar{X}}$-module $\\cs$ is called {\\it coherent} if\n $\\cs_{|V^S}$ is a\ncoherent $\\co_{V^S}$-module for any finite ideal $S\\subset W$ (where a subset $S\\subset W$\nis called an {\\it ideal} if\n for $x\\in S$ and $y\\leq x\\Rightarrow y\\in S$), where $V^S$ is the quasi-compact open subset\n of $\\bar{X}$ defined by\n $$V^S = \\bigcup_{w\\in S} wU^- B\/B.$$\n Let $K^0(\\bar{X})$ denote the Grothendieck group of\n coherent $\\co_{\\bar{X}}$-modules $\\cs$.\n\n Similarly,\ndefine $K_0(X) := \\lim_{n\\to\\infty} K_0(X_n)$, where $\\{\nX_n\\}_{n\\geq 1}$ is the filtration of $X$ giving the ind-projective\nvariety structure (i.e., $X_n = \\bigcup_{\\ell (w)\\leq n} C_w$) and\n$K_0(X_n)$ is the Grothendieck group of coherent\nsheaves on the projective variety $X_n$.\n\nWe also define\n \\[\nK^{\\optop}(X) := \\Invlt_{n\\to\\infty} K^{\\optop}(X_n),\n \\]\nwhere $K^{\\optop}(X_n)$ is the topological $K$-group of the\n projective variety $X_n$.\n\nLet $*:K^{\\optop}(X_n)\\to K^{\\optop}(X_n)$ be the involution induced from\nthe operation which takes a vector bundle to its dual. This,\n of course, induces the involution $*$ on $K^{\\optop}(X)$.\n\nFor any $w\\in W$,\n \\[ [\\co_{X_w}] \\in K_0(X). \\]\n \\end{definition}\n\n \\begin{lemma} $\\bigl\\{ [\\co_{X_w}]\\bigr\\}_{w\\in W}$ forms a basis of $K_0(X)$ as a $\\bz$-module.\n \\end{lemma}\n\n \\begin{proof} By [CG, \\S 5.2.14 and Theorem 5.4.17], the result follows.\n \\end{proof}\n\n For $u\\in W$, by [KS, \\S 2], $\\co_{X^u}$ is a coherent $\\co_{\\bar{X}}$-module.\n In particular, $\\co_{\\bar{X}}$ is a coherent $\\co_{\\bar{X}}$-module.\n\n\n\nDefine a pairing\n$$\n\\langle \\, ,\\, \\rangle : K^0(\\bar{X}) \\otimes K_0(X) \\to \\bz,\\,\\,\n\\langle[\\cs], [\\cf]\\rangle = \\sum_i (-1)^i \\chi \\bigl (X_n, \\tor_\ni^{\\co_{\\bar{X}}}\n (\\cs,\\cf ) \\bigr),$$\nif $\\cs$ is a coherent sheaf on $\\bar{X}$ and $\\cf$\nis a coherent sheaf on ${X}$ supported in $X_n$ (for\nsome $n$), where $\\chi$ denotes the \nEuler-Poincar\\'{e} characteristic.\nThen, as in [K$_4$, Lemma 3.4], \n the above pairing is well defined.\n \nBy [KS, Proof of Proposition 3.4], for any $u\\in W$,\n \\beqn\\label{eq1.0}\n\\ext^k_{\\co_{\\bar{X}}} (\\co_{X^u}, \\co_{\\bar{X}}) =0 \\quad\\forall k\\neq \\ell (u).\n \\eeqn\nDefine the sheaf\n \\[\n\\om_{X^u} := \\ext^{\\ell (u)}_{\\co_{\\bar{X}}}\n\\bigl(\\co_{X^u}, \\co_{\\bar{X}} \\bigr)\\otimes\\cl (-2\\rho ),\n \\]\n which, by the analogy with the Cohen-Macaulay (for short CM) schemes of finite type, will be called\n the {\\it dualizing sheaf} of $X^u$.\n\n\nNow, set the sheaf on $\\bar{X}$\n \\begin{align*}\n\\xi^u &:= \\cl (\\rho )\\om_{X^u} \\\\\n&= \\cl (-\\rho ) \\ext^{\\ell (u)}_{\\co_{\\bar{X}}}\n(\\co_{X^u}, \\co_{\\bar{X}} ).\n \\end{align*}\nThen, as proved in [K$_4$, Proposition 3.5], for any $u,w\\in W$,\n\\beqn\\label{e106}\n\\langle[\\xi^u], [\\co_{X_w}]\\rangle = \\delta_{u,w}.\n\\eeqn\nWith these preliminaries, we are ready to prove the following result.\n \\begin{proposition} \\label{prop5} With the notation as in the proof of Theorem \\ref{thm1},\n$Z = (X^-)^{s}$, if $\\eps^v $ occurs in $\\eps^{u_1}\n\\cdots\\eps^{u_s}$ with\n nonzero coefficient.\n \\end{proposition}\n\n \\begin{proof} We give the proof in the case $s=2$. The proof for general\n $s$ is similar.\n\nFor $u,v\\in W$, express\n \\[\n\\eps^u\\eps^v = \\sum_{ \\substack{w\\\\ \\ell (w)=\\ell (u)+\\ell (v)} } n^w_{u,v} \\eps^w.\n \\]\nExpress the product in topological $K$-theory $K^{\\optop}(X)$ of $X=G^{\\min}\/B$:\n \\[\n\\psi^u_o\\psi^v_o = \\sum_{\\ell (w)\\geq\\ell (u)+\\ell (v)} m^w_{u,v} \\psi^w_o,\n \\]\nwhere $\\psi^w := *\\tau^{w^{-1}}$ ($\\tau^w$ being the Kostant-Kumar `basis'\nof $K^{\\optop}_H(X)$ as in [KK, Remark 3.14]) and $\\{\\psi^{w}_o\\}_{w \\in W}$ is the corresponding `basis' of \n$K^{\\optop}(X)\\simeq \\bz\\otimes_{R(H)}\\,K^{\\optop}_H(X),$ cf. [KK, Proposition 3.25]). \n\nThen, by [KK, Proposition 2.30],\n \\beqn\\label{e102}\nn_{u,v}^w = m_{u,v}^w, \\quad\\text{if }\\ell (w) = \\ell (u)+\\ell (v).\n \\eeqn\nLet $\\Delta: X \\to X\\times X$ be the diagonal map. Then, by [K$_4$, Proposition 4.1] and the identity \\eqref{e106}, for any $u,v,w \\in W$, $g_1,g_2\\in G^{\\min}$,\n \\begin{align*}\nm^w_{u,v} &= \\langle [\\xi^u\\boxtimes\\xi^v], [\\Del_*\\co_{X_w}]\\rangle \\\\\n&= \\langle [\\xi^u\\boxtimes\\xi^v], [(g_1^{-1}, g_2^{-1}) \\cdot (\\Del_* \\co_{X_w})]\\rangle,\n \\end{align*}\nsince $[(g_1^{-1}, g_2^{-1})\\cdot\\Del_*\\co_{X_w}]= [\\Del_*\\co_{X_w}]$ as elements\nof $K_0(X\\times X)$. Thus,\n \\begin{align}\\label{e103}\n m^w_{u,v} &= \\langle [\\xi^u\\boxtimes\\xi^v], [(g_1^{-1}, g_2^{-1}) \\cdot (\\Del_*\\co_{X_w})]\\rangle\n \\\\ &:= \\sum_i (-1)^i \\chi (\\bar{X}\\times \\bar{X}, \\tor_i^{\\co_{\\bar{X}\\times \\bar{X}}} \\Bigl(\\xi^u\\boxtimes\\xi^v,(g_1^{-1}, g_2^{-1}) \\cdot (\\Del_* \\co_{X_w})\\Bigr) .\\notag\n \\end{align}\n\nNow, by definition, the support of $\\xi^u$ is contained in $X^u$ and hence the\nsupport of the sheaf\n \\[\n\\cs_i := \\tor_i^{\\co_{\\bar{X}\\times \\bar{X}}} \\bigl( \\xi^u\\boxtimes\\xi^v, (g_1^{-1}, g_2^{-1})\\cdot \\Del_* \\co_{X_w}\\bigr)\n \\]\nis contained in\n \\beqn\\label{e104}\nX^u\\times X^v \\cap \\bigl((g_1^{-1}, g_2^{-1})\\cdot \\Del (X_w)\\bigr),\n \\eeqn\nwhich is empty if\n \\beqn\\label{e105}\n(g_1X^u) \\cap (g_2X^v) \\cap X_w = \\emptyset .\n \\eeqn\nThus, if the equation \\eqref{e105} is true, then the Tor sheaf $\\cs_i =0$ $\\forall i\\geq 0$. Thus, if \nthe equation \\eqref{e105} is satisfied, \n \\[ m_{u,v}^w =0. \\]\nNow, assume that $\\ell (w) = \\ell (u)+\\ell (v)$. Then, by the equation \\eqref{e102},\n \\[ n^w_{u,v} =0, \\quad\\text{ if the equation \\eqref{e105} is satisfied}. \\]\nBut, since by assumption, $n^w_{u,v} \\neq 0$, we see that\n \\[\n(g_1X^u) \\cap (g_2X^v )\\cap X_w \\neq \\emptyset , \\;\\text{ for any } g_1,g_2\\in G^{\\min}.\n \\]\nBut since $G^{\\min}\/(G^{\\min} \\cap B^-) \\simto X^-$, we get the proposition.\n \\end{proof}\n\n\n\\section{Tensor Product Decomposition for Affine Kac-Moody Lie Algebras}\n\n\\subsection{The Virasoro Algebra}\n\nWe recall the definition of the Virasoro algebra and its basic representation theory, which we need. \nThe {\\it Virasoro algebra} $\\mathrm{Vir}$ has a basis $\\{C,\\, L_{n}\\;:\\; n\\in\\mathbb{Z}\\}$\nover $\\bc$ \nand the Lie bracket is given by \n\\[[L_{m},L_{n}]=(m-n)L_{m+n}+\\frac{1}{12}(m^{3}-m)\\delta_{m,-n}C\\,\\,\n\\text{and}\\, [\\mathrm{Vir,C]=0}.\\] \n\nLet $\\Vir_0:= \\mathbb{C}L_{0}\\oplus\\mathbb{C}C$. Then, a Vir module\n$V$ is said to be a {\\it highest weight representation} if there exists a $\\Vir_0$-eigenvector $v_o\\in V$\nsuch that $L_{n}v_o=0$ for $n\\in\\mathbb{Z}_{>0}$ and $U(\\bigoplus_{n<0}\\mathbb{C}L_{n})v_o=V$.\nSuch a $V$ is said to have {\\it highest weight} $\\lambda\\in \\Vir_0^{*}$\nif $Xv_o=\\lambda(X)v_o$, for all $X\\in \\Vir_0$. (It is easy to see that such a $v_o$ is unique up to a scalar multiple and hence\n$\\lambda$ is unique.)\nThe irreducible\nhighest weight representations of Vir are in 1-1 correspondence with elements\nof $\\Vir_0^{*}$ given by the highest weight. \nDenote the basis of $\\Vir_0^*$ dual to the basis $\\{L_{0},C\\}$ of $\\Vir_0$ as $\\{h,z\\}$. \nFor any $\\mu\\in \\Vir_0^*$, denote the $\\mu$-th weight space \nof $V$ by $V_\\mu$, i.e.,\n\\[V_\\mu:=\\{v\\in V: X\\cdot v=\\mu(X)v\\,\\, \\forall X\\in \\Vir_0\\}.\\]\n\nDefine a Vir module $V$ to be {\\it unitarizable} if there exists a positive\ndefinite Hermitian form $(\\cdot\\,,\\,\\cdot)$ on $V$ so that $(L_{n}v\\,,\\, w)=(v\\,,\\, L_{-n}w)$\nfor all $n\\in\\mathbb{Z}$ and $(Cv\\,,\\, w)=(v\\,,\\, Cw)$. It is easy\nto see that if $M$ is a $\\Vir$-submodule of $V$, then $M^{\\perp}$\nis also a submodule. Hence, any unitarizable representation of Vir\nis completely reducible. Note that for a unitarizable highest weight Vir-representation\n$V$ with highest weight $\\lambda$, if $v_o$ is a highest weight vector,\nthen \n\\beqn \\label{e4.2}\n0\\leq(L_{-n}v_o\\,,\\, L_{-n}v_o)=(L_{n}L_{-n}v_o\\,,\\, v_o)=(2n\\lambda(L_{0})+\\frac{1}{12}(n^{3}-n)\\lambda(C))(v_o\\,,\\, v_o)\n\\eeqn\n for all $n>0$. Therefore, both $\\lambda(L_{0})$ and $\\lambda(C)$\nmust be nonnegative real numbers. \n\n\n\\begin{lem}\\label{virasoro}\nLet $V$ be a unitarizable, highest weight (irreducible) representation of $Vir$\nwith highest weight $\\lambda$. \n\n(a) If $\\lambda(L_0)\\neq 0$, then $V_{\\lambda+nh}\\neq 0$, for any $n\\in \\bz_+$. \n\n(b) If \n$\\lambda(L_0)= 0$ and $\\lambda (C)\\neq 0$, then $V_{\\lambda+nh}\\neq 0$, for any $n\\in \\bz_{>1}$ and \n$V_{\\lambda+h} =0$.\n\n(c) If \n$\\lambda(L_0)= \\lambda (C) = 0$, then $V$ is one dimensional. \n\\end{lem}\n\\begin{proof}\nIf $\\lambda(L_0)\\neq 0$, then by the equation \\eqref{e4.2} (since both of $\\lambda (L_0)$ and $\\lambda(C)\\in \\br_+$), \n$L_{-n}v_o\\neq 0$, for any $n\\in \\bz_+$. \n\nIf $\\lambda(L_0)=0$ and $\\lambda(C)\\neq 0$, then again by the equation \\eqref{e4.2}, \n$L_{-n}v_o\\neq 0$, for any $n\\in \\bz_{>1}$. Also, $L_{-1}v_o=0$. \n\n If $\\lambda(L_0)=\\lambda(C)= 0$, then (by the equation \\eqref{e4.2} again), \n$L_{-n}v_o= 0$, for any $n\\in \\bz_{\\geq 1}$. This shows that $V$ is one dimensional.\n\\end{proof}\n\\subsection{Tensor product decomposition: A general method}\n\nLet $\\mathfrak{g}$ be the untwisted affine Kac-Moody Lie algebra associated to a finite dimensional simple \nLie algebra $\\overset{\\circ}{\\mathfrak{g}}$, i.e.,\n\\[\\fg=\\bigl(\\overset{\\circ}{\\mathfrak{g}}\\otimes \\bc[t,t^{-1}]\\bigr) \\oplus \\bc c\\oplus \\bc d.\\]\nLet $\\overset{\\circ}{\\mathfrak{h}}$ be a Cartan subalgebra of $\\overset{\\circ}{\\mathfrak{g}}$. Then, \n\\[\\fh:=\\overset{\\circ}{\\mathfrak{h}}\\otimes 1\\oplus\\bc c\\oplus \\bc d\\]\nis the standard Cartan subalgebra of $\\fg$.\n Let $\\delta\\in \\fh^*$ be the smallest positive imaginary root of \n$\\fg$ (so that the positive imaginary roots of $\\fg$ are precisely $\\{n\\delta, n\\in \\bz_{\\geq 1}\\}$). Then,\n$\\delta$ is given by $\\delta_{|\\overset{\\circ}{\\mathfrak{h}}\\oplus \\bc c}\\equiv 0$ and $\\delta(d)=1$. \nFor any $\\Lambda \\in P_+$, let $P(\\Lambda)$ \nbe the set of weights of $L(\\Lambda)$ and let $P^o(\\Lambda)$ be the set of $\\delta$-maximal weights of $L(\\Lambda)$, i.e.,\n\\[\nP^o(\\Lambda)=\\left\\{ \\lambda\\in\\mathfrak{h}^{*}:\\lambda \\in P(\\Lambda) \\,\\,\\text{but}\\,\\, \\lambda+n\\delta \\notin P(\\Lambda)\\,\\,\n\\text{for any}\\,\\,n>0\\right\\}.\n\\]\nFor any $\\lambda \\in X(H)$, define the $\\delta$-{\\it character of $L(\\Lambda)$ through} $\\lambda$ by\n\\[c_{\\Lambda,\\lambda}=\\sum_{n\\in\\mathbb{Z}}\\dim L(\\Lambda)_{\\lambda+n\\delta}\\,e^{n\\delta}.\\]\nSince $\\delta$ is $W$-invariant,\n\\beqn \\label{e4.1}\nc_{\\Lambda,\\lambda}=c_{\\Lambda, w\\lambda}, \\,\\,\\text{for any}\\, w\\in W.\n\\eeqn\nMoreover, $P^o(\\Lambda)$ is $W$-stable.\n It\nis obvious that \n\\beqn \\label{e13}\nch\\, L(\\Lambda)=\\sum_{\\lambda\\in P^o(\\Lambda)}c_{\\Lambda,\\lambda}e^{\\lambda}.\n\\eeqn\nBy [K$_3$, Exercise 13.1.E.8], for any $\\lambda\\in P(\\Lambda')$ and $\\Lambda''\\in P_+$, $\\Lambda''+\\lambda+\\rho$ belongs to the Tits cone. Hence,\nthere exists $v\\in W$ such that $v^{-1}(\\Lambda''+\\lambda+\\rho)\\in P_+$. Moreover, if $\\Lambda''+\\lambda+\\rho$ has nontrivial $W$-isotropy, then its isotropy group must contain a reflection (cf. [K$_3$, Proposition 1.4.2(a)]). Thus, for such a $\\lambda\\in P(\\Lambda')$, i.e., if $\\Lambda''+\\lambda+\\rho$ has nontrivial $W$-isotropy,\n\\beqn\\label{e14} \\sum_{w\\in W}\\,\\varepsilon(w) e^{w(\\Lambda''+\\lambda+ \\rho)}=0.\\eeqn\nDefine \n$$ \\bar{P}_+:=\\{\\Lambda \\in P_+: \\Lambda(d)=0\\}.$$\nFor any $m\\in \\bz_+$, let \n\\[P_+^{(m)}:=\\{\\Lambda\\in P_+: \\Lambda (c)=m\\},\\]\nand let \n\\[\\bar{P}_+^{(m)}:=\\bar{P}_+\\cap P_+^{(m)}.\\]\n Then, \n${\\bar{P}}_+^{(m)}$ provides a set of representatives in $P_+^{(m)}$ mod $(P_+\\cap \\bc\\delta)$. \n\nFor any $\\Lambda, \\Lambda',\\Lambda''\\in P_+$, define\n\\begin{align*} T_{\\Lambda}^{\\Lambda',\\Lambda''}=\\{&\\lambda \\in P^o(\\Lambda'): \\exists v_{\\Lambda,\\Lambda'', \\lambda}\\in W\\,\\,\\text{and}\\, \nS_{\\Lambda,\\Lambda'', \\lambda}\\in \\bz \\,\\,\\text{with}\\\\ \n&\\, \\lambda+\\Lambda''+\\rho=v_{\\Lambda,\\Lambda'', \\lambda}(\\Lambda+\\rho)+ \nS_{\\Lambda,\\Lambda'', \\lambda}\\delta\\}.\n\\end{align*}\nObserve that since $\\Lambda+\\rho +n\\delta \\in P_{++}$ for any $n\\in \\bz$, such a $v_{\\Lambda,\\Lambda'', \\lambda}$ and $S_{\\Lambda,\\Lambda'', \\lambda}$ are unique by [K$_3$, Proposition 1.4.2 (a), (b)]\n(if they exist). Also, observe that \n\\beqn \\label{eq1001} T_{\\Lambda}^{\\Lambda',\\Lambda''}=\\emptyset,\\,\\,\\text{unless}\\, \\Lambda (c)=\\Lambda'(c)+\\Lambda''(c)\\,\\,\\,\\text{and}\\,\\, \n\\Lambda'+\\Lambda''-\\Lambda\\in Q,\\eeqn\nwhere $Q$ is the root lattice of $\\fg$.\n\\begin{prop} \\label{tensor} For any $\\Lambda'$ and $\\Lambda''\\in P_+$, \n\\[ch\\,\\bigl( L(\\Lambda')\\otimes L(\\Lambda'')\\bigr) = \\sum_{\\Lambda\\in \\bar{P}_{+}^{(m)}}ch\\, L(\\Lambda)\\bigl(\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}\\bigr),\\]\nwhere $m:=\\Lambda'(c)+\\Lambda''(c)$. \n\nMoreover, $\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}$ is the character of a unitary representation (though, in general, not irreducible) of the Virasoro algebra \n$\\mathrm{Vir}$ with central charge \n$$\\dim \\overset{\\circ}{\\mathfrak{g}}\\cdot\\bigl(\\frac{m'}{m'+g}+\\frac{m''}{m''+g}-\\frac{m}{m+g}\\bigr),$$\nwhere $m':=\\Lambda'(c), m'':=\\Lambda''(c)$ and $g$ is the dual Coxeter number of $\\overset{\\circ}{\\mathfrak{g}}$.\n\\end{prop}\n\\begin{proof}\nBy the Weyl-Kac character formula (cf. [K$_3$, Theorem 2.2.1]) and the identity \\eqref{e13}, for any $\\Lambda', \\Lambda''\\in P_+$, \n\\begin{align*}\n\\left(\\sum_{w\\in W}\\varepsilon(w)e^{w\\rho}\\right)&\\cdot ch\\, L(\\Lambda')\\cdot ch\\, L(\\Lambda'')\\\\\n&=\\left(\\sum_{\\lambda\\in P^o(\\Lambda')}c_{\\Lambda',\\lambda}e^{\\lambda}\\right)\\cdot\\left(\\sum_{w\\in W}\\varepsilon(w)e^{w(\\Lambda''+\\rho)}\\right)\\\\\n&=\\sum_{\\lambda\\in P^o(\\Lambda')}c_{\\Lambda',\\lambda}\\sum_{w\\in W}\\varepsilon(w)e^{w(\\Lambda''+\\lambda+\\rho)},\n\\,\\, \\text{by}\\, \\eqref{e4.1}\\\\\n&= \\sum_{\\Lambda\\in \\bar{P}_{+}^{(m)}}\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}c_{\\Lambda',\\lambda}\\sum_{w\\in W}\\varepsilon(w)e^{w(v_{\\Lambda,\\Lambda'',\\lambda}(\\Lambda+\\rho))+S_{\\Lambda,\\Lambda'',\\lambda}\\delta},\\,\\,\\text{by}\\, \\eqref{e14}\\\\\n &= \\sum_{\\Lambda\\in \\bar{P}_{+}^{(m)}}\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}c_{\\Lambda',\\lambda}\\sum_{w\\in W}\\varepsilon(w)\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})e^{w(\\Lambda+\\rho)}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}\\\\\n &= \\sum_{\\Lambda\\in \\bar{P}_{+}^{(m)}}\\sum_{w\\in W}\\varepsilon(w)e^{w(\\Lambda+\\rho)}\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}.\n\\end{align*}\nThus,\n\\[ch\\, \\bigl(L(\\Lambda')\\otimes L(\\Lambda'')\\bigr) = \\sum_{\\Lambda\\in \\bar{P}_{+}^{(m)}}ch\\, L(\\Lambda)\\bigl(\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}\\bigr).\n\\]\nTo prove the second part of the proposition, use [KR, Proposition 10.3].\nThis proves the proposition.\n\\end{proof}\n\\begin{remark} For an affine Kac-Moody Lie algebra $\\fg$, if we consider the tensor product decomposition of $L(\\Lambda')\\otimes L(\\Lambda'')$ with respect to the derived subalgebra $\\fg'$ (i.e., without the $d$-action), then the components $L(\\Lambda)$ are precisely of the form \n$\\Lambda\\in \\Lambda'+\\Lambda''+\\overset{\\circ}{Q}$, where $\\overset{\\circ}{Q}$ is the root lattice of $\\overset{\\circ}{\\mathfrak{g}}$\n(cf. [KW]). Thus, the determination of the eigen semigroup and the saturated eigen semigroup is fairly easy for $\\fg'$.\n\\end{remark}\n\nLet $\\theta=\\sum_{i=1}^\\ell h_i\\alpha_i$ be the highest root of $\\overset{\\circ}{\\mathfrak{g}}$ (with respect to a choice of the positive roots), written as a linear combination of the simple roots $\\{\\alpha_1, \\dots, \\alpha_\\ell\\}$ of $\\overset{\\circ}{\\mathfrak{g}}$. Let\n\\[ S:= \\{\\sum_{i=0}^\\ell \\, n_i\\alpha_i: n_i\\geq 0 \\,\\,\\,\\text{for any}\\, i\\,\\,\\,\\text{and}\\, 0\\leq n_i< h_i \\,\\,\\text{for some}\\,\\, 0\\leq i\\leq \\ell \\},\\]\nwhere $h_0:=1$.\n\\begin{prop} \\label{prop4.1} Let $\\fg$ be an untwisted affine Kac-Moody Lie algebra as above. Then, \nfor any $\\Lambda \\in P_+$ with $\\Lambda (c)>0$, \n\\[P^o(\\Lambda)_+= S(\\Lambda)\\cap P_+,\\]\nwhere $P^o(\\Lambda)_+:=P^o(\\Lambda)\\cap P_+$ and\n$S(\\Lambda)=\\{\\Lambda-\\beta: \\beta \\in S\\}.$\n\\end{prop}\n\\begin{proof}\nTake $\\lambda \\in S(\\Lambda)$. Then, for any $n\\geq 1$, \n\\[\n\\Lambda-(\\lambda+n\\delta)=\\bigl(\\sum_{i=0}^\\ell\\, n_{i}\\alpha_{i}\\bigr)-n\\delta=(n_{0}-n)\\alpha_{0}+ \\sum_{i=1}^\\ell\\,(n_{i}-nh_{i})\\alpha_{i},\n\\]\n since $\\alpha_{0}:=\\delta-\\theta$. Now, the coefficient of some $\\alpha_{i}$\nin the above sum is negative, for any positive $n$, since $\\lambda \\in S(\\Lambda)$. Thus, \n $\\lambda+n\\delta$\ncould not be a weight of $L(\\Lambda)$ for any positive $n$. Therefore, if $\\lambda \\in P(\\Lambda)\\cap S(\\Lambda)$, then it is $\\delta$-maximal.\n\nBy [Kac, Proposition 12.5(a)], if $\\Lambda(c)\\neq 0$, then $ S(\\Lambda) \\cap P_+\\subset P(\\Lambda).$ Therefore,\n $ S(\\Lambda) \\cap P_+\\subset P^o(\\Lambda)_+.$\n\nConversely, take $\\lambda \\in P^o(\\Lambda)_+$. Then, $\\lambda \\in P(\\Lambda) \\cap P_+$ and \n$\\lambda+\\delta \\notin P(\\Lambda)$. Express $\\lambda=\\Lambda - n_0\\alpha_0-\\sum_{i=1}^\\ell\\, n_i\\alpha_i$, for some \n$n_i\\in \\bz_+$. Then, \n\\[\\lambda+\\delta=\\Lambda - (n_0-1)\\alpha_0-\\sum_{i=1}^\\ell\\, (n_i-h_i)\\alpha_i.\\]\nAgain applying [Kac, Proposition 12.5(a)], $\\lambda+\\delta\\notin P(\\Lambda)$ if and only if $\\lambda+\\delta\\not\\leq \\Lambda $,\ni.e., for some $0\\leq i\\leq \\ell$, $n_i< h_i$. Thus, $\\lambda \\in S(\\Lambda)$. This proves the proposition.\n\\end{proof} \n\n\n\n\\section{$A_{1}^{(1)}$ Case}\n\nIn this section, we consider $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}=\\left(\\bigoplus_{n\\in\\mathbb{Z}}\\mathbb{C}t^{n}\\otimes\n\\mathfrak{sl}_{2}\\right)\\oplus\\mathbb{C}c\\oplus\\bc d$.\nIn this case $\\mathfrak{h}^{*}=\\mathbb{C}\\alpha\\oplus\\mathbb{C}\\delta\\oplus\\mathbb{C}\\Lambda_{0}$, where \n$\\alpha$ is the simple root of $\\mathfrak{sl}_{2}$ and \n${\\Lambda_0}_{|\\overset{\\circ}{\\mathfrak{h}}\\oplus \\bc d}\\equiv 0$ and $\\Lambda_0(c)=1$. Then, $\\Lambda_0$ is a zeroeth fundamental weight. \nThe simple roots of $\\widehat{\\mathfrak{sl}_{2}}$ are \n $\\alpha_0:=\\delta-\\alpha$ and $\\alpha_1:=\\alpha$. The simple coroots are \n $\\alpha_0^\\vee :=c-\\alpha^\\vee$ and $\\alpha_1^\\vee :=\\alpha^\\vee$. It is easy to see that an element of $\\fh^*$ of the form $m\\Lambda_0+\\frac{j}{2}\\alpha$ belongs to $P_+$ if and only if $m,j\\in \\bz_+$ and $m\\geq j$. \n\n\nSpecializing Proposition \\ref{prop4.1} to the case of $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$, we get the following.\n\n\\begin{corollary} \\label{cor5.1} For $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$\nand $\\Lambda=m\\Lambda_{0}+\\frac{j}{2}\\alpha\\in P_+$, \n\\beqn \\label{e5.2}\nP^o(\\Lambda)_+=\\left\\{ \\Lambda-k\\alpha,\\,\\Lambda-l(\\delta -\\alpha)\\;:\\; k,l\\in \\bz_+ \\,\\,\\text{and}\\,\\, k\\leq\\frac{j}{2},\\, l\\leq\\frac{m-j}{2}\\right\\}. \n\\eeqn\n\\end{corollary}\n\\begin{proof} \nThe corollary follows from Proposition \\ref{prop4.1} since $m_1\\Lambda_0+\\frac{m_2}{2}\\alpha +\n m_3 \\delta $ belongs to $P_+$ if and only if $m_1,m_2\\in \\bz_+$ and $m_1\\geq m_2$.\n\\end{proof}\n\nLet $\\pi$ be the projection $\\mathfrak{h}^{*}=\\mathbb{C}\\Lambda_{0}\\oplus \\bc \\alpha\\oplus\\mathbb{C}\\delta\n\\to \\mathbb{C}\\Lambda_{0}\\oplus \\bc \\alpha$. \n\\begin{lemma} \\label{lemma5.1} Let $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$. For $\\Lambda=m\\Lambda_{0}+\\frac{j}{2}\\alpha \\in P_+$ (i.e., \n$m,j\\in \\bz_+$ and $m\\geq j$) such that $m>0$,\n\\beqn\\label{e5.1} \\pi(P^o(\\Lambda))=\\{\\Lambda+k\\alpha: k\\in \\bz\\}.\n\\eeqn\nMoreover, for any $k\\in \\bz$, let $n_k$ be the unique integer such that $\\Lambda+k\\alpha+n_k\\delta\\in P^o(\\Lambda)$. Then, writing\n$k=qm+r, 0\\leq r0$. \nThen,\n\\begin{align*}\n\\pi\\left(T_{\\Lambda}^{\\Lambda',\\,\\Lambda''}\\right)=\\{ \\Lambda'+k\\alpha\\;:\\; k\\in\\mathbb{Z},\\, & k\\equiv\\frac{1}{2}\\left(j-j'-j''\\right) \\\\\n&\\text{or}\\,\\, k\\equiv -\\frac{1}{2}\\left(j+j'+j''\\right)-1\\,\\text{mod}\\,M\\} ,\n\\end{align*}\nwhere $M:=m+2$. In particular, by the equation \\eqref{eq1001}, $T_{\\Lambda}^{\\Lambda',\\,\\Lambda''}$ is nonempty if and\nonly if $\\frac{j-j'-j''}{2}\\in \\bz$. \n\nMoreover, for \n$\\lambda=\\Lambda'+k\\alpha+n_k\\delta \\in T_{\\Lambda}^{\\Lambda',\\Lambda''}$,\n\\[\nv_{\\Lambda,\\Lambda'',\\,\\lambda}=\\begin{cases}\nT_{\\frac{k-\\frac{1}{2}\\left(j-j'-j''\\right)}{M}\\alpha^\\vee}, & \\text{if}\\; k\\equiv\\frac{1}{2}\\left(j-j'-j''\\right)\\,\\mod\\, M\\\\\ns_{1}T_{-\\frac{k+\\frac{1}{2}\\left(j+j'+j''\\right)+1}{M}\\alpha^\\vee },& \\text{if}\\; k\\equiv-\\frac{1}{2}\\left(j+j'+j''\\right)-1\\,\\text{mod}\\, M,\n\\end{cases}\n\\]\nwhere $T_{n\\alpha^\\vee}$ is defined by the equation \\eqref{e5.3}. Further,\n\\[\n S_{\\Lambda,\\Lambda'',\\lambda}=n_k+\\frac{\\left(k-\\frac{1}{2}\\left(j-j'-j''\\right)\\right)\\left(k+\\frac{1}{2}\\left(j+j'+j''\\right)+1\\right)}{M}.\n\\]\n\\end{lemma}\n\\begin{proof}\nFollows from the fact that $W=\\stackrel{\\circ}{W}\\rtimes\\mathbb{Z}\\alpha^\\vee$\nand that $\\rho=2\\Lambda_{0}+\\frac{1}{2}\\alpha$.\\end{proof}\nWe have the following very crucial result.\n\\begin{prop}\\label{maximum}\nFix $\\Lambda, \\Lambda'$ and $\\Lambda''$ as in Lemma \\ref{lemma5.1'} and asume that $\\frac{j-j'-j''}{2}\\in \\bz$ and both of $m',m''>0$. Then, the maximum of \n$\\left\\{ S_{\\Lambda,\\Lambda'',\\lambda}:\\;\\lambda\\in T_{\\Lambda}^{\\Lambda',\\,\\Lambda''}\\,\\,\\text{and}\\, \\,\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=1\\right\\} $\n is achieved precisely when $\\pi(\\lambda)=\\Lambda'+\\frac{1}{2}\\left(j-j'-j''\\right)\\alpha$. \n\\end{prop}\n\\begin{proof} By Lemma \\ref{lemma5.1'} and the explicit description of the length function of $T_{n\\alpha^\\vee}$ (cf. [K$_3$, Exercise 13.1.E.3]), \n$$\\pi\\{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\,\\Lambda''}:\\varepsilon (v_{\\Lambda,\\Lambda'',\\lambda})=1\\}=\\{\\Lambda'+k_l\\alpha:l\\in \\bz\\},$$\nwhere $M:=m+2$ and $k_l:=\\frac{j-j'-j''}{2}+lM$. \nTake $\\lambda=\\Lambda'+k_l\\alpha\\in \\pi(T_{\\Lambda}^{\\Lambda',\\,\\Lambda''})$ for $l\\in \\bz$.\n Write $k_l=q_lm'+r_l$ for $q_l\\in \\bz$ and $0\\leq r_l< m'$. Then, by Lemmas \\ref{lemma5.1}, \\ref{lemma5.2'} and \\ref{lemma5.1'}, for $\\lambda=\\Lambda'+k_l\\alpha$ (setting $J:=\\frac{j-j'-j''}{2}$), \n\\begin{align*}\nS_{\\Lambda,\\Lambda'',\\lambda} & = n_{r_l}-\\frac{(J+j'+lM+r_l)(J+lM-r_l)}{m'}+l(lM+1+j)\\\\\n&=l^2M(1-\\frac{M}{m'})+l(1+j-\\frac{M(j-j'')}{m'})-\\frac{(j-j'')^2-j'^2}{4m'}+\\frac{r_l^2}{m'}\n+\\frac{r_lj'}{m'}+n_{r_l}\\\\\n&= l^2M(1-\\frac{M}{m'})+l(1+j-\\frac{M}{m'}(j-j''))-\\frac{(j-j'')^2-j'^2}{4m'}+p(k_l),\n\\end{align*}\nwhere \n\\[\np(k_l):= \\frac{r_l^2}{m'}+\\frac{r_l}{m'}j' +n_{k_l}.\n\\]\nLet $P=P_{m',j'}:\\mathbb{R}\\rightarrow\\mathbb{R}$ be the following function:\n\\begin{eqnarray*}\n P(s) & := & \\begin{cases}\n\\frac{(s-\\frac{m'}{2}k)^{2}}{m'}-\\frac{(j')^2}{4m'}, & \\text{if}\\;\\left|s-\\frac{m'}{2}k\\right|\\leq\\frac{j'}{2}\\;\\text{for some }k\\in2\\mathbb{Z}\\\\\n\\frac{(s-\\frac{m'}{2}k)^{2}}{m'}-\\frac{(m'-j')^2}{4m'}, & \\text{if}\\;\\left|s-\\frac{m'}{2}k\\right|\\leq\\frac{m'-j'}{2}\\;\\text{for some }k\\in2\\mathbb{Z}+1.\n\\end{cases}\n\\end{eqnarray*}\nLet $k_s\\in \\bz$ be such a $k$. (Of course, $k_s$ depends upon $m'$ and $j'$.)\n\\begin{claim}\n$P(s) = p (s-\\frac{j'}{2})$ for $s \\in \\frac{j'}{2}+\\mathbb{Z}$.\n\\end{claim}\n\\begin{proof}\nClearly, both of $P$ and $p$ are periodic with period $m'$. So, it is enough to show that $P(s) = p (s-\\frac{j'}{2})$, for $s-\\frac{j'}{2}$\nequal to any of the integral points of the interval $[-j', m'-j']$. By Lemma \\ref{lemma5.2'} and the identity \\eqref{ne18}, for\nany integer $-j'\\leq r\\leq 0$,\n$$p (r)= \\frac{1}{m'}r(r+j'),$$\nand for any integer $0\\leq r\\leq m'-j'$, \n$$p (r)= \\frac{r(r+j')}{m'}-r.$$\nFrom this, the claim follows immediately. \n\\end{proof}\n Fix $m'>0$. Let \n\\begin{align*}\nI :=\\{(t,j',m'',j'',j)\\in\\mathbb{R}^5 \\,:\\, &0\\leq j' \\leq m',\\, 1 \\leq m'',\\\\\n &0\\leq j'' \\leq m'',\\, 0\\leq j\\leq m'+m''\\}.\n\\end{align*}\nDefine $F: I \\rightarrow \\mathbb{R}$ by\n\\begin{eqnarray*}\nF: (t,j',m'',j'',j) &\\mapsto& t^{2}M(1-\\frac{M}{m'})+t\\bigl(j(1-\\frac{M}{m'})+1+\\frac{M}{m'}j''\\bigr)\\\\\n&&+\\frac{(j')^2-(j-j'')^{2}}{4m'} +P(\\frac{1}{2}\\left(j-j''\\right)+tM).\n\\end{eqnarray*}\nThus, $F$ is a continuous, piecewise smooth function with failure of differentiability along the set $$\\{(t,j',m'',j'',j)\\in I\\,:\\, \\frac{1}{2}(j\\pm j'-j'')+tM\\in m'\\mathbb{Z} \\}.$$\n\\begin{claim}\\label{claim1}\nLet $\\Delta (t)=\\Delta(t, j',m'',j'', j):=F(t+1,j',m'',j'',j)-F(t,j',m'',j'',j)$. Then, on $I$,\n\\begin{enumerate}\n\\item $\\Delta$ is a nonincreasing function of $t$\n\\item $\\Delta$ is increasing with respect to $j''$\n\\item $\\Delta$ is nonincreasing in $j$\n\\item\\label{delclaimb} $\\Delta(0)$ is decreasing in $m''$\n\\item\\label{delclaima} $\\Delta(-1)$ is nondecreasing in $m''$.\\end{enumerate}\n\\end{claim}\n\n\\begin{proof}\nWe compute and give bounds for the partial derivatives of $\\Delta$, where they exist. \n\\begin{eqnarray*}\n\\Delta(t) & =&\\,2tM(1-\\frac{M}{m'})+\\bigl((j+M)(1-\\frac{M}{m'})+1+\\frac{M}{m'}j''\\bigr)\\\\\n&&+P(tM+M+\\frac{1}{2}(j-j''))-P(tM+\\frac{1}{2}(j-j'')).\n\\end{eqnarray*}\nHence, \n\\begin{eqnarray*}\n\\partial_t\\Delta(t) & = & 2M(1-\\frac{M}{m'})+M\\bigl(P'(tM+M+\\frac{1}{2}(j-j''))-P'(tM+\\frac{1}{2}(j-j''))\\bigr)\\\\\n & = & 2M(1-\\frac{M}{m'})+2\\frac{M}{m'}(M-\\frac{m'}{2}k_{1}+\\frac{m'}{2}k_{0})\\\\\n & = & 2M(1-\\frac{k_{1}-k_{0}}{2}),\n\\end{eqnarray*}\nwhere $k_1:=k_{(t+1)M+\\frac{1}{2}(j-j'')}$ and $k_0:=k_{tM+\\frac{1}{2}(j-j'')}$.\nSince $2\\leq k_{1}-k_{0}$, we see that $\\partial_t\\Delta\\leq0$, wherever $\\partial_t \\Delta$ exists. \n Since $\\Delta$ is continuous everywhere and differentiable \non all but a discrete set, $\\Delta$ is nonincreasing in $t$.\n\\[\n\\partial_{j''}\\Delta(t)=\\frac{M}{m'}-\\frac{1}{2}\\left(P'(tM+M+\\frac{1}{2}(j-j''))-P'(tM+\\frac{1}{2}(j-j''))\\right).\n\\]\nNow, $|P'|\\leq1$, so $\\frac{M}{m'}+1\\geq\\partial_{j''}\\Delta\\geq\\frac{M}{m'}-1=\\frac{m''+2}{m'}>0$.\n\nFor (3):\n\\begin{eqnarray*}\n\\partial_{j}\\Delta(t) & = & 1-\\frac{M}{m'}+\\frac{1}{2}\\bigl(P'(tM+M+\\frac{1}{2}(j-j''))-P'(tM+\\frac{1}{2}(j-j''))\\bigr)\\\\\n & = & 1-\\frac{M}{m'}+\\frac{1}{m'}\\left(M-\\frac{m'}{2}k_{1}+\\frac{m'}{2}k_{0}\\right)\\\\\n & = & 1-\\frac{k_{1}-k_{0}}{2}\\leq 0.\n\\end{eqnarray*}\n (\\ref{delclaimb}) and (\\ref{delclaima}) follow from the following calculation:\n\\begin{align*}\n\\partial_{m''}\\Delta=&2t(1-2\\frac{M}{m'})+(1-2\\frac{M}{m'}+\\frac{1}{m'}(j''-j))\\\\\n&+(t+1)P'(tM+M+\\frac{1}{2}(j-j''))-tP'(tM+\\frac{1}{2}(j-j'')).\n\\end{align*}\nHence, \n\\begin{eqnarray*}\n\\partial_{m''}\\Delta(0) & = & 1-2\\frac{M}{m'}+\\frac{1}{m'}(j''-j)+P'(M+\\frac{1}{2}(j-j''))\\\\\n & \\leq & 1-2\\frac{M}{m'}+\\frac{m''}{m'}+1\\\\\n & = & \\frac{-m''-4}{m'}<0,\n\\end{eqnarray*}\nand \n\\begin{eqnarray*}\n\\partial_{m''}\\Delta(-1) & = & -2(1-2\\frac{M}{m'})+(1-2\\frac{M}{m'}+\\frac{1}{m'}(j''-j))+P'(-M+\\frac{1}{2}(j-j''))\\\\\n & = & -1+2\\frac{M}{m'}+\\frac{1}{m'}(j''-j)+P'(-M+\\frac{1}{2}(j-j''))\\\\\n & = & -1+2\\frac{M}{m'}+\\frac{1}{m'}(j''-j)-2\\frac{M}{m'}+\\frac{1}{m'}(j-j'')-k_{0}\\\\\n & = & -1-k_{0}.\n\\end{eqnarray*}\nNote that $k_{0}\\leq-1$ since $-\\frac{(j-j'')}{2}-M<-\\frac{m'}{2}$.\nThus, $\\partial_{m''}\\Delta(-1)\\geq0$.\\end{proof}\n\\begin{claim}\n{\\it The maximum of $F=F(-, j',m'',j'',j):\\,\\mathbb{Z}\\rightarrow\\mathbb{R}$ occurs at $0$.}\n\\end{claim}\n\\begin{proof}\nWe show that $\\Delta(-1)>0 > \\Delta(0)$. Since $\\Delta$ is\nnonincreasing in $t$, it would follow that $F(0)>F(t)$ for all $t\\in\\mathbb{Z}_{\\neq0}$.\n\nLet us begin with $\\Delta(-1)$. By the previous claim \\ref{claim1}, $\\Delta(-1)$\nis as small as possible when $m''=1$, $j''=0$, and $j=m'+1$. So,\nlet us compute with these values:\n\n\\begin{eqnarray*}\n\\Delta(-1) & \\geq & \\frac{6}{m'}+1+P(\\frac{1}{2}m'+\\frac{1}{2})-P(-2-\\frac{1}{2}m'-\\frac{1}{2})\\\\\n & = & \\frac{6}{m'}+1+\\frac{(\\frac{1}{2}m'+\\frac{1}{2}-\\frac{1}{2}m'k_{1})^{2}}{m'}-\\frac{(2+\\frac{1}{2}m'+\\frac{1}{2}+\n\\frac{1}{2}m'k_{0})^{2}}{m'}\\\\\n&+&\\begin{cases}\n\\frac{m'}{4}-\\frac{j'}{2} & \\text{ if }k_{0}\\text{ odd},\\, k_{1}\\text{ even}\\\\\n0 & \\text{ if }k_{1}-k_{0}\\text{ even}\\\\\n\\frac{j'}{2}-\\frac{m'}{4} & \\text{ if }k_{1}\\text{ odd},\\, k_{0}\\text{ even}.\n\\end{cases}\n\\end{eqnarray*}\nNote that for $m'\\geq 5$, the possible values of $(k_{1},k_{0})$ are\n$(1,-1)$; $(1,-2)$; or $(2,-2)$. So, the result, that $\\Delta (-1)>0$, is established\nby considering such pairs directly and by cases for smaller $m'$.\n\nFor $\\Delta(0)$, we take $m''=1$, $j''=1$, and $j=0$.\n\\begin{eqnarray*}\n\\Delta(0) & = & \\bigl(\\frac{-3(3+m')}{m'}+1+\\frac{3+m'}{m'}\\bigr)+P(\\frac{1}{2}+2+m')-P(-\\frac{1}{2})\\\\\n & =&1 - \\frac{2(3+m')}{m'}+P(\\frac{1}{2}+2+m')-P(-\\frac{1}{2})\\\\\n & = & 1-\\frac{2(3+m')}{m'}+\\frac{(\\frac{1}{2}+2+m'-\\frac{1}{2}m'k_{1})^{2}}{m'}-\\frac{(\\frac{1}{2}+\\frac{1}{2}m'k_{0})^{2}}{m'}\n\\\\ &+&\\begin{cases}\n\\frac{m'}{4}-\\frac{j'}{2} & \\text{ if }k_{0}\\text{ odd},\\, k_{1}\\text{ even}\\\\\n0 & \\text{ if }k_{1}-k_{0}\\text{ even}\\\\\n\\frac{j'}{2}-\\frac{m'}{4} & \\text{ if }k_{1}\\text{ odd},\\, k_{0}\\text{ even}.\n\\end{cases}\n\\end{eqnarray*}\nFor $m'\\geq 5$, the possible values of $(k_{1}, k_{0})$ are\n $(3,-1)$; $(3,0)$; or $(2,0)$. So, again the result, that $\\Delta (0)<0$, is established\nby considering such pairs directly and by cases for smaller $m'$.\n\\end{proof}\n This completes the proof of the proposition.\n\\end{proof}\n\\begin{rem}\\label{newremark}\nWe have shown that $F(l,j',m'',j'',j) = S_{\\Lambda,\\Lambda'',\\lambda}$ for integral values of $l$. If $l$ is not an integer, then $\\lambda_l := \\Lambda' + (lM+J)\\alpha$ may not be in $\\pi(T_\\Lambda^{\\Lambda',\\Lambda''})$, in which case $ S_{\\Lambda,\\Lambda'',\\lambda_l}$ is not defined. On the other hand, if $\\lambda_l \\in \\pi(T_\\Lambda^{\\Lambda',\\Lambda''})$, we note that the equality $F(l,j',m'',j'',j) = S_{\\Lambda,\\Lambda'',\\lambda_l}$ holds, as can be seen by letting $k_l = lM -\\frac{1}{2} (j+j'+j'')-1$ in the above proof.\n\\end{rem}\n\nNow, let us apply the same analysis to the case that $\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=-1$.\nBy Lemma \\ref{lemma5.1'}, this corresponds to $k_l=-\\frac{1}{2}\\left(j+j'+j''\\right)-1+lM$.\n For \n$\\lambda=\\Lambda'+k_l\\alpha$, let us denote the function $S_{\\Lambda,\\Lambda'',\\lambda}$ by $G_\\bz(l)=G_\\bz(l, j',m'',j'',j)$. Thus,\n$G_\\bz:\\bz \\to \\bz$. \n\\begin{lemma} \\label{lemma5.9} Define the function $G=G(-, j',m'',j'',j):\\br \\to \\br$ by \n\\[G(t, j',m'',j'',j)=F(t-\\frac{j+1}{M}, j',m'',j'',j).\\]\nThen, $G_{|\\bz}=G_\\bz.$\n\nHence, $S_{\\Lambda,\\Lambda'',\\lambda}$\nhas a maximum when $l=0$ or $l=1$.\n\\end{lemma}\n\\begin{proof}\nBy the proof of Proposition \\ref{maximum} and Remark \\ref{newremark}, $S_{\\Lambda,\\Lambda'',\\lambda+(j+1)\\alpha} = F(l)$, for $\\lambda=\\Lambda'+k_l\\alpha$. Since $\\lambda = \\Lambda' +(-\\frac{1}{2}\\left(j+j'+j''\\right)-1+lM)\\alpha$, by Proposition \\ref{maximum}, $S_{\\Lambda,\\Lambda'',\\lambda} = F(l-\\frac{j+1}{M})$.\n This proves the lemma.\n\\end{proof}\n\nFrom Lmma \\ref{lemma5.9} and the definition of $F$, it is easy to see that\n\\begin{equation}\\label{eq22}\nG(1-t,m'-j',m'',m''-j'',m'+m''-j)+\\frac{1}{2}(j'+j''-j)=G(t, j',m'', j'',j),\n\\end{equation}\nfor any $t\\in \\br$.\nHence, if the maximum of $G_\\bz$ occurs at\n1, it is equal to \n\\begin{equation}\\label{eq23}\nG(0, m'-j',m'',m''-j'',m'+m''-j)+\\frac{1}{2}(j'+j''-j).\n\\end{equation}\nWe also record the following identity, which is easy to prove from the definition of $F$.\n\\begin{equation}\\label{neq23}\nF (0,m'-j',m'', m''-j'',m'+m''-j)+\\frac{1}{2}(j'+j''-j)=F(0,j',m'', j'',j).\n\\end{equation}\n\nAs a corollary of Proposition \\ref{maximum} and Lemma \\ref{lemma5.9}, we get the following `Non-Cancellation Lemma'.\n\\begin{corollary}\\label{corcancel} Let $\\Lambda, \\Lambda',\\Lambda''$ be as in Proposition \\ref{maximum} and let\n\\begin{align*}\n\\mu^{\\Lambda',\\Lambda''}_{\\Lambda}&:=\\max\\left\\{ S_{\\Lambda,\\Lambda'',\\lambda}:\\;\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}\\,\\,\\,\\text{and}\\,\\,\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=1\\right\\},\\\\\n\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}&:=\\max\\left\\{ S_{\\Lambda,\\Lambda'',\\lambda} :\\;\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}\\,\\,\\,\\text{and}\\,\\,\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=\n-1\\right\\}. \n\\end{align*}\nAssume that ${\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}=\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}$. Then,\n\\[{\\mu}^{\\Lambda'',\\Lambda'}_{\\Lambda}\\neq\\bar{\\mu}^{\\Lambda'',\\Lambda'}_{\\Lambda}.\\]\n\\end{corollary}\n\\begin{proof}\nWe proceed in two cases:\n\n{\\em Case I.} Suppose the maximum $\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}$ \noccurs when $\\pi(\\lambda)=\\Lambda'-(\\frac{1}{2}\\left(j+j'+j''\\right)+1)\\alpha$ (cf. Lemma \\ref{lemma5.9}).\nThis means that the $\\delta$-maximal weights of $L(\\Lambda')$ through\n$\\Lambda'-(\\frac{1}{2}\\left(j+j'+j''\\right)+1)\\alpha$ and through\n$\\Lambda'+\\frac{1}{2}\\left(j-j'-j''\\right)\\alpha$ have the same $\\delta$\ncoordinate (cf. Proposition \\ref{maximum}). By (next) Lemma \\ref{lemma5.11}, \nwe know that this occurs if and only if \none of the following two conditions are satisfied:\n\n(1) $\\left|\\frac{1}{2}\\left(j-j''\\right)\\right|\\leq\\frac{j'}{2}$\nand $\\frac{1}{2}\\left(j+j''\\right)+1\\leq\\frac{j'}{2}$, or \n\n(2) $\\frac{1}{2}\\left(j+j''\\right)+1=\\frac{1}{2}\\left(j-j''\\right)$.\n\nThe latter is clearly impossible, while the former condition is fulfilled\nprecisely when $\\frac{1}{2}\\left(j+j''\\right)+1\\leq\\frac{j'}{2}$.\n\n So, for the equality ${\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}=\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}$ in this case,\nthe neccesary and sufficient condition is: \n\\beqn \\label{e5.10.3} \\frac{1}{2}\\left(j+j''\\right)+1\\leq\\frac{j'}{2}.\n\\eeqn\n\n{\\em Case II.} Suppose the maximum $\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}$ \noccurs when $\\pi(\\lambda )= \\Lambda'-(\\frac{1}{2}\\left(j+j'+j''\\right)+1-M)\\alpha$.\nThen, by the identities \\eqref{eq23} and \\eqref{neq23}, we get \n\\beqn\\label{eq24} G (0,m'-j',m'', m''-j'',m'+m''-j)=F (0,m'-j',m'', m''-j'',m'+m''-j).\n\\eeqn\nSo, from the case I, we get in this case II, \n${\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}=\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}$ if and only if\n\\beqn\\label{eq25}\n\\frac{1}{2}\\bigl((m'+m''-j)+(m''-j'')\\bigr)+1\\leq\\frac{1}{2}(m'-j').\n\\eeqn\nSo, if either of the inequalities \\eqref{e5.10.3} or \\eqref{eq25} is satisfied, then none of them can be satisfied \nfor the triple $(\\Lambda, \\Lambda',\\Lambda'')$ replaced by $(\\Lambda, \\Lambda'',\\Lambda')$. This proves the corollary.\n\\end{proof}\n\\begin{lem}\\label{lemma5.11}\nSuppose $\\Lambda'-(\\frac{1}{2}\\left(j+j'+j''\\right)+1)\\alpha+n_1 \\delta$ and $\\Lambda'+\\frac{1}{2}\\left(j-j'-j''\\right)\\alpha+ n_2 \\delta$ are $\\delta$-maximal weights of $L(\\Lambda')$. Then $n_1 = n_2$ if and only if \n\\[\n\\left|\\frac{1}{2}\\left(j-j''\\right)\\right|\\leq\\frac{j'}{2} \\quad \\text{and}\\,\\,\\,\\,\n\\frac{1}{2}\\left(j+j''\\right)+1\\leq\\frac{j'}{2},\n\\]\nor $\\frac{1}{2}\\left(j+j''\\right)+1=\\frac{1}{2}\\left(j-j''\\right)$.\n\\end{lem}\n\\begin{proof}\nFix an integer $n$ and consider the set $P_n = \\{ \\nu \\in P(\\Lambda') \\, : \\, \\Lambda' - \\nu = k\\alpha + n \\delta,\\, k \\in \\bz \\}$. We give a description of $P_n \\cap P^o(\\Lambda')$. Clearly, $P_n = \\{\\lambda, \\lambda -\\alpha, \\dots, \\lambda - \\langle\\lambda,\\alpha^\\vee\\rangle\\alpha \\}$ for some $\\lambda=\\lambda_n$ and that this $\\lambda$ is uniquely determined by $n$ (cf. [K$_3$, Exercise 2.3.E.2]). Suppose that some $\\mu \\in P_n$ is not $\\delta$-maximal, then none of $\\{\\mu, \\dots, \\mu - \\langle\\mu, \\alpha^\\vee\\rangle\\alpha \\}$ are $\\delta$-maximal, since\nif $\\mu+k\\delta\\in P(\\Lambda')$, then the whole string $\\{\\mu +k\\delta, \\dots, \\mu +k\\delta- \\langle\\mu, \\alpha^\\vee\\rangle\\alpha \\}\n\\subset P(\\Lambda')$. In particular, if $\\lambda - \\alpha$ is $\\delta$-maximal, then so is $\\lambda$. Hence, $\\mathfrak{g}_{\\delta-\\alpha}L(\\Lambda')_\\lambda = 0$ and $\\mathfrak{g}_{\\alpha}L(\\Lambda')_\\lambda = 0$. Therefore, $\\lambda$ is the highest weight $\\Lambda'$. Thus, $P_n \\cap P^o(\\Lambda')$ is either empty, or $\\lambda=\\Lambda'$ (in the case that $n = 0$), or the set $\\{\\lambda,s_1\\lambda\\}$. From this and Corollary \\ref{cor5.1} the lemma follows easily.\n\\end{proof}\n\n\\section{Saturation factor for the $A_{1}^{(1)}$ Case}\n\n We assume that $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$ in this section.\n\\begin{defn}\nLet $\\Lambda'\\in P_{+}^{(m')}, \\Lambda''\\in P_{+}^{(m'')}$ and $\\Lambda\\in P_{+}^{(m'+m'')}$. Then, we call $L(\\Lambda+n\\delta)$\nthe \\textit{$\\delta$-maximal component of $L(\\Lambda')\\otimes L(\\Lambda'')$\nthrough $\\Lambda$} if $L(\\Lambda+n\\delta)$ is a submodule of $L(\\Lambda')\\otimes L(\\Lambda'')$\nbut $L(\\Lambda+m\\delta)$ is not a component for any $m>n$.\\end{defn}\n\\begin{thm} \\label{thm6.2} Let $\\Lambda', \\Lambda'', \\Lambda$ be as in Proposition \\ref{maximum} . Then, \n$L(\\Lambda+n\\delta)$ is a $\\delta$-maximal component of $L(\\Lambda')\\otimes L(\\Lambda'')$\nif $n=\\min(n_{1},n_{2})$, where $n_{1}$ is such that $\\Lambda-\\Lambda''+n_{1}\\delta\\in P^o(\\Lambda')$\nand $n_{2}$ is such that $\\Lambda-\\Lambda'+n_{2}\\delta\\in P^o(\\Lambda'')$.\\end{thm}\n\\begin{proof} This follows immediately by combining Propositions \\ref{tensor}, \\ref{maximum} and Lemma \\ref{lemma5.1'}.\n\\end{proof}\n\\begin{lem} Fix a positive integer $N$. \nLet $\\Lambda\\in \\bar{P}_+$ and let $\\lambda\\in\\Lambda+Q$, where $Q$ is the root lattice $\\mathbb{Z}\\alpha\\oplus\\mathbb{Z}\\delta$ of\n$\\widehat{\\mathfrak{sl}_{2}}$. \nThen, $N\\lambda\\in P^o(N\\Lambda)$ if\nand only if $\\lambda\\in P^o(\\Lambda)$. \n\\end{lem}\n\\begin{proof}\nThe validity of the lemma is clear for $\\lambda\\in P^o(\\Lambda)_+$ from Corollary \\ref{cor5.1}. But since\n$ P^o(\\Lambda)=W\\cdot ( P^o(\\Lambda)_+)$,\n and the action of $W$ on $\\fh^*$ is linear, the lemma follows for any $\\lambda \\in P^o(\\Lambda)$.\\end{proof}\n\\begin{cor}\\label{cor6.4}\nLet $d_o\\in\\mathbb{Z}_{>1}$. Let $\\Lambda$, $\\Lambda'$, $\\Lambda'' \\in P_+$\nbe such that $\\Lambda-\\Lambda'-\\Lambda''\\in Q$ and \n $L(N\\Lambda)$ is a submodule of $L(N\\Lambda')\\otimes L(N\\Lambda'')$, for some $N\\in \\bz_{>0}$. \nThen, $L(d_o\\Lambda)$ is a submodule of $L(d_o\\Lambda')\\otimes L(d_o\\Lambda'')$.\n\nSuch a $d_o$ is called a {\\em saturation factor}. \n\\end{cor}\n\\begin{proof} If $\\Lambda'(c)=0$ or $\\Lambda''(c)=0$, then \n$$L(N\\Lambda')\\otimes L(N\\Lambda'')\\simeq L(N(\\Lambda'+\\Lambda'')),$$\nfor any $N\\geq 1$. Thus, the corollary is clearly true in this case. So, let us assume that both of \n$\\Lambda'(c)>0$ and $\\Lambda''(c)>0$. \nLet $L(N\\Lambda+n\\delta)$ be the $\\delta$-maximal component of $L(N\\Lambda')\\otimes L(N\\Lambda'')$\nthrough $L(N\\Lambda)$, for some $n\\geq 0$. For any $\\Psi\\in P_+$, let $\\bar{\\Psi}\\in \\bar{P}_+$ be the projection $\\pi(\\Psi)$ defined just before Lemma \\ref{lemma5.1}. Applying Theorem \\ref{thm6.2} to $\\bar{\\Lambda}', \\bar{\\Lambda}'', \\bar{\\Lambda}$, and observing that \n\\beqn \\label{eq6.4.0} L(\\bar{\\Psi}+k\\delta)\\simeq L(\\bar{\\Psi})\\otimes L(k\\delta)\\eeqn\nand $L(k\\delta)$ is one dimensional, we get that there is a $\\delta$-maximal component $L(\\Lambda+\\tilde{n}\\delta)$ of \n$L(\\Lambda')\\otimes L(\\Lambda'')$ through $L(\\Lambda)$, for some (unique) $\\tilde{n}\\in \\bz$. \n\nAgain applying Theorem \\ref{thm6.2} to $N\\bar{\\Lambda}', N\\bar{\\Lambda}'', N\\bar{\\Lambda}$, and observing \n(using Corollary \\ref{cor5.1}) that \n\\beqn \\label{eq6.4.1} P^o(N\\bar{\\Psi})\\supset NP^o(\\bar{\\Psi}),\\eeqn\nwe get that $L(N\\Lambda+N\\tilde{n}\\delta)$ is the $\\delta$-maximal component of \n$L(N\\Lambda')\\otimes L(N\\Lambda'')$ through $L(N\\Lambda)$. Thus, $n=N\\tilde{n}$. In particular,\n\\beqn\\label{e6.3} \\tilde{n}\\geq 0.\\eeqn\n Let \n\\beqn\\label{e6.4}\n\\sum_{\\lambda\\in T_{\\bar{\\Lambda}}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\bar{\\Lambda},\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\bar{\\Lambda},\\Lambda'',\\lambda}\\delta}=\\sum_{k\\in\\mathbb{Z}_+}c_{k}e^{(\\Lambda(d)+\\tilde{n}-k)\\delta},\n\\eeqn\nfor some $c_k\\in \\bz_+$ with $c_0$ nonzero. \nBy Proposition \\ref{tensor}, this is the character of a unitarizable \nVirasoro representation with each irreducible component having the same nonzero central charge. Thus, by Lemma \n\\ref{virasoro}, for any $k>1$, we get \n$c_{k}\\neq0$. \n\nBy the above argument, $L(d_o\\Lambda+d_o\\tilde{n}\\delta)$ is the $\\delta$-maximal component of \n$L(d_o\\Lambda')\\otimes L(d_o\\Lambda'')$ through $L(d_o\\Lambda)$. If $\\tilde{n}=0$, we get that \n $$L(d_o\\Lambda) \\subset L(d_o\\Lambda')\\otimes L(d_o\\Lambda'').$$\nIf $\\tilde{n}>0$, then $d_o\\tilde{n}$ being $>1$, by the analogue of \\eqref{e6.4} for \n$d_o\\Lambda', d_o\\Lambda''$ and $d_o\\Lambda$,\n$L(d_o\\Lambda) \\subset L(d_o\\Lambda')\\otimes L(d_o\\Lambda'').$ (Here we have used that \n$L_0=-d+p$ on any $\\fg$-isotypical component of $L(\\Lambda')\\otimes L(\\Lambda'')$ with highest weight in $\\Lambda+\\bz \\delta$, for \na number $p$ depending only upon $\\bar{\\Lambda}, \\Lambda'$ and $\\Lambda''$, cf. [KR, Identity 10.25 on page 116].)\nThis proves the corollary.\n\\end{proof}\n\n\\begin{rem}\nWe note that $L(2\\Lambda_{0}-\\delta)$ is not a component of $L(\\Lambda_{0})\\otimes L(\\Lambda_{0})$\n(cf. [Kac, Exercise 12.16]).\nBut, of course, $L(2\\Lambda_{0})$ is a $\\delta$-maximal component. By the identity \\eqref{e6.4}, we know that $L(2d_o\\Lambda_{0}-d_o\\delta)$\nmust be a component of $L(d_o\\Lambda_{0})\\otimes L(d_o\\Lambda_{0})$, for any $d_o>1$. \nSo $d_o$ can not be taken to be $1$ in Corollary \\ref{cor6.4}.\n\\end{rem}\n\n\\section{A Conjecture}\n\nIn this section, $G$ is any symmetrizable Kac-Moody group. \nWe recall the following definition of the deformed product due to Belkale-Kumar [BK]. (Even though they gave the definition in the finite case, the same definition works in the symmetrizable Kac-Moody case, though with only one parameter.)\n\\begin{definition}\nLet $P$ be any standard parabolic subgroup of $G$. Recall from Section 2 that $\\{\\epsilon^w_P\\}_{w\\in W^P}$ is a\nbasis of the singular cohomology $H^*(X_P, \\bz)$. \nWrite the standard cup product in $H^*(X_P, \\Bbb Z)$ in this basis as follows:\n\\begin{equation}\\label{constants}\n\\epsilon_P^u\\cdot \\epsilon_P^v=\\sum_{w\\in W^P} n^w_{u,v}\\epsilon_P^w,\\,\\,\\,\\text{for some (unique)}\\, n^w_{u,v}\\in \\bz.\n\\end{equation}\nIntroduce the indeterminate ${\\tau}$ and define a deformed cup product $\\odot$\nas follows:\n\\begin{equation}\\label{10n}\n\\epsilon_P^u \\odot \\epsilon_P^v=\n\\sum_{w\\in W^P} \n{\\tau}^{(u^{-1}\\rho+v^{-1}\\rho -w^{-1}\\rho -\\rho)(x_P)}\nn^w_{u,v} \\epsilon_P^w,\n\\end{equation}\nwhere $x_P:=\\sum_{\\alpha_i\\in\n\\Delta\\setminus\\Delta(P)}\\,x_i$, $\\Delta(P)$ is the set of simple roots of the Levi $L$ of $P$ and, as in Section 2, $\\Delta$ is \nthe set of simple roots of $G$. \n\nThe following lemma is a generalization of the corresponding result in the finite case (cf. [BK, Proposition 17]). \n\\begin{proposition} (a) The product $\\odot$ is associative and clearly commutative. \n\n(b) \nWhenever $n^w_{u,v}$ is nonzero,\nthe exponent of $\\tau$ in the above is a nonnegative integer. \n\\end{proposition}\n\\begin{proof} The proof of the associativity of $\\odot$ is identical to the proof given in [BK, Proof of Proposition 17 (b)]. \n\n(b) The proof of this part follows the proof of [BK, Theorem 43]. Consider the\ndecreasing filtration $\\ca =\\{\\ca_m\\}_{m\\geq 0}$ of\n$H^*(X_P, \\bc )$ defined as follows:\n \\[\n\\ca_m := \\bigoplus_{w\\in W^P:(\\rho-w^{-1}\\rho) (x_P)\\geq m} \\bc \\epsilon_P^w.\n \\]\nA priori $\\{\\ca_m\\}_{m\\geq 0}$ may not be a multiplicative filtration. \n\nWe next introduce another filtration $\\{\\bar\\cf_m\\}_{m\\geq 0}$ of\n$H^*(X_P,\\bc )$ in terms of the Lie algebra cohomology. Recall\nthat $H^*(X_P,\\bc )$ can be identified canonically with the Lie\nalgebra cohomology $H^*(\\fg ,\\fl )$, where $\\fl$ is the Lie algebra of the Levi subgroup\n$L$ of $P$ (cf. [K$_2$, Theorem 1.6]). The underlying cochain\ncomplex $C\\u. =C\\u. (\\fg ,\\fl )$ for $H^*(\\fg ,\\fl )$ can be\nwritten as\n \\[\nC\\u. := [\\wed\\u. (\\fg \/\\fl )^*]^{\\fl} = \\Hom_{\\fl} \\bigl( \\wed\\u.\n(\\fu_P)\\otimes\\wed\\u. (\\fu_P^-), \\bc \\bigr),\n \\]\nwhere $\\fu_P$ (resp. $\\fu_P^-$) is the nil-radical of the Lie algebra of $P$ (resp. the opposite parabolic subgroup $P^-$). \nDefine a decreasing multiplicative filtration $\\cf =\\{\\cf_m\\}_{m\\geq 0}$ of the\ncochain\ncomplex $C\\u.$ by subcomplexes:\n \\[\n\\cf_m := \\Hom_{\\fl}\\Biggl( \\frac{\\wed\\u. (\\fu_P)\\otimes\\wed\\u.\n(\\fu^-_P)}{\\bigoplus_{s+t\\leq m-1}\n\\wed\\u._{(s)}(\\fu_P)\\otimes\\wed\\u._{(t)}(\\fu^-_P)}, \\bc\\Biggr) ,\n \\]\nwhere $\\wed\\u._{(s)}(\\fu_P)$ (resp. $\\wed\\u._{(s)}(\\fu^-_P)$) denotes the\nsubspace of $\\wed\\u. (\\fu_P)$ (resp. $\\wed\\u. (\\fu^-_P)$) spanned by the\n$\\fh$-weight vectors of weight $\\beta$ with {\\em $P$-relative height}\n$$\\text{ht}_P (\\beta ) := \\mid\\beta (x_P)\\mid=s.$$\n\nNow, define the filtration $\\bar\\cf =\\{ \\bar\\cf_m\\}_{m\\geq 0}$ of\n$H^*(\\fg ,\\fl )\\simeq H^*(X_P, \\bc)$ by\n \\[\n\\bar\\cf_m := \\text{Image of } H^*(\\cf_m) \\to H^*(C\\u. ).\n \\]\nThe filtration $\\cf$ of $C\\u.$ gives rise to the cohomology\nspectral sequence $\\{ E_r\\}_{r\\geq 1}$ converging to $H^*(C\\u.\n)=H^*(X_P,\\bc )$. By [K$_3$, Proof of Proposition 3.2.11], for any $m\\geq\n0$,\n \\[\nE^m_1 = \\bigoplus_{s+t=m} [H\\u._{(s)}(\\fu_P)\\otimes\nH\\u._{(t)}(\\fu^-_P)]^{\\fl},\n \\]\nwhere $H\\u._{(s)}(\\fu_P)$ denotes the cohomology of the subcomplex\n$(\\wed\\u._{(s)}(\\fu_P))^*$ of the standard cochain complex\n$\\wed\\u. (\\fu_P)^*$ associated to the Lie algebra $\\fu_P$ and\nsimilarly for $H\\u._{(t)}(\\fu^-_P)$. Moreover, by loc. cit., the\nspectral sequence degenerates at the $E_1$ term, i.e.,\n \\begin{equation} \\label{eqn10.1}\nE_1^m = E^m_{\\infty}.\n \\end{equation}\nFurther, by the definition of $P$-relative height,\n \\[\n[H\\u._{(s)}(\\fu_P) \\otimes H\\u._{(t)}(\\fu^-_P)]^{\\fl} \\neq 0 \\Rightarrow\ns=t.\n \\]\nThus,\n \\begin{align*}\nE^m_1 &= 0, \\qquad\\quad\\text{unless $m$ is even and}\\\\\nE_1^{2m} &= [H\\u._{(m)}(\\fu_P) \\otimes H\\u._{(m)}(\\fu^-_P)]^{\\fl} .\n \\end{align*}\nIn particular, from (\\ref{eqn10.1}) and the general properties of spectral sequences\n(cf. [K$_3$, Theorem E.9]), we have a canonical algebra isomorphism:\n \\begin{equation} \\label{eqn10.2}\n\\gr (\\bar\\cf ) \\simeq \\bigoplus_{m\\geq 0} \\bigl[ H\\u._{(m)}(\\fu_P) \\otimes\nH\\u._{(m)}(\\fu^-_P)\\bigr]^{\\fl} ,\n \\end{equation}\nwhere $\\bigl[ H\\u._{(m)}(\\fu_P) \\otimes H\\u._{(m)}(\\fu^-_P)\\bigr]^{\\fl}$\nsits inside $\\gr(\\bar\\cf )$ precisely as the homogeneous part of degree\n$2m$; homogeneous parts of $\\gr(\\bar\\cf )$ of odd degree being zero.\n\nFinally, we claim that, for any $m\\geq 0$,\n \\begin{equation} \\label{eqn10.3}\n\\ca_m = \\bar\\cf_{2m} :\n \\end{equation}\n\nFollowing Kumar [K$_1$], take the d-$\\partial$ harmonic representative\n $\\hat{s}^w$ in $C\\u.$ for the cohomology class $\\epsilon_P^w$. An\nexplicit expression is given in [K$_1$, Proposition 3.17]. From this explicit expression, we easily see that\n \\begin{equation} \\label{eqn10.4}\n\\ca_m \\subset \\bar\\cf_{2m}, \\,\\,\\,\\text{for all}\\,\\, m\\geq 0.\n \\end{equation}\nMoreover, from the definition of $\\ca$, for any $m\\geq 0$,\n \\[\n\\dim \\frac{\\ca_m}{\\ca_{m+1}} = \\# \\bigl\\{ w\\in W^P : (\\rho\n-w^{-1}\\rho) (x_P)=m\\bigr\\} .\n \\]\nAlso, by the isomorphism (\\ref{eqn10.2}) and [K$_3$, Theorem 3.2.7],\n $$\n\\dim \\frac{\\bar\\cf_{2m}}{\\bar\\cf_{2m+1}} = \\# \\bigl\\{ w\\in W^P: (\\rho-w^{-1}\\rho)\n(x_P) = m\\bigr\\}. $$\n\nThus,\n \\beqn \\label{eq77}\n\\dim \\frac{\\ca_m}{\\ca_{m+1}} = \\dim \\frac{\\bar\\cf_{2m}}{\\bar\\cf_{2m+1}} .\n \\eeqn\nOf course, \n\\beqn\\label{eq78} \\ca_0 = \\bar\\cf_0.\n\\eeqn\nThus, combining the equations \\eqref{eqn10.4}, \\eqref{eq77} and \\eqref{eq78}, we get \\eqref{eqn10.3}. \nIt is easy to see that the filtration $\\{\\bar{\\cf}_{2m}\\}_{m\\geq 0}$ is multiplicative and hence so is\n$\\{\\ca_m \\}_{m\\geq 0}$. This proves the (b) part of the proposition.\n \\end{proof}\n\nThe cohomology of $X_P$\n obtained by setting ${\\tau}=0$ in\n$(H^*(X_P, \\Bbb Z)\\otimes\\Bbb{Z}[{\\tau}],\\odot)$ is denoted by\n$(H^*(X_P, \\Bbb Z),\\odot_0)$. Thus, as a $\\Bbb Z$-module, it is the same as the singular cohomology\n $H^*(X_P, \\Bbb Z)$ and under the product $\\odot_0$ it is associative (and commutative).\n\\end{definition}\n\nThe following conjecture is a generalization of the corresponding result in the finite case due to Belkale-Kumar [BK, Theorem 22].\n\\begin{conjecture} \\label{conj1}\nLet $G$ be any indecomposable symmetrizable Kac-Moody group (i.e., its generalized Cartan matrix is indecomposable, cf. [K$_3$, $\\S$ 1.1]) \nand let $(\\lambda_1, \\dots, \\lambda_s, \\mu)\\in P_+^{s+1}$. Assume further that none of $\\lambda_j$ is $W$-invariant and $\\mu-\\sum_{j=1}^s \\lambda_j\\in Q$, where $Q$ is the root lattice of $G$. \nThen, the following are equivalent:\n\n(a) $(\\lambda_1, \\dots, \\lambda_s, \\mu)\\in \\Gamma_s$.\n\n(b) For every standard maximal parabolic subgroup $P$ in $G$ and every choice of\n$s+1$-tuples $(w_1, \\dots, w_s, v)\\in (W^P)^{s+1}$ such that $\\epsilon_P^v$ occurs with coefficient $1$ in \nthe deformed product\n$$\\epsilon_P^{w_1}\\odot_0\\, \\cdots \\,\\odot_0 \\epsilon_P^{w_s}\n\\in \\bigl(H^*(X_P,\\Bbb{Z}), \\odot_0\\bigr),$$\n the following inequality holds:\n \\[\\label{eqn29}\n\\bigl(\\sum_{j=1}^s \\lambda_j(w_jx_{P})\\bigr)-\\mu(vx_P)\\geq 0, \\tag{$I^P_{(w_1,\\dots, w_s,v)}$}\n\\]\nwhere $\\alpha_{i_P}$ is the (unique) simple root in $\\Delta\\setminus \\Delta (P)$\nand $x_P:=x_{i_P}$.\n\\end{conjecture}\n\\begin{remark} (a) By Theorem \\ref{thm1}, the above inequalities $I^P_{(w_1,\\dots, w_s,v)}$ are indeed satisfied for any \n$(\\lambda_1, \\dots, \\lambda_s, \\mu)\\in \\Gamma_s$. \n\n(b) If $G$ is an affine Kac-Moody group, then the condition \nthat $\\lambda\\in P_+$ is $W$-invariant is equivalent to the condition that $\\lambda(c)=0$.\n\\end{remark}\n\n\\begin{theorem} \\label{thm7.5} Let $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$. \nLet $\\lambda,\\mu,\\nu \\in P_+$ be such that $\\lambda+\\mu-\\nu\\in Q$ and both of $\\lambda(c)$ and $\\mu(c)$ are nonzero. Then, the following are equivalent:\n\n(a) $(\\lambda,\\mu,\\nu) \\in \\Gamma_2$.\n\n(b) The following set of inequalities is satisfied for all $w\\in W$ and $i=0,1$.\n\\begin{align*}\n\\lambda(x_i)+\\mu(wx_i)-\\nu(wx_i) &\\geq 0, \\,\\,\\,\\text{and}\\\\\n\\lambda(wx_i)+\\mu(x_i)-\\nu(wx_i) &\\geq 0.\n\\end{align*}\nIn particular, Conjecture \\ref{conj1} is true for $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$ and $s=2$.\n\\end{theorem}\n\\begin{proof}\nBy Lemma \\ref{lemma5.1}, there exist (unique) $n_1, n_2\\in \\bz$ such that \n\\[\n\\nu - \\mu + n_1 \\delta \\in P^o(\\lambda), \\,\\,\\,\\,\\text{and}\\,\\,\\,\n\\nu - \\lambda +n_2 \\delta \\in P^o(\\mu).\n\\]\nLet $n:=$ min $(n_1,n_2)$. \nBy our description of the $\\delta$-maximal components as in Theorem \\ref{thm6.2} applied to $\\bar{\\lambda},\\bar{\\mu}, \\bar{\\nu}$\nand using the identity \\eqref{eq6.4.0}, we see that \n$L(\\nu+n\\delta)$ is a $\\delta$-maximal component of $L(\\lambda)\\otimes L(\\mu)$. Thus, by the equation \\eqref{eq6.4.1}, for any $N\\geq 1$, \n$L(N\\nu+Nn\\delta)$ is a $\\delta$-maximal component of $L(N\\lambda)\\otimes L(N\\mu)$. In particular, by Proposition \\ref{tensor} and Lemma \n\\ref{virasoro}, \n\\beqn\\label{eq7.4.1} L(N\\nu) \\subset L(N\\lambda)\\otimes L(N\\mu) \\,\\,\\,\\text{ for some}\\,\\, N>1\\,\\,\\,\\text{ if and only if}\\,\\, n\\geq 0.\n\\eeqn\nBy [Kac, Proposition 12.5 (a)], if a weight $\\gamma+k\\delta\\in P(\\lambda)$ (for some $k\\in \\bz_+$), then $\\gamma\\in P(\\lambda)$. Thus, \n\\beqn\\label{eq7.4.2} n\\geq 0 \\,\\,\\,\\text{ if and only if}\\,\\, \\nu\\in \\bigl(P(\\lambda)+\\mu\\bigr)\\cap \\bigl(P(\\mu)+\\lambda\\bigr).\n\\eeqn\nWe next show that \n\\beqn\\label{eq7.4.3} P(\\lambda)=(\\lambda + Q)\\cap C_\\lambda,\n\\eeqn\nwhere $C_\\lambda:=\\{\\gamma\\in \\fh^*: \\lambda(x_i)-\\gamma(wx_i)\\geq 0 \\,\\,\\,\\text{for all}\\,\\, w\\in W \\,\\,\\,\\text{and all} \\,\\, x_i\\}$.\nClearly, $$\n P(\\lambda)\\subset(\\lambda + Q)\\cap C_\\lambda.$$\nSince $\\lambda+Q$ and $C_\\lambda$ are $W$-stable, and $\\lambda+Q$ is contained in the Tits cone (by [K$_3$, Exercise 13.1.E.8(a)]),\n$(\\lambda+Q)\\cap C_\\lambda= W\\cdot \\bigl((\\lambda+Q)\\cap C_\\lambda\\cap P_+\\bigr)$.\n\nConversely,\ntake $\\gamma\\in (\\lambda + Q)\\cap C_\\lambda \\cap P_+$. Then, \n$(\\lambda-\\gamma) (x_i)\\geq 0$ and $(\\lambda-\\gamma) (c)= 0$ and hence $\\lambda-\\gamma \\in \\oplus_i\\,\\bz_+\\alpha_i$, i.e., \n$\\lambda \\geq \\gamma$. Thus, by [Kac, Proposition 12.5(a)], $\\gamma \\in P(\\lambda)$. This proves \\eqref{eq7.4.3}. \nNow, combining \\eqref{eq7.4.1}, \\eqref{eq7.4.2} and \\eqref{eq7.4.3}, we get \n$ L(N\\nu) \\subset L(N\\lambda)\\otimes L(N\\mu)$ for some $N>1$ if and only if for all \n $w\\in W$ and $i=0,1$,\n\\[\n\\lambda(x_i)-(\\nu-\\mu)(wx_i)\\geq 0, \\,\\,\\,\\text{and}\\,\\,\\,\n\\mu(x_i)-(\\nu - \\lambda)(wx_i) \\geq 0.\\]\nThis proves the equivalence of (a) and (b) in the theorem. \n\nTo prove the `In particular' statement of the theorem, let $P_0$ (resp. $P_1$) be the maximal parabolic subgroup of $G=\\widehat{\\SL_{2}}$\nwith $\\Delta(P_0)=\\{\\alpha_1\\}$ (resp. $\\Delta(P_1)=\\{\\alpha_0\\}$). For any $n\\geq 0$, let \n$$w_n:=\\dots s_0s_1s_0 \\,\\,(\\text{$n$-factors})\\,\\,\\,\\text{and}\\,\\,\\,v_n:=\\dots s_1s_0s_1 \\,\\,(\\text{$n$-factors}).$$\nThen, by [K$_3$, Exercise 11.3.E.4], $H^*(G\/P_0)$ has a $\\bz$-basis $\\{\\epsilon^{n}_{P_0}\\}_{n\\geq 0}$, where \n$\\epsilon^{n}_{P_0}:=\\epsilon^{w_n}_{P_0}$. Moreover,\n$$\\epsilon^{n}_{P_0}\\cdot \\epsilon^{m}_{P_0}=\\binom {n+m}{n}\\epsilon^{n+m}_{P_0}.$$\nIn particular, $\\epsilon^{n+m}_{P_0}$ appears with coefficient one in $\\epsilon^{n}_{P_0}\\cdot \\epsilon^{m}_{P_0}$ if and only if \nat least one of $n$ or $m$ is $0$. \n\nBy using the diagram automorphism of $\\widehat{\\SL_{2}}$, one similarly gets that $H^*(G\/P_1)$ has a $\\bz$-basis $\\{\\epsilon^{n}_{P_1}\\}_{n\\geq 0}$, where \n$\\epsilon^{n}_{P_1}:=\\epsilon^{v_n}_{P_1}$, with the product given by\n$$\\epsilon^{n}_{P_1}\\cdot \\epsilon^{m}_{P_1}=\\binom {n+m}{n}\\epsilon^{n+m}_{P_1}.$$\nMoreover, from the definition of the deformed product $\\odot_0$, clearly \n$$\\epsilon^{0}_{P_0}\\odot_0 \\epsilon^{m}_{P_0}= \\epsilon^{0}_{P_0}\\cdot \\epsilon^{m}_{P_0},$$\nand similarly for $P_1$. From this the `In particular' statement of the theorem follows.\n\\end{proof}\n\\begin{remark} (1) It is easy to see that if $\\lambda=m\\delta$ for some $m\\in \\bz$, then the equivalence of (a) and (b) in the above theorem breaks down.\n\n(2) Though we have proved Conjecture \\ref{conj1} for $\\widehat{\\SL_{2}}$ only for $s=2$, it is quite likely that a similar proof will prove it for \nany $s$ (for $\\widehat{\\SL_{2}}$).\n\\end{remark}\n\n\\section{The $A_2^{(2)}$ case}\nBy a method similar to that of $A_1^{(1)}$, we handle the $A_2^{(2)}$ case, with minor modifications where necessary. \nWrite $\\mathfrak{h}=\\mathbb{C}c\\oplus\\mathbb{C}\\alpha^{\\vee}\\oplus\\mathbb{C}d$\nand $\\mathfrak{h}^{*}=\\mathbb{C}\\omega_{0}\\oplus\\mathbb{C}\\alpha\\oplus\\mathbb{C}\\delta$,\nwhere $\\alpha(\\alpha^{\\vee})=2$, $\\delta(d)=1$, $\\omega_{0}(c)=1$,\nand all other values are $0$. Then $(\\mathfrak{h},\\{\\alpha_0:=\\delta-2\\alpha, \\alpha_1:=\\alpha\\}, \\{\\alpha_0^\\vee:=c-\\frac{1}{2}\\alpha^{\\vee},\\alpha_1^\\vee:=\\alpha^{\\vee}\\})$ is a realization of the GCM \n\\[ \\left( \\begin{array}{cc}\n2 & -1\\\\\n-4 & 2 \\end{array} \\right)\\] \nof $A_2^{(2)}$.\nThe fundamental weights are $\\omega_{0}$ and $\\omega_{1}=\\frac{1}{2}\\omega_{0}+\\frac{1}{2}\\alpha$. This easily allows one to compute the dominant $\\delta$-maximal weights. Analogous to Corollary \\ref{cor5.1}, we have the following:\n\\begin{lemma}\\label{domdelmaxweights} Let $\\lambda$ be a dominant integral weight. Then, the dominant $\\delta$-maximal\nweights of $L(\\lambda)$ are the dominant weights of the form \n\\[\nP_+\\cap \\left\\{ \\lambda-j\\alpha,\\,\\lambda+k(2\\alpha-\\delta),\\,\\lambda+\\alpha-\\delta+l(2\\alpha-\\delta)\\,:\\, j,k,l\\in\\mathbb{Z}_{\\geq0}\\right\\} .\n\\]\nMoreover, $P^o(\\lambda)$ is the $W$-orbit of the above.\n\\end{lemma}\n\nAgain, to determine the saturated tensor cone, it is enough to describe the $\\delta$-maximal components. Thus, to determine the $\\delta$-maximal components, by virtue of proposition \\ref{tensor}, we must find the highest $\\delta$-degree term in $\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}$. This computation is done in a somewhat similar manner as in the $A_1^{(1)}$ case, but there are some important modifications.\nFirst, we need to use two different piecewise smooth functions to describe the $\\delta$-maximal weights of $L(\\lambda)$. An upper function $A^+$ interpolates the $\\delta$-maximal weights which are in the $W$-orbit of the dominant weights of the form\n\\[\n\\left\\{ \\lambda-j\\alpha,\\,\\lambda+k(2\\alpha-\\delta)\\,:\\, j,k\\in\\mathbb{Z}_{\\geq0}\\right\\}, \n\\]\nwhile another function $A^-$ interpolates the $\\delta$-maximal weights in the $W$-orbit of the dominant weights of the form\n\\[\n\\left\\{ \\lambda-j\\alpha,\\,\\lambda+\\alpha-\\delta+l(2\\alpha-\\delta)\\,:\\, j,l\\in\\mathbb{Z}_{\\geq0}\\right\\}. \n\\]\n\nNow, all of the arguments made in the $\\widehat{\\mathfrak{sl}_2}$ case must be made for two extensions of $S_{\\Lambda,\\Lambda'',\\lambda}$ to non-integral values, using $A^+$ and $A^-$ respectively. Let $\\Lambda :=m_{0}\\omega_{0}+m_{1}\\omega_{1}$, $\\Lambda' :=m'_{0}\\omega_{0}+m'_{1}\\omega_{1}$, and $\\Lambda'' :=m''_{0}\\omega_{0}+m''_{1}\\omega_{1}$. The following result is an analogue of \nProposition \\ref{maximum} and Lemma \\ref{lemma5.9} for the $A_2^{(2)}$ case.\n\\begin{prop}\\label{maxS} Let $\\Lambda, \\Lambda', \\Lambda''$ be as above. Assume that both of $\\Lambda'(c)$ and $\\Lambda''(c)>0$ and $\\Lambda-\\Lambda'-\\Lambda''\\in Q$, where \n$Q=\\bz \\alpha+\\bz \\delta$ is the root lattice of $A_2^{(2)}$ . \nThen, the maximum $\\mu^{\\Lambda',\\Lambda''}_\\Lambda$ of the set\n\\[ \n\\left\\{ S_{\\Lambda,\\Lambda'',\\lambda}:\\;\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''},\\;\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=1\\right\\}\n\\] occurs when $\\lambda\\equiv\\Lambda'+\\frac{1}{2}\\left(m_{1}-m_{1}'-m_{1}''\\right)\\alpha \\mod\\mathbb{C}\\delta$.\nThe maximum $\\bar{\\mu}^{\\Lambda',\\Lambda''}_\\Lambda$ of the set \n\\[\n\\left\\{ S_{\\Lambda,\\Lambda'',\\lambda}:\\;\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''},\\;\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=-1\\right\\}\n\\]\noccurs when $\\lambda\\equiv\\Lambda'-\\bigl(\\frac{1}{2}(m_{1}'+m_{1}''+m_{1})+1\\bigr)\\alpha \\mod \\mathbb{C}\\delta $\nor when $\\lambda\\equiv\\Lambda'-\\bigl(\\frac{1}{2}(m_{1}'+m_{1}''+m_{1})-2(\\Lambda'(c)+\\Lambda''(c)+1)\\bigr)\\alpha \\mod \\mathbb{C}\\delta $. \n\\end{prop}\n\n\\begin{corollary}\\label{a22cancellation}\nLet $\\Lambda, \\Lambda', \\Lambda''$ be as in Proposition \\ref{maxS}. Assume further that \n$\\Lambda'(c) \\geq 2$, $\\Lambda''(c) \\geq 2$, $m'_1,m''_1 \\neq 1$. Then, if ${\\mu}^{\\Lambda',\\Lambda''}_\\Lambda = \\bar{\\mu}^{\\Lambda',\\Lambda''}_\\Lambda$, we have \n\\[ {\\mu}^{\\Lambda'',\\Lambda'}_\\Lambda \\neq \\bar{\\mu}^{\\Lambda'',\\Lambda'}_\\Lambda.\\]\n\\end{corollary}\n\nThe proof of Corollary \\ref{a22cancellation} requires a description of the situations in which \n ${\\mu}^{\\Lambda',\\Lambda''}_\\Lambda = \\bar{\\mu}^{\\Lambda',\\Lambda''}_\\Lambda$. We reduce these situations to certain cases, and show that in most of these cases, if the roles of $\\Lambda'$ and $\\Lambda''$ are interchanged, then (as in the $\\widehat{\\mathfrak{sl}_2}$ case) the equality does not occur. In the remaining cases, we show that $\\Lambda'(c) < 2$, $\\Lambda''(c) < 2$, $m'_1 =1$, or $m''_1 = 1$.\n\n\\begin{thm} Let $\\Lambda,\\Lambda',\\Lambda''$ be as in Proposition \\ref{maxS}. Then, \n$L(\\Lambda+n\\delta)$ is a $\\delta$-maximal component of $L(\\Lambda')\\otimes L(\\Lambda'')$\nif $n=\\min(n_{1},n_{2})$, where $n_{1}$ is such that $\\Lambda-\\Lambda''+n_{1}\\delta\\in P^o(\\Lambda')$ and\n$n_{2}$ is such that $\\Lambda-\\Lambda'+n_{2}\\delta\\in P^o(\\Lambda'')$.\n\\end{thm}\n\\begin{lem} Fix a positive integer $N$. \nLet $\\Lambda\\in \\bar{P}_+$ and let $\\lambda\\in\\Lambda+Q$. \nThen, $N\\lambda\\in P^o(N\\Lambda)$ if\nand only if $\\lambda\\in P^o(\\Lambda)$. \n\\end{lem}\nCombining the above results, we get a description of $\\Gamma_2$, which is identical to that of $\\widehat{\\mathfrak{sl}_2}$ \n(cf. Theorem \\ref{thm7.5}).\n\\begin{theorem} \\label{thm8.6} Let $\\mathfrak{g}=A_2^{(2)}$. \nLet $\\lambda,\\mu,\\nu \\in P_+$ be such that $\\lambda+\\mu-\\nu\\in Q$ and both of $\\lambda(c)$ and $\\mu(c)$ are nonzero. Then, the following are equivalent:\n\n(a) $(\\lambda,\\mu,\\nu) \\in \\Gamma_2$.\n\n(b) The following set of inequalities is satisfied for all $w\\in W$ and $i=0,1$.\n\\begin{align*}\n\\lambda(x_i)+\\mu(wx_i)-\\nu(wx_i) &\\geq 0, \\,\\,\\,\\text{and}\\\\\n\\lambda(wx_i)+\\mu(x_i)-\\nu(wx_i) &\\geq 0.\n\\end{align*}\nIn particular, Conjecture \\ref{conj1} is true for this case as well for $s=2$. \n\\end{theorem}\n The `In particular' statement of the above theorem follows by using the description of the cup product in the cohomology of the full flag variety of $A_2^{(2)}$ given by Kitchloo [Ki]. \n\nIt is clear that if the level of $L(\\Lambda')$ or $L(\\Lambda'')$ is zero, then the tensor product has a single component. Thus, it is already saturated. Assume now that the levels of both of $L(\\Lambda')$ and $L(\\Lambda'')$ are $>0$. Then, since there are representations of level $\\frac{1}{2}$, the conditions of Corollary \\ref{a22cancellation} are satisfied for any $N\\Lambda$, $N\\Lambda'$, $N\\Lambda''$ with $\\Lambda-\\Lambda'-\\Lambda''\\in Q$, provided $N \\geq 4$. Hence:\n\\begin{cor} \\label{cor8.7} For $A_2^{(2)}$, $4$ is a saturation factor.\n\\end{cor}\n\\begin{remark} \nWhen the Kac-Moody Lie algebra $\\fg$ is infinite dimensional, then the saturated tensor semigroup $\\Gamma_s$ is {\\em not} finitely generated,\nfor any $s\\geq 2$. Thus, it is not clear a priori that there exists a saturation factor for such a $\\fg$.\n\\end{remark}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Summary}\n\nSymplectic field theory (SFT), introduced by H. Hofer, A. Givental and Y. Eliashberg in 2000 ([EGH]), is a very large project and can be viewed as a topological quantum field theory approach to Gromov-Witten theory. Besides providing a unified view on established pseudoholomorphic curve theories like symplectic Floer homology, contact homology and Gromov-Witten theory, it leads to numerous new applications and opens new routes yet to be explored. \\\\\n \nWhile symplectic field theory leads to algebraic invariants with very rich algebraic structures, which are currently studied by a large group of researchers, for all the geometric applications found so far it was sufficient to work with simpler invariants like cylindrical contact homology. Although cylindrical contact homology is not always defined, it is much easier to compute, not only since it involves just moduli spaces of holomorphic cylinders but also due to the simpler algebraic formalism. While the rich algebraic formalism of the higher invariants of symplectic field theory seems to be too complicated for concrete geometric applications, it was pointed out by Eliashberg in his ICM 2006 plenary talk ([E]) that the integrable systems of rational Gromov-Witten theory very naturally appear in rational symplectic field theory by using the link between the rational symplectic field theory of circle bundles in the Morse-Bott version and the rational Gromov-Witten potential of the underlying symplectic manifold. Indeed, after introducing gravitational descendants as in Gromov-Witten theory, it is precisely the rich algebraic formalism of SFT with its Weyl and Poisson structures that provides a natural link between symplectic field theory and (quantum) integrable systems. In particular, in the case where the contact manifold is a circle bundle over a closed symplectic manifold, the rich algebraic formalism of symplectic field theory seems to provide the right framework to understand the deep relation between Gromov-Witten theory and integrable systems, at least in the genus zero case. \\\\\n\nWhile in the Morse-Bott case in [E] it follows from the corresponding statements for the Gromov-Witten descendant potential that the sequences of commuting operators and Poisson-commuting functions are independent of auxiliary choices like almost complex structure and abstract perturbations, for the case of general contact manifolds it is well-known that the SFT Hamiltonian however in general explicitly depend on choices like contact form, cylindrical almost complex structure and coherent abstract perturbations and hence is not an invariant for the contact manifold itself. But before we can come down to the question of invariance, we first need to give a rigorous definition of gravitational descendants in the context of symplectic field theory. \\\\\n\nWhile in Gromov-Witten theory the gravitational descendants were defined by integrating powers of the first Chern class of the tautological line bundle over the moduli space, which by Poincare duality corresponds to counting common zeroes of sections in this bundle, in symplectic field theory, more generally every holomorphic curves theory where curves with punctures and\/or boundary are considered, we are faced with the problem that the moduli spaces generically have codimension-one boundary, so that the count of zeroes of sections in general depends on the chosen sections in the boundary. It follows that the integration of the first Chern class of the tautological line bundle over a single moduli space has to be replaced by a construction involving all moduli space at once. Note that this is similar to the choice of coherent abstract perturbations for the moduli spaces in symplectic field theory in order to achieve transversality for the Cauchy-Riemann operator. Keeping the interpretation of descendants as common zero sets of sections in powers of the tautological line bundles (which will turn out to be particularly useful when one studies the topological meaning of descendants by localizing on special divisors, see [FR]), we define in this paper the notion of {\\it coherent collections of sections} in the tautological line bundles over all moduli spaces, which just formalizes how the sections chosen for the lower-dimensional moduli spaces should affect the section chosen for a moduli spaces on its boundary. To be more precise, since the sections should be invariant under obvious symmetries like reordering of the punctures and the marked points, we actually need to work with multi-sections in order to meet both the symmetry and the transversality assumption. We will then define {\\it descendants of moduli spaces} $\\overline{\\operatorname{\\mathcal{M}}}^j\\subset\\overline{\\operatorname{\\mathcal{M}}}$, which we obtain inductively as zero sets of these coherent collections of sections $(s_j)$ in the tautological line bundles over the descendant moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}\\subset\\overline{\\operatorname{\\mathcal{M}}}$, and define {\\it descendant Hamiltonians} $\\operatorname{\\mathbf{H}}^1_{i,j}$ by integrating chosen closed differential forms $\\theta_i$ over $\\overline{\\operatorname{\\mathcal{M}}}^j$. For these we prove the following theorem. \\\\ \n\\\\\n{\\bf Theorem:} {\\it Counting holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements} \n\\begin{equation*} \\operatorname{\\mathbf{H}}^1_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0) \\end{equation*} \n{\\it in the full SFT homology algebra with differential $D^0=[\\operatorname{\\mathbf{H}}^0,\\cdot]: \\hbar^{-1}\\operatorname{\\mathfrak{W}}^0\\to\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0$, which commute with respect to the commutator bracket on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0)$,} \n\\begin{equation*} [\\operatorname{\\mathbf{H}}^1_{i,j},\\operatorname{\\mathbf{H}}^1_{k,\\ell}] = 0,\\; (i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}. \\end{equation*}\n\\\\\nIn contrast to the Morse-Bott case considered in [E] it follows that, when the differential in symplectic field theory counting holomorphic curves without additional marked points is no longer zero, the sequences of generating functions no longer commute with respect to the bracket, but only commute {\\it after passing to homology.} On the other hand, in the same way as the rational symplectic field theory of a contact manifold is defined by counting only curves with genus zero, we immediately obtain a rational version of the above statement by expanding $\\operatorname{\\mathbf{H}}^0$ and the $\\operatorname{\\mathbf{H}}^1_{i,j}$ in powers of the formal variable $\\hbar$ for the genus. \\\\\n\\\\\n{\\bf Corollary:} {\\it Counting rational holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements} \n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^0,d^0), \\end{equation*} \n{\\it in the rational SFT homology algebra with differential $d^0=\\{\\operatorname{\\mathbf{h}}^0,\\cdot\\}: \\operatorname{\\mathfrak{P}}^0\\to\\operatorname{\\mathfrak{P}}^0$, which commute with respect to the Poisson bracket on $H_*(\\operatorname{\\mathfrak{P}}^0,d^0)$,} \n\\begin{equation*} \\{\\operatorname{\\mathbf{h}}^1_{i,j},\\operatorname{\\mathbf{h}}^1_{k,\\ell}\\} = 0,\\; (i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}. \\end{equation*}\n\\\\\nAs we already outlined above, in contrast to the circle bundle case we have to expect that the sequence of descendant Hamiltonians depends on the auxiliary choices like contact form, cylindrical almost complex structure and coherent abstract polyfold perturbations. Here we prove the following natural invariance statements. \\\\\n\\\\\n{\\bf Theorem:} {\\it For different choices of contact form $\\lambda^{\\pm}$, cylindrical almost complex structure $\\underline{J}^{\\pm}$ , abstract polyfold perturbations and sequences of coherent collections of sections $(s^{\\pm}_j)$ the resulting systems of commuting operators $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}$ on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $\\operatorname{\\mathbf{H}}^{1,+}_{i,j}$ on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ are isomorphic, i.e., there exists an isomorphism of the Weyl algebras $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ which maps $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ to $\\operatorname{\\mathbf{H}}^{1,+}_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$.}\\\\ \n\\\\\nNote that this theorem is an extension of the theorem in [EGH] stating that for different choices of auxiliary data the Weyl algebras $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ are isomorphic. As above we clearly also get a rational version of the invariance statement: \\\\\n\\\\\n{\\bf Corollary:} {\\it For different choices of contact form $\\lambda^{\\pm}$, cylindrical almost complex structure $\\underline{J}^{\\pm}$, abstract polyfold perturbations and sequences of coherent collections of sections $(s^{\\pm}_j)$ the resulting system of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^{1,-}_{i,j}$ on $H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ and $\\operatorname{\\mathbf{h}}^{1,+}_{i,j}$ on $H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$ are isomorphic, i.e., there exists an isomorphism of the Poisson algebras $H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ and $H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$ which maps $\\operatorname{\\mathbf{h}}^{1,-}_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ to $\\operatorname{\\mathbf{h}}^{1,+}_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$.} \\\\\n\nAs concrete example beyond the case of circle bundles discussed in [E] we consider the symplectic field theory of a closed geodesic. For this recall that in [F2] the author introduces the symplectic field theory of a closed Reeb orbit $\\gamma$, which is defined by counting only those holomorphic curves which are branched covers of the orbit cylinder $\\operatorname{\\mathbb{R}}\\times\\gamma$ in $\\operatorname{\\mathbb{R}}\\times V$. In [F2] we prove that these orbit curves do not contribute to the algebraic invariants of symplectic field theory as long as they do not carry additional marked points. Our proof explicitly uses that the subset of orbit curves over a fixed orbit is closed under taking boundaries and gluing, which follows from the fact that they are also trivial in the sense that they have trivial contact area and that this contact area is preserved under taking boundaries and gluing. It follows that every algebraic invariant of symplectic field theory has a natural analog defined by counting only orbit curves. In particular, in the same way as we define sequences of descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_{i,j}$ and $\\operatorname{\\mathbf{h}}^1_{i,j}$ by counting general curves in the symplectization of a contact manifold, we can define sequences of descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_{\\gamma,i,j}$ and $\\operatorname{\\mathbf{h}}^1_{\\gamma,i,j}$ by just counting branched covers of the orbit cylinder over $\\gamma$ with signs (and weights), where the preservation of the contact area under splitting and gluing of curves proves that for every theorem from above we have a version for $\\gamma$. We further prove that for branched covers of orbit cylinders over any closed Reeb orbit the gravitational descendants indeed have a geometric interpretation in terms of branching conditions, which generalizes the work of [OP] used in [E] for the circle. \\\\\n\nSince all the considered holomorphic curves factor through the embedding of the closed Reeb orbit into the contact manifold, it follows that it only makes sense to consider differential forms of degree zero or one. While it follows from the result $\\operatorname{\\mathbf{h}}^0_{\\gamma}=0$ in [F2] that the sequences $\\operatorname{\\mathbf{h}}^1_{\\gamma,i,j}$ indeed commute with respect to the Poisson bracket (before passing to homology), the same proof as in [F2] shows that every descendant Hamiltonian in the sequence vanishes if the differential form is of degree zero. For differential forms of degree one the strategy of the proof however no longer applies and it is indeed shown in [E] that for $\\gamma=V=S^1$ and $\\theta=dt$ we get nontrivial contributions from branched covers. In this paper we want to determine the corresponding Poisson-commuting sequence in the special case where the contact manifold is the unit cotangent bundle $S^*Q$ of a ($m$-dimensional) Riemannian manifold $Q$, so that every closed Reeb orbit $\\gamma$ on $V=S^*Q$ corresponds to a closed geodesic $\\bar{\\gamma}$ on $Q$. For this we denote by $\\operatorname{\\mathfrak{W}}^0_{\\gamma}$ be the graded Weyl subalgebra of the Weyl algebra $\\operatorname{\\mathfrak{W}}^0$, which is generated only by those $p$- and $q$-variables $p_n=p_{\\gamma^n}$, $q_n=q_{\\gamma^n}$ corresponding to Reeb orbits which are multiple covers of the fixed orbit $\\gamma$ {\\it and which are good in the sense of [BM]}. In the same way we further introduce the Poisson subalgebra $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ of $\\operatorname{\\mathfrak{P}}^0$. Setting $q_{-n}=p_n$ we prove the following \\\\\n\\\\\n{\\bf Theorem:} {\\it Assume that the contact manifold is the unit cotangent bundle $V=S^*Q$ of a Riemannian manifold $Q$, so that the closed Reeb orbit $\\gamma$ corresponds to a closed geodesic $\\bar{\\gamma}$ on $Q$, and that the string of differential forms just consists of a single one-form which integrates to one around the orbit. Then the resulting system of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ is isomorphic to the system of Poisson-commuting functions $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\bar{\\gamma}}=\\operatorname{\\mathfrak{P}}^0_{\\gamma}$, where for every $j\\in\\operatorname{\\mathbb{N}}$ the descendant Hamiltonian $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ is given by} \n\\begin{equation*} \n \\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j} \\;=\\; \\sum \\epsilon(\\vec{n})\\;\\frac{q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}}{(j+2)!} \n\\end{equation*}\n{\\it where the sum runs over all ordered monomials $q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}$ with $n_1+...+n_{j+2} = 0$ \\textbf{and which are of degree $2(m+j-3)$}. Further $\\epsilon(\\vec{n})\\in\\{-1,0,+1\\}$ is fixed by a choice of coherent orientations in symplectic field theory and is zero if and only if one of the orbits $\\gamma^{n_1},...,\\gamma^{n_{j+2}}$ is bad.} \\\\\n\nNote that in the case of the circle $\\bar{\\gamma}=Q=S^1$ the degree condition is automatically fulfilled and we just get back the sequence of descendant Hamiltonians for the circle in [E], which agrees with the sequence of Poisson-commuting integrals of the dispersionless KdV integrable hierarchy. Forgetting about the appearing sign issues, it follows that the sequence $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ is obtained from the sequence for the circle by removing all summands with the wrong, that is, not maximal degree, so that the system is completely determined by the KdV hierarchy and the Morse indices of the closed geodesic and its iterates. \\\\\n\n\\noindent{\\bf Remark:} Note that the signs in the formula for $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ are determined by the linearized Reeb flow around $\\gamma$ and a choice of \norientations for all multiples of $\\gamma$. For this recall from [BM] that in order to orient moduli spaces in symplectic field theory one additionally needs to choose orientations for all occuring Reeb orbits, while the resulting invariants are independent of these auxiliary choices. While the precise formula for the functions $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ depends on these choices, the resulting systems of Poisson-commuting functions for different choices are indeed isomorphic, since changing the orientations for some orbits $\\gamma^k$ leads to an automorphism of the underlying Poisson algebra. Apart from the fact that the commutativity condition $\\{\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j},\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},k}\\}=0$ clearly leads to relations between the different $\\epsilon(\\vec{n})$, observe that a choice of orientation for $\\gamma$ does {\\it not} lead to a canonical choice of orientations for its multiples $\\gamma^k$. While we expect that it is in general very hard to write down a set of signs $\\epsilon(\\vec{n})$ explicitly, for all the geometric applications we have in mind and the educational purposes as a test model beyond the Gromov-Witten case we are rather interested in proving vanishing results as the one below than giving precise formulas. \\\\\n\nWhile in the case of the circle we obtain a complete set of integrals, it is important that our theorem allows us to prove the following vanishing result, which simply follows from the fact that for hyperbolic Reeb orbits the Conley-Zehnder index is multiplicative. \\\\\n\\\\\n{\\bf Corollary:} {\\it Assume that the closed geodesic $\\bar{\\gamma}$ represents a hyperbolic Reeb orbit in the unit cotangent bundle of a surface $Q$. Then $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}=0$ and hence $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}=0$ for all $j>0$.} \\\\\n\nNote that this result is actually true for $\\dim Q > 1$. While for $\\dim Q > 2$ the result directly follows from index reasons, in the case when $\\dim Q=2$ a simple computation shows that all moduli spaces with $2j+1$ punctures possibly contribute to the descendant Hamiltonian $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$. Since in this case the Fredholm index is $2j-1$ and hence for $j>0$ strictly smaller than the dimension of the underlying nonregular moduli space of branched covers, which is $4j-2$, transversality cannot be satisfied but the cokernels of the linearized operators fit together to give an obstruction bundle of rank $2j-1$. \\\\\n\nApart from using the geometric interpretation of gravitational descendants for branched covers of orbit cylinders over a closed Reeb orbit in terms of branching conditions mentioned above, the second main ingredient for the proof is the idea in [CL] to compute the symplectic field theory of $V=S^*Q$ from the string topology of the underlying Riemannian manifold $Q$ by studying holomorphic curves in the cotangent bundle $T^*Q$. More precisely, we compute the symplectic field theory of a closed Reeb orbit $\\gamma$ in $S^*Q$ including differential forms and gravitational descendants by studying branched covers of the trivial half-cylinder connecting the closed Reeb orbit in the unit cotangent bundle with the underlying closed geodesic in the cotangent bundle $T^*Q$ with special branching data, where the latter uses the geometric interpretation of gravitational descendants. In order to give a complete proof we also prove the neccessary transversality theorems using finite-dimensional obstruction bundles over the underlying nonregular moduli spaces. While on the SFT side one has very complicated obstruction bundles over nonregular moduli spaces of arbitary large dimension, on the string side all relevant nonregular moduli spaces already turn out to be discrete, so that the obstruction bundles disappear if the Fredholm index is right. It follows that the system of Poisson-commuting function for a closed geodesic is completely determined by the KdV hierarchy and the Morse indices of the closed geodesic and its iterates. \\\\\n\nThis paper is organized as follows. \\\\\n\nSection one is concerned with the definition and the basic results about gravitational descendants in symplectic field theory. After we recalled the basic definitions of symplectic field theory in subsection 1.1, we define gravitational descendants in subsection 1.2 using the coherent collections of sections and prove that the resulting sequences of descendant Hamiltonians commute after passing to homology. In subsection 1.3 we prove the desired invariance statement and discuss the important case of circle bundles in the Morse-Bott setup outlined in [E] in 1.4. \\\\\n\nAfter we treated the general case in section one, section two is concerned with a concrete example beyond the case of circle bundles, the symplectic field theory of a closed geodesic, which naturally generalizes the case of the circle in [E]. After we have recalled the definition of symplectic field theory for a closed Reeb orbit including the results from [F2] in subsection 2.1, we show in subsection 2.2 that for branched covers of orbit cylinders the gravitational descendants have a geometric interpretation in terms of branching conditions. After outlining that there exists a version of the isomorphism in [CL] involving the symplectic field theory of a closed Reeb orbit in the unit cotangent bundle, we study the moduli space of branched covers of the corresponding trivial half-cylinder in the cotangent bundle in subsection 2.3. Since we meet the same transversality problems as in [F2], we study the neccessary obstruction bundle setup including Banach manifolds and Banach space bundles in subsection 2.3. In subsection 2.4 we finally prove the above theorem by studying branched covers of the trivial half-cylinder with special branching behavior. \\\\ \n\\\\\n{\\bf Acknowledgements:} This research was supported by the German Research Foundation (DFG). \nThe author thanks K. Cieliebak, Y. Eliashberg, K. Fukaya, M. Hutchings and P. Rossi for useful discussions.\n \n\\section{Symplectic field theory with gravitational descendants}\n\n\\subsection{Symplectic field theory}\nSymplectic field theory (SFT) is a very large project, initiated by Eliashberg,\nGivental and Hofer in their paper [EGH], designed to describe in a unified way \nthe theory of pseudoholomorphic curves in symplectic and contact topology. \nBesides providing a unified view on well-known theories like symplectic Floer \nhomology and Gromov-Witten theory, it shows how to assign algebraic invariants \nto closed contact manifolds $(V,\\xi=\\{\\lambda=0\\})$: \\\\\n \nRecall that a contact one-form $\\lambda$ defines a vector field $R$ on $V$ by \n$R\\in\\ker d\\lambda$ and $\\lambda(R)=1$, which \nis called the Reeb vector field. We assume that \nthe contact form is Morse in the sense that all closed orbits of the \nReeb vector field are nondegenerate in the sense of [BEHWZ]; in particular, the set \nof closed Reeb orbits is discrete. The invariants are defined by counting \n$\\underline{J}$-holomorphic curves in $\\operatorname{\\mathbb{R}}\\times V$ which are asymptotically cylindrical over \nchosen collections of Reeb orbits $\\Gamma^{\\pm}=\\{\\gamma^{\\pm}_1,...,\n\\gamma^{\\pm}_{n^{\\pm}}\\}$ as the $\\operatorname{\\mathbb{R}}$-factor tends to $\\pm\\infty$, see [BEHWZ]. \nThe almost complex structure $\\underline{J}$ on the cylindrical \nmanifold $\\operatorname{\\mathbb{R}}\\times V$ is required to be cylindrical in the sense that it is \n$\\operatorname{\\mathbb{R}}$-independent, links the two natural vector fields on $\\operatorname{\\mathbb{R}}\\times V$, namely the \nReeb vector field $R$ and the $\\operatorname{\\mathbb{R}}$-direction $\\partial_s$, by $\\underline{J}\\partial_s=R$, and turns \nthe distribution $\\xi$ on $V$ into a complex subbundle of $TV$, \n$\\xi=TV\\cap \\underline{J} TV$. We denote by $\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)$ the corresponding compactified\nmoduli space of genus $g$ curves with $r$ additional marked points ([BEHWZ],[EGH]). \nPossibly after choosing abstract perturbations using polyfolds (see [HWZ]), obstruction \nbundles ([F2]) or domain-dependent structures ([F1]) following the ideas in [CM] we get that \n$\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)$ is a branched-labelled orbifold with boundaries and corners of dimension \nequal to the Fredholm index of the Cauchy-Riemann operator for $\\underline{J}$. \n{\\it Note that in the same way as we will not discuss transversality for the general case but just refer to the upcoming papers on polyfolds by Hofer and his co-workers, in what follows we will for simplicity assume that every moduli space is indeed a manifold with boundaries and corners, since we expect that all the upcoming constructions can be generalized in an appropriate way.} \\\\ \n \nLet us now briefly introduce the algebraic formalism of SFT as described in [EGH]: \\\\\n \nRecall that a multiply-covered Reeb orbit $\\gamma^k$ is called bad if \n$\\operatorname{CZ}(\\gamma^k)\\neq\\operatorname{CZ}(\\gamma)\\mod 2$, where $\\operatorname{CZ}(\\gamma)$ denotes the \nConley-Zehnder index of $\\gamma$. Calling a Reeb orbit $\\gamma$ {\\it good} if it is not bad we assign to every good Reeb orbit $\\gamma$ two formal \ngraded variables $p_{\\gamma},q_{\\gamma}$ with grading \n\\begin{equation*} \n|p_{\\gamma}|=m-3-\\operatorname{CZ}(\\gamma),|q_{\\gamma}|=m-3+\\operatorname{CZ}(\\gamma) \n\\end{equation*} \nwhen $\\dim V = 2m-1$. In order to include higher-dimensional moduli spaces we further assume that a string \nof closed (homogeneous) differential forms $\\Theta=(\\theta_1,...,\\theta_N)$ on $V$ is chosen and assign to \nevery $\\theta_i\\in\\Omega^*(V)$ a formal variables $t_i$ \nwith grading\n\\begin{equation*} |t_i|=2 -\\deg\\theta_i. \\end{equation*} \nFinally, let $\\hbar$ be another formal variable of degree $|\\hbar|=2(m-3)$. \\\\\n\nLet $\\operatorname{\\mathfrak{W}}$ be the graded Weyl algebra over $\\operatorname{\\mathbb{C}}$ of power series in the variables \n$\\hbar,p_{\\gamma}$ and $t_i$ with coefficients which are polynomials in the \nvariables $q_{\\gamma}$, which is equipped with the associative product $\\star$ in \nwhich all variables super-commute according to their grading except for the \nvariables $p_{\\gamma}$, $q_{\\gamma}$ corresponding to the same Reeb orbit $\\gamma$, \n\\begin{equation*} [p_{\\gamma},q_{\\gamma}] = \n p_{\\gamma}\\star q_{\\gamma} -(-1)^{|p_{\\gamma}||q_{\\gamma}|} \n q_{\\gamma}\\star p_{\\gamma} = \\kappa_{\\gamma}\\hbar.\n\\end{equation*}\n($\\kappa_{\\gamma}$ denotes the multiplicity of $\\gamma$.) Following [EGH] we further introduce \nthe Poisson algebra $\\operatorname{\\mathfrak{P}}$ of formal power series in the variables $p_{\\gamma}$ and $t_i$ with \ncoefficients which are polynomials in the variables $q_{\\gamma}$ with Poisson bracket given by \n\\begin{equation*} \n \\{f,g\\} = \\sum_{\\gamma}\\kappa_{\\gamma}\\Bigl(\\frac{\\partial f}{\\partial p_{\\gamma}}\\frac{\\partial g}{\\partial q_{\\gamma}} -\n (-1)^{|f||g|}\\frac{\\partial g}{\\partial p_{\\gamma}}\\frac{\\partial f}{\\partial q_{\\gamma}}\\Bigr). \n\\end{equation*}\n\nAs in Gromov-Witten theory we want to organize all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)$\ninto a generating function $\\operatorname{\\mathbf{H}}\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}$, called {\\it Hamiltonian}. In order to include also higher-dimensional \nmoduli spaces, in [EGH] the authors follow the approach in Gromov-Witten theory to integrate the chosen differential forms \n$\\theta_1,...,\\theta_N$ over the moduli spaces after pulling them back under the evaluation map from target manifold $V$. The \nHamiltonian $\\operatorname{\\mathbf{H}}$ is then defined by\n\\begin{equation*}\n \\operatorname{\\mathbf{H}} = \\sum_{\\Gamma^+,\\Gamma^-} \\int_{\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}}\n \\operatorname{ev}_1^*\\theta_{i_1}\\wedge...\\wedge\\operatorname{ev}_r^*\\theta_{i_r}\\; \\hbar^{g-1}t^Ip^{\\Gamma^+}q^{\\Gamma^-}\n\\end{equation*}\nwith $t^I=t_{i_1}...t_{i_r}$, $p^{\\Gamma^+}=p_{\\gamma^+_1}...p_{\\gamma^+_{n^+}}$ and $q^{\\Gamma^-}=q_{\\gamma^-_1}...q_{\\gamma^-_{n^-}}$. \nExpanding \n\\begin{equation*} \\operatorname{\\mathbf{H}}=\\hbar^{-1}\\sum_g \\operatorname{\\mathbf{H}}_g \\hbar^g \\end{equation*} \nwe further get a rational Hamiltonian $\\operatorname{\\mathbf{h}}=\\operatorname{\\mathbf{H}}_0\\in\\operatorname{\\mathfrak{P}}$, which counts only curves with genus zero. \\\\\n\nWhile the Hamiltonian $\\operatorname{\\mathbf{H}}$ explicitly depends on the chosen contact form, the cylindrical almost complex structure, the differential forms and abstract polyfold perturbations making all moduli spaces regular, it is outlined in [EGH] how to construct algebraic invariants, which just depend on the contact structure and the cohomology classes of the differential forms. \\\\\n \n\\subsection{Gravitational descendants}\nFor the relation to integrable systems it is outlined in [E] that, as in Gromov-Witten theory, symplectic field theory must be enriched by considering so-called {\\it gravitational descendants} of the {\\it primary} Hamiltonian $\\operatorname{\\mathbf{H}}$. \\\\\n\nBefore we give a rigorous definition of gravitational descendants in SFT, we recall the definition from \nGromov-Witten theory. Denote by $\\overline{\\operatorname{\\mathcal{M}}}_r=\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(X,J)$ the compactified moduli space of closed $J$-holomorphic curves in the closed symplectic manifold $X$ of genus $g$ with $r$ marked points (and fixed homology class). Following [MDSa] we introduce over $\\overline{\\operatorname{\\mathcal{M}}}_r$ so-called {\\it tautological line bundles} $\\operatorname{\\mathcal{L}}_1,...,\\operatorname{\\mathcal{L}}_r$, where the fibre of $\\operatorname{\\mathcal{L}}_i$ over a punctured curve $(u,z_1,...,z_r)\\in\\overline{\\operatorname{\\mathcal{M}}}_r$ in the noncompactified moduli space is given by the cotangent line to the underlying, possibly unstable closed nodal Riemann surface $S$ at the $i$.th marked point,\n\\begin{equation*} (\\operatorname{\\mathcal{L}}_i)_{(u,z_1,...,z_r)} = T^*_{z_i}S,\\;\\;i=1,...,r. \\end{equation*}\nTo be more formal, observe that there exists a canonical map $\\pi: \\overline{\\operatorname{\\mathcal{M}}}_{r+1}\\to\\overline{\\operatorname{\\mathcal{M}}}_r$ by forgetting the $(r+1)$.st marked point and stabilizing the map, where the fibre over the curve $(u,z_1,...,z_r)$ agrees with the curve itself. Then the tautological line bundle $\\operatorname{\\mathcal{L}}_i$ can be defined as the pull-back of the vertical cotangent line bundle of $\\pi: \\overline{\\operatorname{\\mathcal{M}}}_{r+1}\\to\\overline{\\operatorname{\\mathcal{M}}}_r$ under the canonical section $\\sigma_i:\\overline{\\operatorname{\\mathcal{M}}}_r\\to\\overline{\\operatorname{\\mathcal{M}}}_{r+1}$ mapping to the $i$.th marked point in the fibre. Note that while the vertical cotangent line bundle is rather a sheaf than a true bundle since it becomes singular at the nodes in the fibres, the pull-backs under the canonical sections are indeed true line bundles as the marked points are different from the nodes and hence these sections avoid the singular loci. \\\\\n\nDenoting by $c_1(\\operatorname{\\mathcal{L}}_i)$ the first Chern class of the complex line bundle $\\operatorname{\\mathcal{L}}_i$, one then considers for the descendant potential of Gromov-Witten theory integrals of the form\n\\begin{equation*} \\int_{\\overline{\\operatorname{\\mathcal{M}}}_r} \\operatorname{ev}_1^*\\theta_{i_1}\\wedge c_1(\\operatorname{\\mathcal{L}}_1)^{j_1}\\wedge ... \\wedge \\operatorname{ev}_r^*\\theta_{i_r}\\wedge c_1(\\operatorname{\\mathcal{L}}_r)^{j_r}, \\end{equation*}\nwhere $(i_k,j_k)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$, which can again be organized into a generating function. \\\\\n\nLike pulling-back cohomology classes from the target manifold, the introduction of the tautological line bundles hence \nhas the effect that the generating function also sees the higher-dimensional moduli spaces. On the other hand, \nin contrast to the former, the latter refers to partially fixing the complex structure on the underlying punctured \nRiemann surface. \\\\\n\nBefore we can turn to the definition of gravitational descendants in SFT, it will turn out to be useful to give an alternative definition, where the integration of the powers of the first Chern classes is replaced by considering zero sets of sections. Restricting for notational simplicity to the case with one marked point, we can define by induction over $j\\in\\operatorname{\\mathbb{N}}$ a nested sequence of moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{j+1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^j_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ such that \n\\begin{equation*} \\int_{\\overline{\\operatorname{\\mathcal{M}}}_1} \\operatorname{ev}^*\\theta_i \\wedge c_1(\\operatorname{\\mathcal{L}})^j = \\frac{1}{j!} \\cdot \\int_{\\overline{\\operatorname{\\mathcal{M}}}^j_1} \\operatorname{ev}^*\\theta_i. \\end{equation*}\n\nFor $j=1$ observe that, since the first Chern class of a line bundle agrees with its Euler class, the homology class obtained by integrating $c_1(\\operatorname{\\mathcal{L}})$ over the compactified moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$ can be represented by the zero set of a generic section $s_1$ in $\\operatorname{\\mathcal{L}}$. Note that here we use that $\\overline{\\operatorname{\\mathcal{M}}}_1$ represents a pseudo-cycle and hence has no codimension-one boundary strata. In other words, we find that \n\\begin{equation*} \\int_{\\overline{\\operatorname{\\mathcal{M}}}_1} \\operatorname{ev}^*\\theta_i \\wedge c_1(\\operatorname{\\mathcal{L}}) \\;=\\; \\int_{\\overline{\\operatorname{\\mathcal{M}}}^1_1} \\operatorname{ev}^*\\theta_i, \\end{equation*}\nwhere $\\overline{\\operatorname{\\mathcal{M}}}^1_1=s_1^{-1}(0)$. \\\\\n\nNow consider the restriction of the tautological line bundle $\\operatorname{\\mathcal{L}}$ to $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$. Instead of describing the integration of powers of the first Chern class in terms of common zero sets of sections in the same line bundle $\\operatorname{\\mathcal{L}}$, it turns out to be more geometric (see 2.2) to choose a section $s_j$ not in $\\operatorname{\\mathcal{L}}$ but in its $j$-fold (complex) tensor product $\\operatorname{\\mathcal{L}}^{\\otimes j}$ and define \n\\begin{equation*} \\overline{\\operatorname{\\mathcal{M}}}^j_1 = s_j^{-1}(0) \\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1. \\end{equation*} \nSince $c_1(\\operatorname{\\mathcal{L}}^{\\otimes j}) = j\\cdot c_1(\\operatorname{\\mathcal{L}})$ it follows that \n\\begin{equation*} \\int_{\\overline{\\operatorname{\\mathcal{M}}}^j_1} \\operatorname{ev}^*\\theta_i = j\\cdot \\int_{\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1} \\operatorname{ev}^*\\theta_i\\wedge c_1(\\operatorname{\\mathcal{L}}) \\end{equation*}\nso that by induction\n\\begin{equation*} \\int_{\\overline{\\operatorname{\\mathcal{M}}}_1} \\operatorname{ev}^*\\theta_i \\wedge c_1(\\operatorname{\\mathcal{L}})^j = \\frac{1}{j!} \\cdot \\int_{\\overline{\\operatorname{\\mathcal{M}}}^j_1} \\operatorname{ev}^*\\theta_i \\end{equation*}\nas desired. \\\\\n\nWhile the result of the integration is well-known to be independent of the choice of the almost complex structure and the abstract polyfold perturbations, it also follows that the result is independent of the precise choice of the sequence of sections $s_1,...,s_j$. Like for the almost complex structure and the perturbations this results from the fact that the moduli spaces studied in Gromov-Witten theory have no codimension-one boundary. \\\\\n\nOn the other hand, it is well-known that the moduli spaces in SFT typically have codimension-one boundary, so that now the result of the integration will not only depend on the chosen contact form, cylindrical almost complex structure and abstract polyfold perturbations, but also additionally explicitly depend on the chosen sequences of sections $s_1,...,s_j$. While the Hamiltonian is hence known to depend on all extra choices, it is well-known from Floer theory that we can expect to find algebraic invariants independent of these choices. \\\\\n\nWhile the problem of dependency on contact form, cylindrical almost complex structure and abstract polyfold perturbations is sketched in [EGH], we will now show how to include gravitational descendants into their algebraic constructions. For this we will define {\\it descendants of moduli spaces}, which we obtain as zero sets of {\\it coherent collections of sections} in the tautological line bundles over all moduli spaces. \\\\\n\nFrom now on let $\\overline{\\operatorname{\\mathcal{M}}}_r$ denote the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}$ studied in SFT for chosen collections \nof Reeb orbits $\\Gamma^+,\\Gamma^-$. In complete analogy to Gromov-Witten theory we can introduce $r$ tautological line \nbundles $\\operatorname{\\mathcal{L}}_1,...,\\operatorname{\\mathcal{L}}_r$, where the fibre of $\\operatorname{\\mathcal{L}}_i$ over a punctured curve $(u,z_1,...,z_r)\\in\\overline{\\operatorname{\\mathcal{M}}}_r$ is again given \nby the cotangent line to the underlying, possibly unstable nodal Riemann surface (without ghost components) at the \n$i$.th marked point and which again formally can be defined as the pull-back of the vertical cotangent line \nbundle of $\\pi: \\overline{\\operatorname{\\mathcal{M}}}_{r+1}\\to\\overline{\\operatorname{\\mathcal{M}}}_r$ under the canonical section $\\sigma_i: \\overline{\\operatorname{\\mathcal{M}}}_r\\to\\overline{\\operatorname{\\mathcal{M}}}_{r+1}$ mapping to the $i$.th marked \npoint in the fibre. Note again that while the vertical cotangent line bundle is rather a sheaf than a true bundle since \nit becomes singular at the nodes in the fibres, the pull-backs under the canonical sections are still true line bundles \nas the marked points are different from the nodes and hence these sections avoid the singular loci. \\\\\n\nFor notational simplicity let us again restrict to the case $r=1$. Following the compactness statement in [BEHWZ], the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}_1$ consists of curves with two levels (in the sense of [BEHWZ]), whose moduli spaces can be represented as products $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ or $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ of moduli spaces of strictly lower dimension, where the marked point sits on the first or the second level. As we want to keep the notation as simple as possible, note that here and in what follows for product moduli spaces {\\it the first index refers to the level} and not to the genus of the curve. To be more precise, after introducing asymptotic markers as in [EGH] for orientation issues, one obtains a fibre rather than a direct product, see also [F2]. However, since all the bundles and sections we will consider do or should not depend on these asymptotic markers, we will forget about this issue in order to keep the notation as simple as possible. \\\\ \n\nOn the other hand, it directly follows from the definition of the tautological line bundle $\\operatorname{\\mathcal{L}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_1$ that over the boundary components $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ it is given by \n\\begin{equation*} \\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}} = \\pi_1^*\\operatorname{\\mathcal{L}}_1,\\;\\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}} = \\pi_2^*\\operatorname{\\mathcal{L}}_2, \\end{equation*}\nwhere $\\operatorname{\\mathcal{L}}_1$, $\\operatorname{\\mathcal{L}}_2$ denotes the tautological line bundle over the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}$, $\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ and $\\pi_1$, $\\pi_2$ is the projection onto the first or second factor, respectively. \\\\\n\nWith this we can now introduce the notion of coherent collections of sections in (tensor products of) tautological line bundles. \\\\\n\\\\\n{\\bf Definition 1.1:} {\\it Assume that we have chosen sections $s$ in the tautological line bundles $\\operatorname{\\mathcal{L}}$ over all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_1$ of $\\underline{J}$-holomorphic curves with one additional marked point. Then this collection of sections $(s)$ is called} coherent {\\it if for every section $s$ in $\\operatorname{\\mathcal{L}}$ over a moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$ the following holds: Over every codimension-one boundary component $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ of $\\overline{\\operatorname{\\mathcal{M}}}_1$ the section $s$ agrees with the pull-back $\\pi_1^*s_1$, $\\pi_2^*s_2$ of the chosen section $s_1$, $s_2$ in the tautological line bundle $\\operatorname{\\mathcal{L}}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}$, $\\operatorname{\\mathcal{L}}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$, respectively.} \\\\\n\n\\noindent{\\bf Remark:} Since in the end we will again be interested in the zero sets of these sections, {\\it we will assume that all occuring sections are transversal to the zero section.} Furthermore, we want to assume that all the chosen sections are indeed {\\it invariant under the obvious symmetries like reordering of punctures and marked points.} In order to meet both requirements, it follows that actually need to employ {\\it multi-sections} as in [CMS], which we however want to suppress for the rest of this exposition. \\\\\n\nThe important observation is clearly that one can always find coherent collections of (transversal) sections $(s)$ by using induction on the dimension of the underlying moduli space. While for the induction start it suffices to choose a non-vanishing section in the tautological line bundle over the moduli space of orbit cylinders with one marked point, for the induction step observe that the coherency condition fixes the section on the boundary of the moduli space. Here it is important to remark that the coherency condition further ensures that two different codimension-one boundary components actually agree on their common boundary strata of higher codimension. On the other hand, we can use our assumption that every moduli space is indeed a manifold with corners to obtain the desired section by simply extending the section from the boundary to the interior of the moduli space in an arbitrary way. \\\\\n \nFor a given coherent collection of transversal sections $(s)$ we will again define for every moduli space \n\\begin{equation*} \\overline{\\operatorname{\\mathcal{M}}}^1_1 = s^{-1}(0) \\subset \\overline{\\operatorname{\\mathcal{M}}}_1. \\end{equation*}\nAs an immediate consequence of the above definition we find that $\\overline{\\operatorname{\\mathcal{M}}}^1_1$ is a neat submanifold (with corners) of $\\overline{\\operatorname{\\mathcal{M}}}_1$, i.e., the components of the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}^1_1$ are given by products $\\overline{\\operatorname{\\mathcal{M}}}^1_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^1_{2,1}$, where $\\overline{\\operatorname{\\mathcal{M}}}^1_{1,1} = s_1^{-1}(0)$, $\\overline{\\operatorname{\\mathcal{M}}}^1_{2,1} = s_2^{-1}(0)$ for the section $s_1$ in $\\operatorname{\\mathcal{L}}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}$, $s_2$ in $\\operatorname{\\mathcal{L}}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$, respectively. To be more precise, since we actually need to work with multi-sections rather than sections in the usual sense, the zero set is indeed a branched-labelled manifold. On the other hand, since we already suppressed the fact that our moduli spaces are indeed branched and labelled, we want to continue ignoring this technical aspect. On the other hand, we can use the above result as an induction start to obtain for every moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$ a sequence of nested subspaces $\\overline{\\operatorname{\\mathcal{M}}}^j_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ as in Gromov-Witten theory. \\\\\n\\\\\n{\\bf Definition 1.2:} {\\it Let $j\\in\\operatorname{\\mathbb{N}}$. Assume that for all moduli spaces we have chosen $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ such that the components of the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ are given by products of the form $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{2,0}$. Then we again call a collection of transversal sections $(s_j)$ in the j-fold tensor products $\\operatorname{\\mathcal{L}}^{\\otimes j}$ of the tautological line bundles over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset \\overline{\\operatorname{\\mathcal{M}}}_1$} coherent {\\it if for every section $s_j$ the following holds: Over every codimension-one boundary component $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{2,1}$ of $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ the section $s_j$ agrees with the pull-back $\\pi_1^*s_{1,j}$, $\\pi_2^*s_{2,j}$ of the section $s_{1,j}$, $s_{2,j}$ in the line bundle $\\operatorname{\\mathcal{L}}^{\\otimes j}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{1,1}$, $\\operatorname{\\mathcal{L}}^{\\otimes j}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{2,1}$, respectively.} \\\\\n\nWith this we will now introduce {\\it (gravitational) descendants of moduli spaces.} \\\\\n\\\\\n{\\bf Definition 1.3:} {\\it Assume that we have inductively defined a subsequence of nested subspaces $\\overline{\\operatorname{\\mathcal{M}}}^j_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ by requiring that \n$\\overline{\\operatorname{\\mathcal{M}}}^j_1 = s_j^{-1}(0)\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ for a coherent collection of sections $s_j$ in the line bundles $\\operatorname{\\mathcal{L}}^{\\otimes j}$ over the moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$. Then we call $\\overline{\\operatorname{\\mathcal{M}}}^j_1$ the $j$.th (gravitational) descendant of $\\overline{\\operatorname{\\mathcal{M}}}_1$.} \\\\\n\nLet $\\operatorname{\\mathfrak{W}}^0$ be the graded Weyl algebra over $\\operatorname{\\mathbb{C}}$ of power series in the variables $\\hbar$ and $p_{\\gamma}$ with coefficients which are polynomials in the \nvariables $q_{\\gamma}$, which is obtained from the big Weyl algebra $\\operatorname{\\mathfrak{W}}$ by setting all variables $t_i$ equal to zero. In the same way define the subalgebra $\\operatorname{\\mathfrak{P}}^0$ of the Poisson algebra $\\operatorname{\\mathfrak{P}}$. Apart from the Hamiltonian $\\operatorname{\\mathbf{H}}^0\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0$ counting only curves with no additional marked points,\n\\begin{equation*} \\operatorname{\\mathbf{H}}^0 = \\sum_{\\Gamma^+,\\Gamma^-} \\#\\overline{\\operatorname{\\mathcal{M}}}_{g,0}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}\\; \\hbar^{g-1} p^{\\Gamma^+}q^{\\Gamma^-}, \\end{equation*}\nwe now want to use the chosen differential forms $\\theta_i\\in\\Omega^*(V)$, $i=1,...,N$ and the sequences $\\overline{\\operatorname{\\mathcal{M}}}^j_1 = \\overline{\\operatorname{\\mathcal{M}}}^j_{g,1}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}$ of gravitational descendants to define sequences of new SFT Hamiltonians $\\operatorname{\\mathbf{H}}^1_{i,j}\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0$, $(i,j)\\in\\{1,...,N\\}\\times \\operatorname{\\mathbb{N}}$, by\n\\begin{equation*} \\operatorname{\\mathbf{H}}^1_{i,j} = \\sum_{\\Gamma^+,\\Gamma^-} \\int_{\\overline{\\operatorname{\\mathcal{M}}}^j_{g,1}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}}\\operatorname{ev}^*\\theta_i\\; \\hbar^{g-1} p^{\\Gamma^+}q^{\\Gamma^-}. \\end{equation*}\n\nWe want to emphasize that the following statement is not yet a theorem in the strict mathematical sense as the analytical foundations of symplectic field theory, in particular, the neccessary transversality theorems for the Cauchy-Riemann operator, are not yet fully established. Since it can be expected that the polyfold project by Hofer and his collaborators sketched in [HWZ] will provide the required transversality theorems, we follow other papers in the field in proving everything up to transversality and state it nevertheless as a theorem. \\\\\n\\\\\n{\\bf Theorem 1.4:} {\\it Counting holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements} \n\\begin{equation*} \\operatorname{\\mathbf{H}}^1_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0) \\end{equation*} \n{\\it in the full SFT homology algebra with differential $D^0=[\\operatorname{\\mathbf{H}}^0,\\cdot]: \\hbar^{-1}\\operatorname{\\mathfrak{W}}^0\\to\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0$, which commute with respect to the bracket on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0)$,} \n\\begin{equation*} [\\operatorname{\\mathbf{H}}^1_{i,j},\\operatorname{\\mathbf{H}}^1_{k,\\ell}] = 0,\\; (i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}. \\end{equation*}\n\\\\\n{\\it Proof:} While the boundary equation $D^0\\circ D^0=0$ is well-known to follow from the identity $[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^0]=0$, the fact that every $\\operatorname{\\mathbf{H}}^1_{i,j}$, $(i,j)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$ defines an element in the homology $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0)$ follows from the identity\n\\begin{equation*} [\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^1_{i,j}]=0,\\end{equation*} \nsince this proves $\\operatorname{\\mathbf{H}}^1_{i,j}\\in\\ker D^0$. On the other hand, in order to see that any two $\\operatorname{\\mathbf{H}}^1_{i,j}$, $\\operatorname{\\mathbf{H}}^1_{k,\\ell}$ commute {\\it after passing to homology} it suffices to prove the identity\n\\begin{equation*} [\\operatorname{\\mathbf{H}}^1_{i,j},\\operatorname{\\mathbf{H}}^1_{k,\\ell}]\\pm[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^2_{(i,j),(k,\\ell)}]=0 \\end{equation*} \nfor any $(i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$, where the new Hamiltonian $\\operatorname{\\mathbf{H}}^2_{(i,j),(k,\\ell)}$ is defined below using descendant moduli spaces with two additional marked points. \\\\\n\nThe latter two identities directly follow from our definition of gravitational descendants of moduli spaces based on the definition of coherent sections in tautological line bundles and the compactness theorem in [BEHWZ]. Indeed, in the same way as the identity $[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^0]=0$ follows from the fact that the codimension-one boundary of every moduli space $\\overline{\\operatorname{\\mathcal{M}}}_0$ is formed by products of moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, the second identity $[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^1_{i,j}]=0$ follows from the fact that the codimension-one boundary of a descendant moduli space $\\overline{\\operatorname{\\mathcal{M}}}^j_1$ is given by products of the form $\\overline{\\operatorname{\\mathcal{M}}}^j_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^j_{2,1}$. \\\\\n\nIn order to prove the third identity $[\\operatorname{\\mathbf{H}}^1_{i,j},\\operatorname{\\mathbf{H}}^1_{k,\\ell}]\\pm[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^2_{(i,j),(k,\\ell)}]=0$ for every $(i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$, we slightly have to enlarge our definition of gravitational descendants in order to include moduli spaces with two additional marked points. For this observe that for every pair $j,k\\in\\operatorname{\\mathbb{N}}$ we can define decendants $\\overline{\\operatorname{\\mathcal{M}}}^{(j,k)}_2$ of $\\overline{\\operatorname{\\mathcal{M}}}_2$ by setting $\\overline{\\operatorname{\\mathcal{M}}}^{(j,k)}_2 = \\overline{\\operatorname{\\mathcal{M}}}^{(j,0)}_2\\cap\\overline{\\operatorname{\\mathcal{M}}}^{(0,k)}_2$, where $\\overline{\\operatorname{\\mathcal{M}}}^{(j,0)}_2, \\overline{\\operatorname{\\mathcal{M}}}^{(0,k)}_2\\subset\\overline{\\operatorname{\\mathcal{M}}}_2$ are defined in the same way as $\\overline{\\operatorname{\\mathcal{M}}}^j_1,\\overline{\\operatorname{\\mathcal{M}}}^k_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ by simply forgetting the second or first additional marked point, respectively. Since the boundary of a moduli space of curves with two marked points consists of products of the form $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,2}$, $\\overline{\\operatorname{\\mathcal{M}}}_{1,2}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, it follows that the boundary of $\\overline{\\operatorname{\\mathcal{M}}}^{(j,0)}_2$ consists of products $\\overline{\\operatorname{\\mathcal{M}}}^j_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^j_{2,1}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{(j,0)}_{2,2}$, $\\overline{\\operatorname{\\mathcal{M}}}^{(j,0)}_{1,2}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$. Together with the similar result about the boundary of $\\overline{\\operatorname{\\mathcal{M}}}^{(0,k)}_2$ and using the inclusions we hence obtain that the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}^{(j,k)}_2$ is given by products of the form $\\overline{\\operatorname{\\mathcal{M}}}^j_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^k_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^k_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^j_{2,1}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{(j,k)}_{2,2}$, $\\overline{\\operatorname{\\mathcal{M}}}^{(j,k)}_{1,2}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$. While summing over the first two products (with signs) we obtain $[\\operatorname{\\mathbf{H}}^1_{i,j},\\operatorname{\\mathbf{H}}^1_{k,\\ell}]$, summing over the latter two we get $[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^2_{(i,j),(k,\\ell)}]$, which hence sum up to zero. $\\qed$ \\\\ \n\\\\\n{\\bf Remark:} While the proof suggests that for the above algebraic relations one only has to care about the codimension-one boundary strata of the moduli spaces, it is actually even more important that the coherency condition further ensures that two different codimension-one boundary components can be glued along their common boundary strata of higher codimension. \\\\\n\nAs above we further again obtain a rational version of the above statement by expanding $\\operatorname{\\mathbf{H}}^0$ and the $\\operatorname{\\mathbf{H}}^1_{i,j}$ in powers \nof $\\hbar$. \\\\\n\\\\\n{\\bf Corollary 1.5:} {\\it Counting rational holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements} \n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^0,d^0), \\end{equation*} \n{\\it in the rational SFT homology algebra with differential $d^0=\\{\\operatorname{\\mathbf{h}}^0,\\cdot\\}: \\operatorname{\\mathfrak{P}}^0\\to\\operatorname{\\mathfrak{P}}^0$, which commute with respect to the Poisson bracket on $H_*(\\operatorname{\\mathfrak{P}}^0,d^0)$,} \n\\begin{equation*} \\{\\operatorname{\\mathbf{h}}^1_{i,j},\\operatorname{\\mathbf{h}}^1_{k,\\ell}\\} = 0,\\; (i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}. \\end{equation*}\n\nSo far we have only considered the case with one additional marked point. On the other hand, the general case with $r$ additional marked points is just notationally more involved. Indeed, as we did in the proof of the above theorem we can easily define for every moduli space $\\overline{\\operatorname{\\mathcal{M}}}_r$ with $r$ additional marked points and every $r$-tuple of natural numbers $(j_1,...,j_r)$ descendants $\\overline{\\operatorname{\\mathcal{M}}}^{(j_1,...,j_r)}_r\\subset\\overline{\\operatorname{\\mathcal{M}}}_r$ by setting\n\\begin{equation*} \\overline{\\operatorname{\\mathcal{M}}}^{(j_1,...,j_r)}_r = \\overline{\\operatorname{\\mathcal{M}}}^{(j_1,0,...,0)}_r\\cap ... \\cap \\overline{\\operatorname{\\mathcal{M}}}^{(0,...,0,j_r)}_r, \\end{equation*}\nwhere the descendant moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{(0,...,0,j_k,0,...,0)}_r\\subset\\overline{\\operatorname{\\mathcal{M}}}_r$ are defined in the same way as the one-point descendant $\\overline{\\operatorname{\\mathcal{M}}}^{j_k}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ by looking at the $r$ tautological line bundles over the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_r = \\overline{\\operatorname{\\mathcal{M}}}_r(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}$ separately and forgetting about the other points. \\\\\n\nWith this we can define the descendant Hamiltonian of SFT, which we will continue denoting by $\\operatorname{\\mathbf{H}}$, while the Hamiltonian defined in [EGH] will from now on be called {\\it primary}. In order to keep track of the descendants we will assign to every chosen differential form $\\theta_i$ now a sequence of formal variables $t_{i,j}$ with grading \n\\begin{equation*} |t_{i,j}|=2(1-j) -\\deg\\theta_i. \\end{equation*} \nThen the descendant Hamiltonian $\\operatorname{\\mathbf{H}}$ of SFT is defined by \n\\begin{equation*}\n \\operatorname{\\mathbf{H}} = \\sum_{\\Gamma^+,\\Gamma^-,I} \\int_{\\overline{\\operatorname{\\mathcal{M}}}^{(j_1,...,j_r)}_{g,r}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}}\n \\operatorname{ev}_1^*\\theta_{i_1}\\wedge...\\wedge\\operatorname{ev}_r^*\\theta_{i_r}\\; \\hbar^{g-1}t^Ip^{\\Gamma^+}q^{\\Gamma^-},\n\\end{equation*}\nwhere $p^{\\Gamma^+}=p_{\\gamma^+_1} ... p_{\\gamma^+_{n^+}}$, $q^{\\Gamma^-}=q_{\\gamma^-_1} ... q_{\\gamma^-_{n^-}}$ and $t^I=t_{i_1,j_1} ... t_{i_r,j_r}$ for $I=((i_1,j_1),...,(i_r,j_r))$. \\\\\n\nNote that expanding the Hamiltonian $\\operatorname{\\mathbf{H}}$ in powers of the formal variables $t_{i,j}$,\n\\begin{equation*} \n \\operatorname{\\mathbf{H}} = \\operatorname{\\mathbf{H}}^0 + \\sum_{i,j} t_{i,j}\\operatorname{\\mathbf{H}}^1_{i,j} + o(t^2), \n\\end{equation*}\nwe get back our Hamiltonians $\\operatorname{\\mathbf{H}}^0$ and the sequences of descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_{i,j}$ from above and it is easy to see that the primary Hamiltonian from [EGH] is recovered by setting all formal variables $t_{i,j}$ with $j>0$ equal to zero. \\\\\n\nIn the same way as it was shown for the primary Hamiltonian in [EGH], the descendant Hamiltonian continues to satisfy the master equation $[\\operatorname{\\mathbf{H}},\\operatorname{\\mathbf{H}}]=0$, which is just a generalization of the identities for $\\operatorname{\\mathbf{H}}^0$, $\\operatorname{\\mathbf{H}}^1_{i,j}$ and hence can be shown along the same lines by studying the codimension-one boundaries of descendant moduli spaces. On the other hand, expanding $\\operatorname{\\mathbf{H}}\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}$ in terms of powers of $\\hbar$, \n\\begin{equation*} \\operatorname{\\mathbf{H}}=\\sum_g \\hbar^{g-1}\\operatorname{\\mathbf{H}}_g, \\end{equation*} \nnote that for the rational descendant Hamiltonian $\\operatorname{\\mathbf{h}}=\\operatorname{\\mathbf{H}}_0\\in\\operatorname{\\mathfrak{P}}$ we still have $\\{\\operatorname{\\mathbf{h}},\\operatorname{\\mathbf{h}}\\}=0$. \\\\\n\n\\subsection{Invariance statement}\n\nWe now turn to the question of independence of these nice algebraic structures from the choices like contact form, cylindrical almost complex structure, abstract polyfold perturbations and, of course, the choice of the coherent collection of sections. This is the content of the following theorem, where we however again want to emphasize that the following statement is not yet a theorem in the strict mathematical sense as the analytical foundations of symplectic field theory, in particular, the neccessary transversality theorems for the Cauchy-Riemann operator, are not yet fully established. \\\\ \n\\\\\n{\\bf Theorem 1.6:} {\\it For different choices of contact form $\\lambda^{\\pm}$, cylindrical almost complex structure $\\underline{J}^{\\pm}$ , abstract polyfold perturbations and sequences of coherent collections of sections $(s^{\\pm}_j)$ the resulting systems of commuting operators $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}$ on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $\\operatorname{\\mathbf{H}}^{1,+}_{i,j}$ on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ are isomorphic, i.e., there exists an isomorphism of the Weyl algebras $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ which maps $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ to $\\operatorname{\\mathbf{H}}^{1,+}_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$.} \\\\\n\nAs above we clearly also get a rational version of the invariance statement: \\\\\n\\\\\n{\\bf Corollary 1.7:} {\\it For different choices of contact form $\\lambda^{\\pm}$, cylindrical almost complex structure $\\underline{J}^{\\pm}$, abstract polyfold perturbations and sequences of coherent collections of sections $(s^{\\pm}_j)$ the resulting system of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^{1,-}_{i,j}$ on $H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ and $\\operatorname{\\mathbf{h}}^{1,+}_{i,j}$ on $H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$ are isomorphic, i.e., there exists an isomorphism of the Poisson algebras $H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ and $H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$ which maps $\\operatorname{\\mathbf{h}}^{1,-}_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ to $\\operatorname{\\mathbf{h}}^{1,+}_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$.} \\\\\n\nThis theorem is an extension of the theorem in [EGH] which states that for different choices of auxiliary data the small Weyl algebras $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ are isomorphic. On the other hand, assuming that the contact form, the cylindrical almost complex structure and also the abstract polyfold sections are fixed to have well-defined moduli spaces, the isomorphism of the homology algebras is the identity and hence the theorem states the sequence of commuting operators is indeed independent of the chosen sequences of coherent collections of sections $(s^{\\pm}_j)$, \n\\begin{equation*} \\operatorname{\\mathbf{H}}^{1,-}_{i,j} = \\operatorname{\\mathbf{H}}^{1,+}_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0). \\end{equation*}\n$ $\\\\\n\nFor the proof we have to extend the proof in [EGH] to include gravitational descendants. To this end we have to study sections in the tautological line bundles over moduli spaces of holomorphic curves in symplectic manifolds with cylindrical ends. \\\\\n \nLet $(W,\\omega)$ be a symplectic manifold with cylindrical ends $(\\operatorname{\\mathbb{R}}^+\\times V^+,\\lambda^+)$ and \n$(\\operatorname{\\mathbb{R}}^-\\times V^-,\\lambda^-)$ in the sense of [BEHWZ] which is equipped with an almost complex structure \n$\\underline{J}$ which agrees with the cylindrical almost complex structures $\\underline{J}^{\\pm}$ on $\\operatorname{\\mathbb{R}}^+\\times V^+$. Then we \nstudy $\\underline{J}$-holomorphic curves in $W$ which are asymptotically cylindrical over \nchosen collections of orbits $\\Gamma^{\\pm}=\\{\\gamma^{\\pm}_1,...,\n\\gamma^{\\pm}_{n^{\\pm}}\\}$ of the Reeb vector fields $R^{\\pm}$ in $V^{\\pm}$ as the $\\operatorname{\\mathbb{R}}^{\\pm}$-factor tends \nto $\\pm\\infty$, see [BEHWZ], and denote by $\\operatorname{\\mathcal{M}}_{g,r}(\\Gamma^+,\\Gamma^-)$ the corresponding \nmoduli space of genus $g$ curves with $r$ additional marked points ([BEHWZ],[EGH]). \nPossibly after choosing abstract perturbations using polyfolds, obstruction \nbundles or domain-dependent structures, which agree with chosen abstract perturbations in the boundary as described above, we find that \n$\\operatorname{\\mathcal{M}}_{g,r}(\\Gamma^+,\\Gamma^-)$ is a weighted branched manifold of dimension equal to the Fredholm index of the Cauchy-Riemann operator for $\\underline{J}$. Note that as remarked above we will for simplicity assume that moduli space is indeed a manifold with corners, since this will be sufficient for our example and we expect that all the upcoming \nconstructions can be generalized in an appropriate way. We further extend the chosen differential forms $\\theta^{\\pm}_1,...\\theta^{\\pm}_N$ on $V^{\\pm}$ to differential forms $\\theta_1,...,\\theta_N$ on $W$ as described in [EGH]. \\\\\n\nFrom now on let $\\overline{\\operatorname{\\mathcal{M}}}_r$ denote the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)$ of holomorphic curves in $W$ for chosen collections of Reeb orbits $\\Gamma^+,\\Gamma^-$. Note in particular that there is no longer an $\\operatorname{\\mathbb{R}}$-action on the moduli space which we have to quotient out. In order to distinguish these moduli spaces in non-cylindrical manifolds from those of holomorphic curves in the cylindrical manifolds, we will use the short-hand notation $\\overline{\\operatorname{\\mathcal{M}}}^{\\pm}_r$ for moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}$ of holomorphic curves in $\\operatorname{\\mathbb{R}}\\times V^{\\pm}$, respectively. Like in Gromov-Witten theory we can introduce $r$ tautological line bundles $\\operatorname{\\mathcal{L}}_1,...,\\operatorname{\\mathcal{L}}_r$, where the fibre of $\\operatorname{\\mathcal{L}}_i$ over a punctured curve $(u,z_1,...,z_r)\\in\\operatorname{\\mathcal{M}}_r$ in the noncompactified moduli space is again given by the cotangent line to the underlying closed Riemann surface at the $i$.th marked point and which formally can be defined as the pull-back of the vertical cotangent line bundle under the canonical section $\\sigma_i$ of $\\pi: \\overline{\\operatorname{\\mathcal{M}}}_{r+1}\\to\\overline{\\operatorname{\\mathcal{M}}}_r$ mapping to the $i$.th marked point in the fibre. \\\\\n\nFor notational simplicity let us again restrict to the case $r=1$. Following the compactness statement in [BEHWZ] the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}_1$ now consists of curves with one non-cylindrical level and one cylindrical level (in the sense of [BEHWZ]), whose moduli spaces can now be represented as products $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ or $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ of moduli spaces of strictly lower dimension, where the marked point sits on the first or the second level. Again note that here and in what follows for product moduli spaces the first index refers to the level and not to the genus of the curve. Furthermore it follows from the definition of the tautological line bundle $\\operatorname{\\mathcal{L}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_1$ that over the boundary components $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ it is given by \n\\begin{eqnarray*} \n \\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}} = \\pi_1^*\\operatorname{\\mathcal{L}}_1,&& \\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,1}} = \\pi_2^*\\operatorname{\\mathcal{L}}^+_2, \\\\\n \\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}^-_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}} = \\pi_1^*\\operatorname{\\mathcal{L}}^-_1,&&\\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}} = \\pi_2^*\\operatorname{\\mathcal{L}}_2, \n\\end{eqnarray*}\nwhere $\\operatorname{\\mathcal{L}}^{(-)}_1$, $\\operatorname{\\mathcal{L}}^{(+)}_2$ denotes the tautological line bundle over the moduli space $\\overline{\\operatorname{\\mathcal{M}}}^{(-)}_{1,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^{(+)}_{2,1}$ and $\\pi_1$, $\\pi_2$ is the projection onto the first or second factor, respectively. \\\\\n\nWith this we can now introduce collections of sections in (tensor products of) tautological line bundles coherently connecting two chosen coherent collections of sections. \\\\\n\\\\\n{\\bf Definition 1.8:} {\\it Let $W$ be a symplectic manifold with cylindrical ends $V^{\\pm}$ and let $(s_{\\pm})$ be two coherent collections of sections in the tautological line bundles $\\operatorname{\\mathcal{L}}^{\\pm}$ over all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{\\pm}_1$ of $\\underline{J}$-holomorphic curves with one additional marked point in the cylindrical manifolds $\\operatorname{\\mathbb{R}}\\times V^{\\pm}$. Assume that we have chosen transversal sections $s$ in the tautological line bundles $\\operatorname{\\mathcal{L}}$ over all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_1$ of $\\underline{J}$-holomorphic curves in the non-cylindrical manifold $W$ with one additional marked point. Then this collection of sections $(s)$ is called} coherently connecting $(s_-)$ and $(s_+)$ {\\it if for every section $s$ in $\\operatorname{\\mathcal{L}}$ over a moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$ the following holds: Over every codimension-one boundary component $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ of $\\overline{\\operatorname{\\mathcal{M}}}_1$ the section $s$ agrees with the pull-back $\\pi_1^*s_1$, $\\pi_1^*s^-_1$ or $\\pi_2^*s^+_2$, $\\pi_2^*s_2$ of the chosen sections $s_{1,(-)}$, $s_{2,(+)}$ in the tautological line bundles $\\operatorname{\\mathcal{L}}^{(-)}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}^{(-)}_{1,1}$, $\\operatorname{\\mathcal{L}}^{(+)}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}^{(+)}_{2,1}$, respectively.} \\\\\n\nNote that one can always find collections of sections $(s)$ coherently connecting given coherent collections of sections $(s_+)$ and $(s_-)$ as before by using induction on the dimension of the underlying moduli space. Indeed, for the induction step observe that the coherency condition again fixes the section on the boundary of the moduli space, so that the desired section can be obtained by simply extending the section from the boundary to the interior of the moduli space in an arbitrary way. \\\\\n \nFor a given coherently connecting collection of sections $(s)$ we will again define for every moduli space \n\\begin{equation*} \\overline{\\operatorname{\\mathcal{M}}}^1_1 = s^{-1}(0) \\subset \\overline{\\operatorname{\\mathcal{M}}}_1. \\end{equation*}\nAs an immediate consequence of the above definition we find that the components of the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}^1_1$ are given by products \n$\\overline{\\operatorname{\\mathcal{M}}}^1_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^{1,-}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{1,+}_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^1_{2,1}$, where $\\overline{\\operatorname{\\mathcal{M}}}^{1,(-)}_{1,1} = s_{1,(-)}^{-1}(0)$, $\\overline{\\operatorname{\\mathcal{M}}}^{1,(+)}_{2,1} = s_{2,(+)}^{-1}(0)$ for the section $s_{1,(-)}$ in $\\operatorname{\\mathcal{L}}^{(-)}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}^{(-)}_{1,1}$, $s_{2,(+)}$ in $\\operatorname{\\mathcal{L}}^{(+)}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}^{(+)}_{2,1}$, respectively. As before we can use this result as an induction start to obtain for every moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$ a sequence of nested subspaces $\\overline{\\operatorname{\\mathcal{M}}}^j_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$. \\\\\n\\\\\n{\\bf Definition 1.9:} {\\it Let $j\\in\\operatorname{\\mathbb{N}}$ and let $(s_{j,\\pm})$ be two coherent collections of sections in the $j$-fold tensor products $\\operatorname{\\mathcal{L}}^{\\pm,\\otimes j}$ of the tautological line bundles over the $j-1$.st gravitational descendants $\\overline{\\operatorname{\\mathcal{M}}}^{j-1,\\pm}_1 \\subset\\overline{\\operatorname{\\mathcal{M}}}^{\\pm}_1$ of all moduli spaces of curves in the cylindrical manifolds $\\operatorname{\\mathbb{R}}\\times V^{\\pm}$. Assume that for all moduli spaces of curves in the non-cylindrical manifold $W$ we have chosen $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ such that the components of the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ are given by products of the form $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^{j-1,-}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1,+}_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{2,0}$. Then we again call a collection of transversal sections $(s_j)$ in the j-fold tensor products $\\operatorname{\\mathcal{L}}^{\\otimes j}$ of the tautological line bundles over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset \\overline{\\operatorname{\\mathcal{M}}}_1$} coherently connecting $(s_{j,-})$ and $(s_{j,+})$ {\\it if for every section $s_j$ the following holds: Over every codimension-one boundary component $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^{j-1,-}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1,+}_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{2,1}$ of $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ the section $s_j$ agrees with the pull-back $\\pi_1^*s_{1,j}$, $\\pi_1^*s_{1,j,-}$ or $\\pi_2^*s_{2,j,+}$, $\\pi_2^*s_{2,j}$ of the section $s_{1,j,(-)}$, $s_{2,j,(+)}$ in the line bundle $\\operatorname{\\mathcal{L}}^{(-),\\otimes j}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1,(-)}_{1,1}$, $\\operatorname{\\mathcal{L}}^{(+),\\otimes j}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1,(+)}_{2,1}$, respectively.} \\\\\n\nWith this we can now introduce gravitational descendants of moduli spaces for symplectic manifolds with cylindrical ends. \\\\\n\\\\\n{\\bf Definition 1.10:} {\\it Assume that we have inductively defined subsequence of nested subspaces $\\overline{\\operatorname{\\mathcal{M}}}^j_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ by requiring that \n$\\overline{\\operatorname{\\mathcal{M}}}^j_1 = s_j^{-1}(0)\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ for a collection of sections $s_j$ in the line bundles $\\operatorname{\\mathcal{L}}^{\\otimes j}$ over the moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ coherently connecting the coherent collections of sections $(s_{j,-})$ and $(s_{j,+})$. Then we call $\\overline{\\operatorname{\\mathcal{M}}}^j_1$ the $j$.th (gravitational) descendant of $\\overline{\\operatorname{\\mathcal{M}}}_1$.} \\\\\n \nIn order to prove the above invariance theorem we now recall the extension of the algebraic formalism of SFT from cylindrical manifolds to symplectic cobordisms with cylindrical ends as described in [EGH]. \\\\\n\nLet $\\operatorname{\\mathfrak{D}}^0$ be the space of formal power series in the variables $\\hbar,p^+_{\\gamma}$ with coefficients which are polynomials in the \nvariables $q^-_{\\gamma}$. Elements in $\\operatorname{\\mathfrak{W}}^{0,\\pm}$ then act as differential operators from the right\/left on \n$\\operatorname{\\mathfrak{D}}^0$ via the replacements \n\\begin{equation*} \n q^+_{\\gamma}\\mapsto \\kappa_{\\gamma}\\hbar\\overleftarrow{\\frac{\\partial}{\\partial p^+_{\\gamma}}},\\;\\;\n p^-_{\\gamma}\\mapsto\\kappa_{\\gamma}\\hbar\\overrightarrow{\\frac{\\partial}{\\partial q^-_{\\gamma}}}.\n\\end{equation*}\n\nApart from the potential $\\operatorname{\\mathbf{F}}^0\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0$ counting only curves in $W$ with no additional marked points,\n\\begin{equation*} \\operatorname{\\mathbf{F}}^0 = \\sum_{\\Gamma^+,\\Gamma^-} \\#\\overline{\\operatorname{\\mathcal{M}}}_{g,0}(\\Gamma^+,\\Gamma^-)\\; \\hbar^{g-1} p^{\\Gamma^+}q^{\\Gamma^-}, \\end{equation*}\nwe now want to use the extensions $\\theta_i$, $i=1,...,N$ on $W$ of the chosen differential forms $\\theta^{\\pm}_1,...\\theta^{\\pm}_N$ on $V^{\\pm}$ and these sequences \n$\\overline{\\operatorname{\\mathcal{M}}}^j_1 = \\overline{\\operatorname{\\mathcal{M}}}^j_{g,1}(\\Gamma^+,\\Gamma^-)$ of gravitational descendants to define sequences of new SFT potentials $\\operatorname{\\mathbf{F}}^1_{i,j}$, $(i,j)\\in\\{1,...,N\\}\\times \\operatorname{\\mathbb{N}}$, by\n\\begin{equation*} \\operatorname{\\mathbf{F}}^1_{i,j} = \\sum_{\\Gamma^+,\\Gamma^-} \\int_{\\overline{\\operatorname{\\mathcal{M}}}^j_{g,1}(\\Gamma^+,\\Gamma^-)}\\operatorname{ev}^*\\theta_i\\; \\hbar^{g-1} p^{\\Gamma^+}q^{\\Gamma^-}. \\end{equation*}\n\nFor the potential counting curves with no additional marked points we have the following identity, where we however again want to emphasize that the following statement should again be understood as a theorem up to the transversality problem in SFT. \\\\\n$ $\\\\\n{\\bf Theorem ([EGH]):} {\\it The potential $\\operatorname{\\mathbf{F}}^0\\in\\hbar^{-1}\\operatorname{\\mathfrak{D}}$ satisfies the master equation}\n\\begin{equation*} e^{\\operatorname{\\mathbf{F}}^0}\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}} - \\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}e^{\\operatorname{\\mathbf{F}}^0} = 0. \\end{equation*}\n\nIn [EGH] it is shown that this implies that \n\\begin{equation*}\n D^{\\operatorname{\\mathbf{F}}^0}: \\hbar^{-1}\\operatorname{\\mathfrak{D}}^0\\to\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,\\; \n D^{\\operatorname{\\mathbf{F}}^0}g = e^{-\\operatorname{\\mathbf{F}}^0}\\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}(ge^{\\operatorname{\\mathbf{F}}^0}) - (-1)^{|g|}(ge^{\\operatorname{\\mathbf{F}}^0})\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}}e^{-\\operatorname{\\mathbf{F}}^0} \n\\end{equation*}\nsatisfies $D^{\\operatorname{\\mathbf{F}}^0}\\circ D^{\\operatorname{\\mathbf{F}}^0} = 0$ and hence can be used to define the homology algebra \n$H_*(\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,D^{\\operatorname{\\mathbf{F}}^0})$. Furthermore it is shown that the maps \n\\begin{eqnarray*}\n &&F^{0,-}: \\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-}\\to\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,\\; f\\mapsto e^{-\\operatorname{\\mathbf{F}}^0}\\overrightarrow{f}e^{+\\operatorname{\\mathbf{F}}^0}, \\\\\n &&F^{0,+}: \\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+}\\to\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,\\; f\\mapsto e^{+\\operatorname{\\mathbf{F}}^0}\\overleftarrow{f}e^{-\\operatorname{\\mathbf{F}}^0}\n\\end{eqnarray*}\ncommute with the boundary operators,\n\\begin{equation*} F^{0,\\pm}\\circ D^{0,\\pm} = D^{\\operatorname{\\mathbf{F}}^0}\\circ F^{0,\\pm}, \\end{equation*} \nand hence descend to maps between the homology algebras\n\\begin{equation*} F^{0,\\pm}_*: H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,\\pm},D^{0,\\pm})\\to H_*(\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,D^{\\operatorname{\\mathbf{F}}^0}). \\end{equation*}\n\nNow assume that the contact forms $\\lambda^+$ and $\\lambda^-$ are chosen such that they define the same contact structure $(V^+,\\xi^+)=(V^-,\\xi^-)=:(V,\\xi)$ and let $W=\\operatorname{\\mathbb{R}}\\times V$ be the topologically trivial cobordism. Then in [EGH] the authors prove (up to transversality) the following fundamental theorem. \\\\\n\\\\\n{\\bf Theorem ([EGH]):} {\\it The map}\n\\begin{equation*} (F^{0,+}_*)^{-1}\\circ F^{0,-}_*: H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})\\to H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+}) \\end{equation*}\n{\\it is an isomorphism of graded Weyl algebras.}\\\\\n \nFor the proof of the invariance statement we want to show that this map identifies the sequences $\\operatorname{\\mathbf{H}}^{1,\\pm}_{i,j}$, $(i,j)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$ on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,\\pm},D^{0,\\pm})$. In order to get the right idea for the proof, it turns out to be useful to even enlarge the picture as follows. \\\\\n\nPrecisely in the same way as for cylindrical manifolds we can define for every tuple $(j_1,...,j_r)$ of natural numbers gravitational descendants $\\overline{\\operatorname{\\mathcal{M}}}^{(j_1,...,j_r)}\\subset \\overline{\\operatorname{\\mathcal{M}}}_1$ of moduli spaces of curves in non-cylindrical manifolds with more than one additional marked point, which are collected in the {\\it descendant potential} $\\operatorname{\\mathbf{F}}\\in\\hbar^{-1}\\operatorname{\\mathfrak{D}}$, where $\\operatorname{\\mathfrak{D}}$ is again obtained from $\\operatorname{\\mathfrak{D}}^0$ by considering coefficients which are formal powers in the graded formal variables $t_{i,j}$, $(i,j)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$. \\\\\n\nAssuming for the moment that we have proven the fundamental identity \n\\begin{equation*} e^{\\operatorname{\\mathbf{F}}}\\overleftarrow{\\operatorname{\\mathbf{H}}^+} - \\overrightarrow{\\operatorname{\\mathbf{H}}^-}e^{\\operatorname{\\mathbf{F}}} = 0 \\end{equation*}\nand expanding the potential $\\operatorname{\\mathbf{F}}\\in\\hbar^{-1}\\operatorname{\\mathfrak{D}}$ and the two Hamiltonians $\\operatorname{\\mathbf{H}}^{\\pm}\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{\\pm}$ in \npowers of the $t$-variables,\n\\begin{equation*} \n \\operatorname{\\mathbf{F}} = \\operatorname{\\mathbf{F}}^0 + \\sum_{i,j} t_{i,j}\\operatorname{\\mathbf{F}}^1_{i,j} + o(t^2), \\;\\;\n \\operatorname{\\mathbf{H}}^{\\pm} = \\operatorname{\\mathbf{H}}^{0,\\pm} + \\sum_{i,j} t_{i,j}\\operatorname{\\mathbf{H}}^{1,\\pm}_{i,j} + o(t^2), \\\\\n\\end{equation*}\nwe can deduce besides the master equation for $\\operatorname{\\mathbf{F}}^0$,\n\\begin{equation*} \n e^{\\operatorname{\\mathbf{F}}^0}\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}} - \\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}e^{\\operatorname{\\mathbf{F}}^0} = 0 \n\\end{equation*}\nand other identities also the identity \n\\begin{equation*} \n e^{\\operatorname{\\mathbf{F}}^0}\\overleftarrow{\\operatorname{\\mathbf{H}}^{1,+}_{i,j}} - \\overrightarrow{\\operatorname{\\mathbf{H}}^{1,-}_{i,j}}e^{\\operatorname{\\mathbf{F}}^0} = \n \\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}(e^{\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j}) - (e^{\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j})\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}}, \n\\end{equation*} \nabout $\\operatorname{\\mathbf{F}}^0$, $\\operatorname{\\mathbf{F}}^1_{i,j}$ and $\\operatorname{\\mathbf{H}}^{0,\\pm}$, $\\operatorname{\\mathbf{H}}^{1,\\pm}_{i,j}$, where we used that\n\\begin{equation*} \n e^{\\operatorname{\\mathbf{F}}}= e^{\\operatorname{\\mathbf{F}}^0}\\cdot(1+\\sum_{i,j} t_{i,j}\\operatorname{\\mathbf{F}}^1_{i,j}) + o(t^2). \n\\end{equation*}\n\n{\\it Proof of the theorem:} Instead of proving the master equation for the full descendant potential $\\operatorname{\\mathbf{F}}$, we first show that it suffices to prove \n\\begin{equation*} \n e^{\\operatorname{\\mathbf{F}}^0}\\overleftarrow{\\operatorname{\\mathbf{H}}^{1,+}_{i,j}} - \\overrightarrow{\\operatorname{\\mathbf{H}}^{1,-}_{i,j}}e^{\\operatorname{\\mathbf{F}}^0} = \n \\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}(e^{\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j}) - (e^{\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j})\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}}. \n\\end{equation*} \nIndeed, it is easy to see that the desired identity implies that\n\\begin{equation*} \n F^{0,+}(\\operatorname{\\mathbf{H}}^{1,+}_{i,j}) - F^{0,-}(\\operatorname{\\mathbf{H}}^{1,-}_{i,j}) = \n e^{+\\operatorname{\\mathbf{F}}^0}\\overleftarrow{\\operatorname{\\mathbf{H}}^{1,+}_{i,j}}e^{-\\operatorname{\\mathbf{F}}^0} - e^{-\\operatorname{\\mathbf{F}}^0}\\overrightarrow{\\operatorname{\\mathbf{H}}^{1,-}_{i,j}}e^{+\\operatorname{\\mathbf{F}}^0} \n\\end{equation*}\nis equal to \n\\begin{equation*}\n e^{-\\operatorname{\\mathbf{F}}^0}\\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}(e^{+\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j}) - (e^{+\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j})\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}}e^{-\\operatorname{\\mathbf{F}}^0} \n = D^{\\operatorname{\\mathbf{F}}^0}(\\operatorname{\\mathbf{F}}^1_{i,j}),\n\\end{equation*}\nso that, after passing to homology, we have\n\\begin{equation*} F^{0,+}_*(\\operatorname{\\mathbf{H}}^{1,+}_{i,j}) = F^{0,-}_*(\\operatorname{\\mathbf{H}}^{1,-}_{i,j}) \\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,D^{\\operatorname{\\mathbf{F}}^0}) \\end{equation*}\nas desired. \\\\\n\nOn the other hand, the above identity directly follows from our definition of gravitational descendants of moduli spaces based on the definition of coherently connecting sections in tautological line bundles and the compactness theorem in [BEHWZ]. Indeed, in the same way as it is shown in [EGH] that the master equation for $\\operatorname{\\mathbf{F}}^0$ and $\\operatorname{\\mathbf{H}}^{0,\\pm}$ follows from the fact that the codimension-one boundary of every moduli space $\\overline{\\operatorname{\\mathcal{M}}}_0$ is formed by products of moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, the desired identity relating $\\operatorname{\\mathbf{F}}^0$, $\\operatorname{\\mathbf{F}}^1_{i,j}$ and $\\operatorname{\\mathbf{H}}^{0,\\pm}$, $\\operatorname{\\mathbf{H}}^{1,\\pm}_{i,j}$ can be seen to follow from the fact that the codimension-one boundary of a descendant moduli space $\\overline{\\operatorname{\\mathcal{M}}}^j_1$ is given by products of the form $\\overline{\\operatorname{\\mathcal{M}}}^j_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^{j,-}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j,+}_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^j_{2,1}$: While the two summands involving $\\operatorname{\\mathbf{F}}^0$ and $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}$, $\\operatorname{\\mathbf{H}}^{1,+}_{i,j}$ on the left-hand-side of the equation collect all boundary components of the form $\\overline{\\operatorname{\\mathcal{M}}}^{j,-}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j,+}_{2,1}$, the two summands involving $\\operatorname{\\mathbf{F}}^1_{i,j}$ and $\\operatorname{\\mathbf{H}}^{0,-}$, $\\operatorname{\\mathbf{H}}^{0,+}$ on the right-hand-side of the equation collect all boundary components of the form $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^j_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^j_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, respectively. Note that as for the master equation for $\\operatorname{\\mathbf{F}}^0$ and $\\operatorname{\\mathbf{H}}^{0,\\pm}$ the appearance of $\\operatorname{\\mathbf{F}}^0$ in the exponential follows from the fact that there corresponding curves may appear with an arbitrary number of connected components, while the curves counted for in $\\operatorname{\\mathbf{H}}^{0,\\pm}$, $\\operatorname{\\mathbf{H}}^{1,\\pm}_{i,j}$, $\\operatorname{\\mathbf{F}}^1_{i,j}$ can only appear once due to index reasons or since there is just one additional marked point. \\\\\n\nFinally, in order to see why we actually have $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}=\\operatorname{\\mathbf{H}}^{1,+}_{i,j}$ on homology if we fixed $\\lambda^-=\\lambda^+=\\lambda$, $\\underline{J}^-=\\underline{J}^+=\\underline{J}$ and the abstract polyfold perturbations to have well-defined moduli spaces, observe that in this case $\\operatorname{\\mathbf{F}}^0$ just counts orbit cylinders, so that $F^{0,\\pm}$ and hence $(F^{0,\\pm})_*$ is the identity. $\\qed$ \n \n\n\\subsection{The circle bundle case}\n\nIn this subsection we briefly want to discuss the important case of circle bundles over closed symplectic manifolds, which links our constructions to gravitational descendants in Gromov-Witten theory, see also [R]. \\\\\n\nFor this recall that to any closed symplectic manifold $(M,\\omega)$ with integral symplectic form $[\\omega]\\in H^2(M,\\operatorname{\\mathbb{Z}})$ one can canonically assign a principal circle bundle $\\pi: V\\to M$ over $(M,\\omega)$ by requiring that $c_1(V)=[\\omega]$. Furthermore, it is easy to see that an $S^1$-connection form $\\lambda$ with curvature $\\omega$ on $\\pi:V\\to M$ is a contact form on the total space $V$, where the underlying contact structure agrees with the corresponding horizontal plane field $\\xi=\\ker\\lambda$, while the Reeb vector field $R$ agrees with the infinitesimal generator of the $S^1$-action. Observe that a $\\omega$-compatible almost complex structure $J$ on $M$ naturally equips $\\operatorname{\\mathbb{R}}\\times V$ with a cylindrical almost complex structure by requiring that $\\underline{J}$ maps the Reeb vector field to the $\\operatorname{\\mathbb{R}}$-direction and agrees with $J$ on the horizontal plane field $\\xi$, which is naturally identified with $TM$. \\\\\n\nSince every fibre of the circle bundle is hence a closed Reeb orbit for the contact form $\\lambda$, it follows that the space of orbits is given by $M\\times\\operatorname{\\mathbb{N}}$, where the second factor just refers to the multiplicity of the orbit. Hence, while every contact form in this class is not Morse as long as the symplectic manifold is not a point, it is still of Morse-Bott type. \\\\\n\nFollowing [EGH] the Weyl algebra $\\operatorname{\\mathfrak{W}}^0$ in this Morse-Bott case is now generated by sequences of graded formal variables $p_{\\alpha,k}$, $q_{\\alpha,k}$, $k\\in\\operatorname{\\mathbb{N}}$ assigned to cohomology classes $\\alpha$ forming a basis of $H^*(M,\\operatorname{\\mathbb{Z}})$. For circle bundles in the Morse-Bott setup we now show that the general theorem from above leads the following stronger statement. Note that in the following theorem we do {\\it not} assume that the sequences of coherent collections of sections are neccessarily $S^1$-invariant. \\\\\n\\\\\n{\\bf Theorem 1.11:} {\\it For circle bundles over symplectic manifolds, which are equipped with $S^1$-invariant contact forms, cylindrical almost complex structures (and abstract polyfold perturbations) as described above, the descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_{i,j}$ define a sequence of commuting operators on $\\operatorname{\\mathfrak{W}}^0$, which is independent of the auxiliary data.} \\\\\n\\\\\n{\\it Proof:} Observing that a map $\\tilde{u}:(\\Sigma,j)\\to (\\operatorname{\\mathbb{R}}\\times V,\\underline{J})$ from a punctured Riemann sphere to the cylindrical manifold $\\operatorname{\\mathbb{R}}\\times V$, which is equipped with the canonical cylindrical almost complex structure $\\underline{J}$ defined by the $\\omega$-compatible almost complex structure $J$ on $M$, can be viewed as tuple $(h,u)$, where $u:(\\Sigma,j)\\to(M,J)$ is a $J$-holomorphic curve in $M$ and $h$ is a holomorphic section in $\\operatorname{\\mathbb{R}}\\times u^*V\\to\\Sigma$, it is easy to see that every moduli spaces studied in SFT for the contact manifold $V$ carries a natural circle bundle structure after quotienting out the natural $\\operatorname{\\mathbb{R}}$-action. It follows that $D^0=0$, so that by our first theorem the $\\operatorname{\\mathbf{H}}^1_j$ already commute as elements in $\\operatorname{\\mathfrak{W}}^0$. On the other hand, as long as the two different collections of auxiliary structures for $V$ are actually obtained as pull-backs of the corresponding auxiliary structures on $M$, it follows in the same way that the only rigid holomorphic curves in the resulting cobordisms are the orbit cylinders, so that the resulting automorphism is indeed the identity. $\\qed$ \\\\ \n \nFor $S^1$ and $S^3$ Eliashberg already pointed out in his ICM 2006 talk, see [E], that the corresponding sequences $\\operatorname{\\mathbf{h}}^1_j$ counting only genus zero curves lead to classical integrable systems, while the sequences of commuting operators $\\operatorname{\\mathbf{H}}^1_j$ provide deformation quantizations for these hierarchies. This is based on the surprising fact that the sequence $\\operatorname{\\mathbf{h}}^1_j$ of Poisson-commuting functions actually agrees with integrable system for genus zero from Gromov-Witten theory obtained using the underlying Frobenius manifold structure. In particular, for $V=S^1$ it follows that that the resulting system of Poisson-commuting functions are precisely the commuting integrals of the dispersionless KdV hierarchy,\n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_j = \\oint_{S^1} \\frac{u^{j+2}(x)}{(j+2)!}\\,dx,\\;\\;u(x)=\\sum_{n\\in\\operatorname{\\mathbb{N}}} p_n\\;e^{+2\\pi inx}+ q_n\\;e^{-2\\pi inx}, \\end{equation*} \nwhile in the case of the Hopf fibration $V=S^3$ over $M=S^2$ one arrives at the Poisson-commuting integrals of the continuous limit \nof the Toda lattice. \\\\\n\nIn order to see why in genus zero the SFT of the circle bundle $V$ is so closely related to the Gromov-Witten theory of its symplectic base $M$, we recall from the proof of the theorem that every $\\underline{J}$-holomorphic curve $\\tilde{u}$ can be identified with a tuple $(h,u)$, where $u$ is a $J$-holomorphic curve in $M$ and $h$ is a holomorphic section in $\\operatorname{\\mathbb{R}}\\times u^*V\\to\\Sigma$, whose poles and zeroes correspond to the positive and negative punctures with multiplicities. Since the zeroth Picard group of $S^2$ is trivial and hence every degree zero divisor is indeed a principal divisor, it follows that for every map $u$ the space of sections is isomorphic to $\\operatorname{\\mathbb{C}}$ and hence that the SFT moduli space of $\\underline{J}$-holomorphic curves in $\\operatorname{\\mathbb{R}}\\times V$ is indeed a circle bundle over the corresponding Gromov-Witten moduli space of $J$-holomorphic curves in $M$. \\\\\n\nWhile this explains the close relation of SFT of circle bundles and Gromov-Witten theory in the genus zero case, the non-triviality of the Picard group for nonzero genus implies that the relation gets much more obscure when we allow for curves of arbitrary genus. Indeed, while in the case of $V=S^1$ the sequence $\\operatorname{\\mathbf{H}}^1_j$ defined by counting curves of arbitrary genus in $\\operatorname{\\mathbb{R}}\\times V$ leads to the deformation quantization of the dispersionless KdV hierarchy, in particular, a quantum integrable system, counting curves of all genera in the underlying symplectic manifold, that is, the point, leads by Witten's conjecture to the classical integrable system given by the full KdV hierarchy as proven by Kontsevich . \\\\\n\nAt the end of this subsection we again want to emphasize that the above statement crucially relies on the fact that $V$ is equipped with a $S^1$-invariant contact form, cylindrical almost complex structure and abstract polyfold perturbations. Assuming for the moment that the sequences of coherent collections of sections are also chosen to be $S^1$-invariant, note that in this case the above invariance statement can directly be deduced from the independence of the descendant Gromov-Witten potential of the auxiliary data used to define it, which essentially relies on the fact that all moduli spaces have only boundary components of codimension greater or equal to two, so that absolute rather than relative virtual classes are defined. In particular, the gravitational descendants can be defined by integrating powers of the first Chern class over the absolute moduli cycle. On the other hand, recall that for the above theorem we did {\\it not} require that the sequences of coherent collections of sections are neccessarily $S^1$-invariant. While our definition of coherent collections of sections seems to be very weak, our above theorem shows that the nice invariance property continues to hold even for a larger class of sections. \n\n\n\\section{Example: Symplectic field theory of closed geodesics} \n\n\\subsection{Symplectic field theory of a single Reeb orbit}\nWe are now going to consider a concrete example, which actually formed the starting point for the formal discussion from above. \\\\\n\nAs above consider a closed contact manifold $V$ with chosen contact form $\\lambda\\in\\Omega^1(V)$ and let $\\underline{J}$ be a compatible cylindrical almost complex structure on $\\operatorname{\\mathbb{R}}\\times V$. For any closed orbit $\\gamma$ of the corresponding Reeb vector field $R$ on $V$ the \norbit cylinder $\\operatorname{\\mathbb{R}}\\times\\gamma$ together with its branched covers are the basic examples of $\\underline{J}$-holomorphic curves in $\\operatorname{\\mathbb{R}}\\times V$. \\\\\n \nIn [F2] we prove that these {\\it orbit curves} do not contribute to the algebraic invariants of symplectic field theory as long as they do not carry additional marked points. Our proof explicitly uses that the orbit curves (over a fixed orbit) are closed under taking boundaries and gluing, which follows from the fact that orbit curves are also trivial in the sense that they have trivial contact area and that this contact area is preserved under taking boundaries and gluing. In particular, it follows, see [F2], that every algebraic invariant of symplectic field theory has a natural analog defined by counting only orbit curves. Further specifying the underlying Reeb orbit let us hence introduce the {\\it symplectic field theory of the Reeb orbit $\\gamma$:} \\\\ \n\nFor this denote by $\\operatorname{\\mathfrak{W}}^0_{\\gamma}$ be the graded Weyl subalgebra of the Weyl algebra $\\operatorname{\\mathfrak{W}}$, which is generated only by those $p$- and \n$q$-variables $p_n=p_{\\gamma^n}$, $q_n=q_{\\gamma^n}$ corresponding to Reeb orbits which are multiple covers of the fixed orbit $\\gamma$ {\\it and which are good in the sense of [BM]}. In the same way we further introduce the Poisson subalgebra $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ of $\\operatorname{\\mathfrak{P}}^0$. It will become important that the natural identification of the formal variables $p_n$ and $q_n$ does {\\it not} lead to an isomorphism of the graded algebras $\\operatorname{\\mathfrak{W}}^0_{\\gamma}$ and $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ with the corresponding graded algebras $\\operatorname{\\mathfrak{W}}^0_{S^1}$ and $\\operatorname{\\mathfrak{P}}^0_{S^1}$ for $\\gamma=V=S^1$, not only since the gradings of $p_n$ and $q_n$ are different and hence even the commutation rules may change but also that variables $p_n$ and $q_n$ may not be there since they would correspond to bad orbits. \\\\\n\nIn the same way as we introduced the (rational) Hamiltonian $\\operatorname{\\mathbf{H}}^0$ and $\\operatorname{\\mathbf{h}}^0$ as well as sequences of descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_j$ and $\\operatorname{\\mathbf{h}}^1_j$ by counting general curves in the symplectization of a contact manifold, we can define distinguished elements $\\operatorname{\\mathbf{H}}^0_{\\gamma}\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0_{\\gamma}$ and $\\operatorname{\\mathbf{h}}^0_{\\gamma}\\in\\operatorname{\\mathfrak{P}}^0_{\\gamma}$, as well as sequences of descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_{\\gamma,j}$ and $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ by just counting branched covers of the orbit cylinder over $\\gamma$ with signs \n(and weights), where the preservation of the contact area under splitting and gluing of curves proves that for every theorem from above we have a version for $\\gamma$. \\\\\n\nWhile for the general part described above we have already emphasized that the theorems are not yet theorems in the strict mathematical sense since the neccessary transversality theorems for the Cauchy-Riemann operator are part of the on-going polyfold project by Hofer and his collaborators and we further used the assumption that all occuring moduli spaces are manifolds with corners, for the rest of this paper we will {\\it restrict to the rational case}, i.e., we will only be interested in the Poisson-commuting sequences $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ on $H_*(\\operatorname{\\mathfrak{P}}^0_{\\gamma},d^0_{\\gamma})$, but in return solve the occuring analytical problems in all detail. In particular, we have already proven in the paper [F2] that for (rational) orbit curves the transversality problem can indeed be solved using finite-dimensional obstruction bundles instead of infinite-dimensional polybundles. In order to see why this is even neccessary, observe that while in the case when $\\gamma=V=S^1$ the Fredholm index equals the dimension of the moduli space, for general $\\gamma\\subset V$ the Fredholm index of a true branched cover is in general strictly smaller than the dimension of the moduli space of branched covers, so that transversality for the Cauchy-Riemann operator can in general not be satisfied. \\\\\n\nSo let us recall the main results about obstruction bundle transversality for orbit curves, where we refer to [F2] for all details. The first observation for orbit curves is that the cokernels of the linearized Cauchy-Riemann operators indeed fit together to give a smooth vector bundle $\\overline{\\operatorname{Coker}} \\bar{\\partial}_{\\underline{J}}$ over the compactified (nonregular) moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}$ of orbit curves (of constant rank). It follows that every transveral section $\\bar{\\nu}$ of this cokernel bundle leads to a compact perturbation making the Cauchy-Riemann operator transversal to the zero section in the underlying polyfold setup. \\\\\n\nIn Gromov-Witten theory we would hence obtain the contribution of the regular perturbed moduli space by integrating the Euler class of the finite-dimensional obstruction bundle over the compactified moduli space. On the other hand, passing from Gromov-Witten theory back to symplectic field theory again, we see that we just arrive at the same problem we had to face with when we wanted to define gravitational descendants in symplectic field theory. Indeed, as for the tautological line bundles, the presence of codimension-one boundary of the (nonregular) moduli spaces of branched covers implies that Euler numbers for sections in the cokernel bundles are not defined in general, since the count of zeroes depends on the compact perturbations chosen for the moduli spaces in the boundary. \\\\\n\nInstead of looking at a single moduli space, we hence again have to consider all moduli spaces at once. Replacing the tautological line bundle $\\operatorname{\\mathcal{L}}$ by the cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ and considering the nonregular moduli space of branched covers instead of the regular moduli space itself, we hence now define {\\it coherent} collections of sections in the obstruction bundles $\\overline{\\operatorname{Coker}}\\bar{\\partial}_J$ over all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}$ as follows. \\\\\n\nFollowing the compactness statement in [BEHWZ] for the contact manifold $S^1$ the codimension-one boundary of every moduli space of branched covers $\\overline{\\operatorname{\\mathcal{M}}}$ again consists of curves with two levels (in the sense of [BEHWZ]), whose moduli spaces can be represented as products $\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2$ of moduli spaces of strictly lower dimension, where the first index again refers to the level. On the other hand, it follows from the linear gluing result in [F2] that over the boundary component $\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2$ the cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ is given by \n\\begin{equation*} \\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}|_{\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2} = \\pi_1^*\\overline{\\operatorname{Coker}}^1\\bar{\\partial}_{\\underline{J}}\\oplus\\pi_2^*\\overline{\\operatorname{Coker}}^2\\bar{\\partial}_{\\underline{J}}, \\end{equation*}\nwhere $\\overline{\\operatorname{Coker}}^1\\bar{\\partial}_{\\underline{J}}$, $\\overline{\\operatorname{Coker}}^2\\bar{\\partial}_{\\underline{J}}$ denotes the cokernel bundle over the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$, $\\overline{\\operatorname{\\mathcal{M}}}_2$ and $\\pi_1$, $\\pi_2$ is the projection onto the first or second factor, respectively. \\\\\n\nAssuming that we have chosen sections $\\bar{\\nu}$ in the cokernel bundles $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ over all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}$ of branched covers, we again call this collection of sections $(\\bar{\\nu})$ {\\it coherent} if over every codimension-one boundary component $\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2$ of a moduli space $\\overline{\\operatorname{\\mathcal{M}}}$ the corresponding section $\\bar{\\nu}$ agrees with the pull-back $\\pi_1^*\\bar{\\nu}_1\\oplus\\pi_2^*\\bar{\\nu}_2$ of the chosen sections $\\bar{\\nu}_1$, $\\bar{\\nu}_2$ in the cokernel bundles $\\overline{\\operatorname{Coker}}^1\\bar{\\partial}_{\\underline{J}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_1$, $\\overline{\\operatorname{Coker}}^2\\bar{\\partial}_{\\underline{J}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_2$, respectively. \\\\\n\nSince in the end we will again be interested in the zero sets of these sections, we will again assume that all occuring sections are transversal to the zero section.\nAs before it is not hard to see that one can always find such coherent collections of (transversal) sections in the cokernel bundles by using induction on the dimension of the underlying nonregular moduli space of branched covers. Note that the latter is {\\it not} equal to the Fredholm index. \\\\\n\nIn [F2] we prove the following result about orbit curves with {\\it no} additional marked points. \\\\\n\\\\\n{\\bf Theorem ([F2]):} {\\it For the cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_J$ over the compactification $\\overline{\\operatorname{\\mathcal{M}}}$ of every moduli space \nof branched covers over an orbit cylinder with $\\dim\\operatorname{\\mathcal{M}}-\\operatorname{rank}\\operatorname{Coker}\\bar{\\partial}_J=0$ the following holds:}\n\\begin{itemize} \n\\item {\\it For every pair $\\bar{\\nu}^0$, $\\bar{\\nu}^1$ of coherent and transversal sections in $\\overline{\\operatorname{Coker}}\\bar{\\partial}_J$ the algebraic count of zeroes \n of $\\bar{\\nu}^0$ and $\\bar{\\nu}^1$ are finite and agree, so that we can define an Euler number \n $\\chi(\\overline{\\operatorname{Coker}}\\bar{\\partial}_J)$ for coherent sections in $\\overline{\\operatorname{Coker}}\\bar{\\partial}_J$ by} \n \\begin{equation*} \\chi(\\overline{\\operatorname{Coker}}\\bar{\\partial}_J) \\,:=\\, \\sharp (\\bar{\\nu}^0)^{-1}(0) \\,=\\, \\sharp (\\bar{\\nu}^1)^{-1}(0). \n \\end{equation*} \n\\item {\\it This Euler number is $\\chi(\\overline{\\operatorname{Coker}}\\bar{\\partial}_J) = 0$.} \n\\end{itemize}\n$ $\\\\\nThis theorem in turn has the following consequence.\\\\\n\\\\\n{\\bf Corollary 2.1:} { \\it For every closed Reeb orbit $\\gamma$ the Hamiltonian $\\operatorname{\\mathbf{h}}^0_{\\gamma}$ vanishes independently of the chosen coherent collection of sections $(\\bar{\\nu})$ in the cokernel bundles over all moduli spaces of branched covers,}\n\\begin{equation*} \\operatorname{\\mathbf{h}}^0 = \\operatorname{\\mathbf{h}}^{0,\\bar{\\nu}}= 0. \\end{equation*}\n{\\it In particular, the sequences of descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ already Poisson-commute as elements in $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$.}\\\\\n\nNote that the latter statement is obvious in the case $\\gamma=V=S^1$. While it directly follows from index reasons that $\\operatorname{\\mathbf{h}}^1_{S^1,j}=0$ when the string of differential forms just consists of the zero-form $1$ on $S^1$, it is shown in [E] using the results from Okounkov and Pandharipande in [OP] that for the one-form $dt$ on $S^1$ the system of Poisson commuting functions on $\\operatorname{\\mathfrak{P}}^0_{S^1}$ is given by\n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{S^1,j} = \\oint_{S^1} \\frac{u^{j+2}(x)}{(j+2)!}\\,dx,\\;\\;u(x)=\\sum_{n\\in\\operatorname{\\mathbb{N}}} p_n\\;e^{+2\\pi inx}+ q_n\\;e^{-2\\pi inx}, \\end{equation*} \ni.e., hence agrees with the {\\it dispersionless KdV (or Burger) integrable hierarchy.} \\\\\n \nGoing back from $\\gamma=V=S^1$ to the case of orbit curves over general Reeb orbits $\\gamma$, observe that, since for the orbit curves the evaluation map to $V$ factors through the inclusion map $\\gamma\\subset V$, it follows that it again only makes sense to consider zero- or one-forms, where we can assume without loss of generality that the zero-form agrees with $1\\in\\Omega^0(V)$ and that the integral of the one-form $\\theta\\in\\Omega^1(V)$ over the Reeb orbit is one,\n\\begin{equation*} \\int_{\\gamma}\\theta = 1. \\end{equation*}\nFor the case with no gravitational descendants, note that it follows from index reasons that the only curves to be considered are orbit cylinders with one marked point, since introducing an additional marked point adds two or one to the Fredholm index. Since orbit cylinders are always regular and their contribution hence just equals the integral of the form $\\theta$ over the closed orbit $\\gamma$, we hence get just like in the case of $\\gamma=V=S^1$ that the zeroth descendant Hamiltonian $\\operatorname{\\mathbf{h}}^1_{\\gamma,0}$ vanishes if $\\deg\\theta = 0$ and \n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{\\gamma,0} = \\oint_{S^1} \\frac{u^2(x)}{2!}\\,dx = \\sum p_n q_n \\end{equation*}\nif $\\deg\\theta = 1$ with the normalization from above. For the sum note that we only assigned formal variables $p_n$, $q_n$ to Reeb orbits which are good in the sense of [BM]. \\\\\n\nWhile the Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,0}$ hence agree with the Hamiltonian $\\operatorname{\\mathbf{h}}^1_{S^1,0}$ for $\\gamma=V=S^1$ up to the problem of bad orbits, since no obstruction bundles have to be considered, it is easy to see that the argument breaks down when gravitational descendants are introduced, since the underlying orbit curve then has non-zero Fredholm index $1+2(j-1)+\\deg \\theta$ and hence need not be an orbit cylinder anymore. While for the case of a one-form we can hence expect to find new integrals for the nontrivial Hamiltonian $\\operatorname{\\mathbf{h}}^1_{S^1,0}=\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$, we first show that in the case of a zero-form not only the zeroth Hamiltonian but even the whole sequence of descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ is trivial. \\\\\n\\\\ \n{\\bf Theorem 2.2:} {\\it Let $\\gamma$ be a Reeb orbit in any contact manifold $V$ and assume that the string of differential forms on $V$ just consists of the zero-form $1\\in\\Omega^0(V)$. Then the sequence of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ on $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ is trivial,}\n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{\\gamma,j} = 0,\\;j\\in\\operatorname{\\mathbb{N}} \\end{equation*}\n{\\it just like in the case of $\\gamma=V=S^1$.}\\\\\n\\\\\n{\\it Proof:} Since the proof of this theorem follows from completely the same arguments as the proof of our theorem in [F2] about Euler numbers of coherent sections in obstruction bundles from above, we shortly give the main idea for the proof in [F2] about orbit curves without additional marked points and then discuss its generalization to orbit curves with zero-forms and gravitational descendants. \\\\\n \nAfter proving that we can work with finite-dimensional obstruction bundles instead of infinite-dimensional polybundles, recall that the main problem lies in the presence of codimension-one boundary of the (nonregular) moduli space, so that Euler numbers of Fredholm problems are not defined in general, since the count of zeroes in general depends on the compact perturbations chosen for the moduli spaces in the boundary. In [F2] we prove the existence of the Euler number for moduli spaces of orbit curves without additional marked points by induction on the number of punctures. For the induction step we do not only use that there exist Euler numbers for the moduli spaces in the boundary, but it is further important that all these Euler numbers are in fact trivial. The vanishing of the Euler number in turn is deduced from the different parities of the Fredholm index of the Cauchy-Riemann operator and the actual dimension of the moduli space of branched covers following the idea for the vanishing of the Euler characteristic for odd-dimensional manifolds. \\\\\n\nFor the generalization to the case of additional marked points and gravitational descendants, it is clear that it still suffices to work with finite-dimensional obstruction bundles. On the other hand, recall that the only further ingredient to our proof in [F2] was that the Fredholm index and the dimension of the moduli spaces always have different parity. Hence it follows that the proof in [F2] also works for the case when $\\theta$ is a zero-form as the actual dimension of the moduli spaces is still even, while it breaks down in the case when $\\theta$ is a one-form. $\\qed$ \\\\\n\nObserve that for one-forms it is indeed no longer clear that the every Euler number has to be zero, as we for $\\gamma=V=S^1$ and $\\theta=dt$ we get nontrivial contributions from true branched covers. While at first glance the major problem seems to be the truely complicated computation of the Euler number (see [HT1], [HT2] for related results), we further have the problem that Euler numbers need no longer exist for all Fredholm problems. {\\it For the rest of this paper we will hence only be interested in the case where the chosen differential form has degree one,} $\\deg\\theta =1$.\\\\\n\nWhile for $\\gamma=V=S^1$ we actually get a unique sequence of Poisson-commuting functions, observe that for general fixed Reeb orbits $\\gamma$ in contact manifolds $V$ the descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j} = \\operatorname{\\mathbf{h}}^{1,\\bar{\\nu}}_{\\gamma,j}$ may indeed depend on the chosen collection of sections in the cokernel bundles $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$. Hence the invariance statement is no longer trivial, but implies that for different choices of coherent abstract perturbations $\\bar{\\nu}^{\\pm}$ for the moduli spaces the resulting system of commuting elements $\\operatorname{\\mathbf{h}}^{1,-}_{\\gamma,j}$, $j=0,1,2,..$ and $\\operatorname{\\mathbf{h}}^{1,+}_{\\gamma,j}$, $j=0,1,2,..$ on $\\operatorname{\\mathfrak{P}}_{\\gamma}^0$ are just isomorphic, i.e., there exists an {\\it auto}morphism of the Poisson algebra $\\operatorname{\\mathfrak{P}}_{\\gamma}^0$ which identifies $\\operatorname{\\mathbf{h}}^{1,-}_{\\gamma,j}\\in \\operatorname{\\mathfrak{P}}_{\\gamma}^0$ with $\\operatorname{\\mathbf{h}}^{1,+}_{\\gamma,j}\\in \\operatorname{\\mathfrak{P}}_{\\gamma}^0$ for all $j\\in\\operatorname{\\mathbb{N}}$. \\\\\n\nThe above discussion hence shows that the computation of the symplectic field theory of a closed Reeb orbit gets much more difficult when gravitational descendants are considered. In what follows we want to determine it in the special case where the contact manifold is the unit cotangent bundle $S^*Q$ of a ($m$-dimensional) Riemannian manifold $Q$, so that every closed Reeb orbit $\\gamma$ on $V=S^*Q$ corresponds to a closed geodesic $\\bar{\\gamma}$ on $Q$. \\\\\n\nBefore we can state the theorem we first want to expand the descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{S^1,j}$ in terms of the $p_n$- and $q_n$-variables, where set $p_n=q_{-n}$. Abbreviating $u_n(x) = q_n e^{inx}$ for every nonzero integer $n$ it follows from $u=\\sum_n u_n$ that \n\\begin{equation*} \n \\operatorname{\\mathbf{h}}^1_{S^1,j} = \\oint_{S^1} \\frac{u^{j+2}(x)}{(j+2)!}\\,dx\n = \\oint_{S^1} \\sum \\frac{u_{n_1}(x)\\cdot ... \\cdot u_{n_{j+2}}(x)}{(j+2)!}\\, dx\n\\end{equation*}\nOn the other hand, note that the integration around the circle corresponds to selecting only those sequences of multiplicities $(n_1,...,n_{j+2})$, whose sum is equal to zero, so that \n\\begin{eqnarray*} \n\\operatorname{\\mathbf{h}}^1_{S^1,j} \\,=\\, \\sum_{n_1+...+n_{j+2} = 0} \\frac{q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}}{(j+2)!}.\n\\end{eqnarray*} \nApart from the sequence of Poisson-commuting functions for the circle, the grading of the functions given by the grading of $p_n$- and $q_n$-variables will play a central role for the upcoming theorem. For this observe that it follows from the grading conventions in symplectic field theory that the grading of the full Hamiltonian $\\operatorname{\\mathbf{H}}^0$ is $-1$, so that by $\\operatorname{\\mathbf{H}}^0 = \\sum_g \\hbar^{g-1} \\operatorname{\\mathbf{H}}^0_g$ the grading for the rational Hamiltonian $\\operatorname{\\mathbf{h}}^0=\\operatorname{\\mathbf{H}}^0_0$ is given by $|\\operatorname{\\mathbf{h}}^0| = |\\operatorname{\\mathbf{H}}^0|+|\\hbar| = -1+2(m-2)$. Since this grading has to agree with the grading of $t_j\\operatorname{\\mathbf{h}}^1_j$ with $|t_j|=2(1-j)-\\deg\\theta = 1-2j$, it follows that for every Reeb orbit $\\gamma\\subset V$ we have\n\\begin{equation*} |\\operatorname{\\mathbf{h}}^1_{\\gamma,j}| = -1+2(m-2)-1+2j = 2(m+j-3). \\end{equation*}\n\nWe already mentioned that the natural identification of the formal variables $p_n$ and $q_n$ does {\\it not} lead to an isomorphism of the graded algebras $\\operatorname{\\mathfrak{W}}^0_{\\gamma}$ and $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ with the corresponding graded algebras $\\operatorname{\\mathfrak{W}}^0_{S^1}$ and $\\operatorname{\\mathfrak{P}}^0_{S^1}$ for $\\gamma=V=S^1$, not only since the gradings of $p_n$ and $q_n$ are different and hence even the commutation rules may change but even that variables $p_n$ and $q_n$ may not be there since they would correspond to bad orbits. While for the grading of $\\gamma=V=S^1$ given by $|p_n|=|q_n|=-2$ in the descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{S^1,j}$ every summand indeed has the same degree $2(m+j-3)$, passing over to a general Reeb orbit $\\gamma$ with the new grading given by $|p_n|=m-3-\\operatorname{CZ}(\\gamma^n)$, $|q_n|=m-3+\\operatorname{CZ}(\\gamma^n)$ the descendant Hamiltonian $\\operatorname{\\mathbf{h}}^1_{S^1,j}$ is no longer of pure degree, i.e., different summands of the same descendant Hamiltonian usually have different degree. While the Poisson-commuting sequence for the circle seems not to be related to the sequence of descendant Hamiltonians for general Reeb orbits $\\gamma$, we prove the following result in the case when the Reeb orbit corresponds to a closed geodesic. \\\\\n\\\\\n{\\bf Theorem 2.3:} {\\it Assume that the contact manifold is the unit cotangent bundle $V=S^*Q$ of a Riemannian manifold $Q$, so that the closed Reeb orbit $\\gamma$ corresponds to a closed geodesic $\\bar{\\gamma}$ on $Q$, and that the string of differential forms just consists of a single one-form which integrates to one around the orbit. Then the resulting system of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ is isomorphic to the system of Poisson-commuting functions $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\bar{\\gamma}}=\\operatorname{\\mathfrak{P}}^0_{\\gamma}$, where for every $j\\in\\operatorname{\\mathbb{N}}$ the descendant Hamiltonian $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ is given by} \n\\begin{equation*} \n \\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j} \\;=\\; \\sum \\epsilon(\\vec{n})\\frac{q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}}{(j+2)!} \n\\end{equation*}\n{\\it where the sum runs over all ordered monomials $q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}$ with $n_1+...+n_{j+2} = 0$ \\textbf{and which are of degree $2(m+j-3)$}. Further $\\epsilon(\\vec{n})\\in\\{-1,0,+1\\}$ is fixed by a choice of coherent orientations in symplectic field theory and is zero if and only if one of the orbits $\\gamma^{n_1},...,\\gamma^{n_{j+2}}$ is bad.} \\\\\n\nWe have the following immediate corollary, which immediately follows from the behavior of the Conley-Zehnder index for multiple covers. \\\\\n\\\\\n{\\bf Corollary 2.4:} {\\it Assume that the closed geodesic $\\bar{\\gamma}$ represents a hyperbolic Reeb orbit in the unit cotangent bundle and $\\dim Q>1$. Then $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}=0$ and hence $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}=0$ for all $j>0$.} \\\\\n\nIndeed, since for hyperbolic Reeb orbits the Conley-Zehnder index $\\operatorname{CZ}(\\gamma^n)$ of $\\gamma^n$ is given by $\\operatorname{CZ}(\\gamma^n)=n\\cdot\\operatorname{CZ}(\\gamma)$, an easy computation shows that there are {\\it no} products of the above form of the desired degree. On the other hand, note that without the degree condition we would just get back the sequence of descendant Hamiltonians for the circle. Forgetting about orientation issues, in simple words we can hence say that the sequence $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ is obtained from the sequence for $\\bar{\\gamma}=Q=S^1$ by removing all summands with the wrong, that is, not maximal degree, where the latter can explicitely be computed using the formulas in [Lo] but also follows from our proof. \\\\\n\nThe proof relies on the observation that for orbit curves the gravitational descendants indeed have a geometric meaning in terms of branching conditions, which is a slight generalization of the result for the circle shown by Okounkov and Pandharipande in [OP]. Applying (and generalizing) the ideas of Cieliebak and Latschev in [CL] for relating the symplectic field theory of $V=S^*Q$ to the string topology of the underlying Riemannian manifold $Q$, we then study branched covers of the corresponding trivial half-cylinders in the cotangent bundle connecting the Reeb orbit $\\gamma$ with the underlying geodesic $\\bar{\\gamma}$ to prove that the sequence of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ is isomorphic to a sequence of Poisson-commuting functions $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$. {\\it While the descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ on the SFT side are defined using very complicated obstruction bundles over (nonregular) moduli spaces of arbitary large dimension, the key observation is that for the descendant Hamiltonians $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ on the string side we indeed only have to study obstruction bundles over discrete sets, which clearly disappear if the Fredholm index is right.} With this we get that the Poisson-commuting sequences for the closed geodesics can be computed from the sequences for the circle {\\it and} the Morse indices of the geodesic and its iterates as stated in the theorem. \\\\\n\n\\subsection{Gravitational descendants = branching conditions}\nRecall that by the above theorem from the last subsection we only have to consider the case where $\\theta$ is a one-form on $V$, where we still assume without loss of generality that the integral of $\\theta$ over $\\gamma$ is one. It follows that integrating the pullback of $\\theta$ under the evaluation map over the moduli space of orbit curves with one additional marked point and dividing out the natural $\\operatorname{\\mathbb{R}}$-action on the target $\\operatorname{\\mathbb{R}} \\times S^1\\cong\\operatorname{\\mathbb{R}}\\times\\gamma$ is equivalent to restricting to orbit curves where the additional marked point is mapped to a special point on $\\operatorname{\\mathbb{R}} \\times S^1$. In other words, in what follows we will view $h^1_{\\gamma,j}$ no longer as part of the Hamiltonian for $\\gamma$ but as part of the potential for the cylinder over $\\gamma$ equipped with a non-translation-invariant two-form. In order to save notation, $\\overline{\\operatorname{\\mathcal{M}}}_1=\\overline{\\operatorname{\\mathcal{M}}}_1(\\Gamma^+,\\Gamma^-)$ will from now on denote the corresponding moduli space. On the other hand, after introducing coherent collections $(\\bar{\\nu})$ of obstruction bundle sections, it is easy to see that the tautological line bundle $\\operatorname{\\mathcal{L}}^{\\bar{\\nu}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_1^{\\bar{\\nu}}$ is just the restriction of the tautological line bundle $\\operatorname{\\mathcal{L}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_1$ to $\\overline{\\operatorname{\\mathcal{M}}}_1^{\\bar{\\nu}}=\\bar{\\nu}^{-1}(0)\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$. \\\\\n\nFor the orbit curves we now want to give a geometric interpretation of gravitational descendants in terms of branching conditions over the special point on $\\operatorname{\\mathbb{R}} \\times S^1$. Before we state the corresponding theorem and give a rigorous proof using the stretching-of-the-neck procedure from SFT, we first informally describe a naive direct approach based on our definition of gravitational descendants from above, which should illuminate the underlying geometric ideas. \\\\\n\nRecall that if $(h,z)$ is an element in the non-compactified moduli space $\\operatorname{\\mathcal{M}}^{\\nu}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^{\\bar{\\nu}}_1$ the fibre of the canonical line bundle $\\operatorname{\\mathcal{L}}$ over $(h,z)$ is given by $\\operatorname{\\mathcal{L}}_{(h,z)}=T_z^*S$. Identifying the tangent space to the cylinder at the special point with $\\operatorname{\\mathbb{C}}$ it follows that $s(h,z)=\\frac{\\partial h}{\\partial z}(z)\\in T_z^*S$ is a section in the restriction of $\\operatorname{\\mathcal{L}}$ to $\\operatorname{\\mathcal{M}}^{\\nu}_1$. Since $s$ is a transversal section in the tautological line bundle over $\\operatorname{\\mathcal{M}}_1^{\\nu}$ if and only if it extends to a section over $\\operatorname{\\mathcal{M}}_1$ such that $s\\oplus\\nu$ is transversal to the zero section in $\\operatorname{\\mathcal{L}}\\oplus\\operatorname{Coker}\\bar{\\partial}_{\\underline{J}}$ over $\\operatorname{\\mathcal{M}}_1$, we may assume after possibly perturbing $\\nu$ that $s$ is indeed transversal. On the other hand, since $\\frac{\\partial h}{\\partial z}(z)=0$ is equivalent to saying that $z\\in S$ is a branch point of the holomorphic map $h:S\\to \\mathbb{C}\\mathbb{P}^1$, it follows that $\\operatorname{\\mathcal{M}}_1^1:=s^{-1}(0)\\subset\\operatorname{\\mathcal{M}}_1$ indeed agrees with the space of all orbit curves $(h,z)$ with one additional marked point, where $z$ is a branch point of $h$. \\\\\n\nFurther moving on to the case $j=2$ observe that a natural candidate for a generic section $s_2$ in the restriction of the product line bundle $\\operatorname{\\mathcal{L}}^{\\otimes 2}$ to $\\operatorname{\\mathcal{M}}^1_1\\subset\\operatorname{\\mathcal{M}}_1$ is given by $s_2(h,z)=\\frac{\\partial^2 h}{\\partial z^2}(z)\\in (T_z^*S)^{\\otimes 2}$, for which $\\operatorname{\\mathcal{M}}_1^2=s_2^{-1}(0)\\subset\\operatorname{\\mathcal{M}}_1^1$ agrees with the space of holomorphic maps where $z\\in S$ is now a branch point of order at least two. For general $j$ we can hence proceed by induction and define the section $s_j$ in $\\operatorname{\\mathcal{L}}^{\\otimes j}$ over $\\operatorname{\\mathcal{M}}^{j-1}_1:=s_{j-1}^{-1}(0)\\subset\\operatorname{\\mathcal{M}}_1$ by $s_j(h,z)=\\frac{\\partial^j h}{\\partial z^j}(z)$, so that $\\operatorname{\\mathcal{M}}^j_1$ agrees with the space of holomorphic maps where $z\\in S$ is a branch point of order at least $j$. \\\\\n\nIf the chosen sections $s_1,...,s_j$ over the non-compactified moduli spaces would extend in the same way to a coherent collection of sections in the tautological line bundles over the compactified moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_1^{\\bar{\\nu}}$, the above would show that in the case of orbit curves considering the $j$.th descendant moduli space is equivalent after passing to homology to requiring that the underlying additional marked point is a branch point of order $j$. In [OP] it was however shown that already for the case of the circle $\\gamma=V=S^1$ the latter assumption is not entirely true, but that one instead additionally obtains corrections from the boundary $\\overline{\\operatorname{\\mathcal{M}}}_1-\\operatorname{\\mathcal{M}}_1$. \\\\\n\nTo this end, we define a {\\it branching condition} to be a tuple of natural numbers $\\mu=(\\mu_1,...,\\mu_{\\ell(\\mu)})$ of length $\\ell(\\mu)$ and total branching order $|\\mu|=\\mu_1+...+\\mu_{\\ell(\\mu)}$. Then the moduli space $\\overline{\\operatorname{\\mathcal{M}}}^{\\mu}=\\overline{\\operatorname{\\mathcal{M}}}^{\\mu}(\\Gamma^+,\\Gamma^-)$ consists of orbit curves with $\\ell(\\mu)$ connected components, where every connected component carries one additional marked point $z_i$, which is mapped to the special point on $\\operatorname{\\mathbb{R}}\\times\\gamma$ and is a branch point of order $\\mu_i-1$ for $i=1,...,\\ell(\\mu)$. For every branching condition $\\mu=(\\mu_1,...,\\mu_{\\ell(\\mu)})$ we then define new Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,\\mu}=\\operatorname{\\mathbf{h}}^{1,\\bar{\\nu}}_{\\gamma,\\mu}$ by setting \n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{\\gamma,\\mu} = \\sum_{\\Gamma^+,\\Gamma^-} \\#\\overline{\\operatorname{\\mathcal{M}}}_1^{\\mu}(\\Gamma^+,\\Gamma^-)\\;p^{\\Gamma^+}q^{\\Gamma^-}. \\end{equation*} \n\nWith the following theorem we will prove that the abstract descendants-branching correspondence from [OP] holds for every closed Reeb orbit $\\gamma\\subset V$. \nFor every $j\\in\\operatorname{\\mathbb{N}}$ and every branching condition $\\mu$ we let $\\rho^0_{j,\\mu}$ be the number given by integrating the $j$.th power of the first Chern class of the tautological line bundle over the moduli space of connected rational curves over $\\mathbb{C}\\mathbb{P}^1$ with one marked point mapped to $0$ and $\\ell(\\mu)$ additional marked points $z_i$ mapped to $\\infty$ which are branch points of order $\\mu_i-1$, $i=1,...,\\ell(\\mu)$. \\\\ \n\\\\\n{\\bf Lemma 2.5:} {\\it Each of the descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ can be written as a sum,} \n\\begin{equation*} \n \\operatorname{\\mathbf{h}}^1_{\\gamma,j} \\;=\\; \\frac{1}{j!}\\;\\cdot\\;\\operatorname{\\mathbf{h}}^1_{\\gamma,(j+1)} \\;+\\;\\sum_{|\\mu|2$ let $H^{1,p}(h^*\\xi)\\subset C^0(h^*\\xi)$ denote the space of $H^{1,p}$-sections in $h^*\\xi$ which over every boundary component $C_k\\subset\\partial\\dot{S}$ restrict to a section in $C^0((\\bar{\\gamma}^{n_k^-})^*N)$. Furthermore we will consider the subspace $H^{1,p}_{\\bar{\\Gamma}}(h^*\\xi)\\subset H^{1,p}(h^*\\xi)$ consisting of all sections in $h^*\\xi$, which over every boundary circle $C_k$ restrict to sections in the subspace $TW^+(\\bar{\\gamma}^{n_k^-})\\subset C^0((\\bar{\\gamma}^{n_k^-})^*N)$. While the latter Sobolev spaces describe the normal deformations of the branched covering, we introduce similar as in [F2] for sufficiently small $d>0$ a Sobolev space with asymptotic weights $H^{1,p,d}_{\\operatorname{const}}(\\dot{S},\\operatorname{\\mathbb{C}})$ in order to keep track of tangential deformations, where, additionally to the definitions in [F2], we impose the natural constraint that the function is real-valued over the boundary. In the same way we define the Banach spaces $L^p(\\Lambda^{(0,1)}\\dot{S}\\otimes_{j,\\underline{J}_{\\xi}} h^*\\xi)$ and $L^{p,d}(\\Lambda^{(0,1)}\\dot{S}\\otimes_{j,i}\\operatorname{\\mathbb{C}})$. Further we denote by $\\operatorname{\\mathcal{M}}_{0,s^-,s^+}$ the moduli space of Riemann surfaces with $s^-$ boundary circles, $s^+$ punctures and genus zero. \\\\ \n\nFollowing [F2], [BM] for the general case and [W] for the case with boundary, there exists a Banach space bundle $\\operatorname{\\mathcal{E}}$ over a Banach manifold of maps $\\operatorname{\\mathcal{B}}$ in which the Cauchy-Riemann operator $\\bar{\\partial}_J$ extends to a smooth section. In our special case it follows as in [F2] that the fibre is given by \n\\begin{equation*} \n \\operatorname{\\mathcal{E}}_{h,j} = L^{p,d}(\\Lambda^{0,1}\\dot{S}\\otimes_{j,i}\\operatorname{\\mathbb{C}}) \\oplus L^p(\\Lambda^{0,1}\\dot{S}\\otimes_{j,\\underline{J}_{\\xi}}h^*\\xi),\n\\end{equation*}\nwhile the tangent space to the Banach manifold of maps $\\operatorname{\\mathcal{B}}= \\operatorname{\\mathcal{B}}_{0,s^-}(\\Gamma)$ at $(h,j) \\in \\operatorname{\\mathcal{M}} = \\operatorname{\\mathcal{M}}_{0,s^-}(\\Gamma)$ is given by \n\\begin{equation*}\n T_{h,j}\\operatorname{\\mathcal{B}} = H^{1,p,d}_{\\operatorname{const}}(\\dot{S},\\operatorname{\\mathbb{C}})\\oplus H^{1,p}(h^*\\xi)\\oplus T_j\\operatorname{\\mathcal{M}}_{0,n}.\n\\end{equation*} \n\nIt follows that the linearization $D_{h,j}$ of the Cauchy-Riemann operator $\\bar{\\partial}_{\\underline{J}}$ is a linear map from $T_{h,j}\\operatorname{\\mathcal{B}}$ to $\\operatorname{\\mathcal{E}}_{h,j}$, which is surjective in the case when transversality for $\\bar{\\partial}_{\\underline{J}}$ is satisfied. In this case it follows from the implicit function theorem that $\\ker D_{h,j}=T_{h,j}\\operatorname{\\mathcal{M}}$. In order to prove that the dimension of the desired moduli space $\\operatorname{\\mathcal{M}}_{\\bar{\\Gamma}}=\\operatorname{\\mathcal{M}}(\\Gamma,\\bar{\\Gamma})=\\operatorname{ev}^{-1}(W^+(\\bar{\\Gamma}))\\subset\\operatorname{\\mathcal{M}}(\\Gamma)$ agrees with the virtual dimension expected by the Fredholm index, it remains to prove that the evaluation map $\\operatorname{ev}:\\operatorname{\\mathcal{M}}\\to\\Sigma Q^{s^-}$ is transversal to the product stable manifold $W^+(\\bar{\\Gamma})$. \\\\\n\nIn order to deal with this additional transversality problem, we introduce the Banach submanifold of maps $\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}}=\\operatorname{ev}^{-1}(W^+(\\bar{\\Gamma}))\\subset\\operatorname{\\mathcal{B}}$ with tangent space \n\\begin{eqnarray*}\n T_{h,j}\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}} &=& H^{1,p,d}_{\\operatorname{const}}(\\dot{S},\\operatorname{\\mathbb{C}})\\oplus H^{1,p}_{\\bar{\\Gamma}}(h^*\\xi)\\oplus T_j\\operatorname{\\mathcal{M}}_{0,n} \\\\\n &=&\\{v\\in T_{h,j}\\operatorname{\\mathcal{B}}: v|_{\\partial\\dot{S}}\\in TW^+(\\bar{\\Gamma})\\} \n\\end{eqnarray*} \nand view the Cauchy-Riemann operator as a smooth section in $\\operatorname{\\mathcal{E}}\\to\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}}$. Then we have the following nice transversality lemma. \\\\\n\\\\\n{\\bf Lemma 2.6:} {\\it Assume that $D_{h,j}: T_{h,j}\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}} \\to \\operatorname{\\mathcal{E}}_{h,j}$ is surjective. Then the linearization of the evaluation map \n$d_{h,j}\\operatorname{ev}: T_{h,j}\\operatorname{\\mathcal{M}}\\to TW^-(\\bar{\\Gamma})=C^0(\\bar{\\Gamma}^*N)\/TW^+(\\bar{\\Gamma})$ is surjective.} \\\\\n\\\\\n{\\it Proof:} Given $v_0\\in TW^-(\\bar{\\Gamma})$, choose $\\tilde{v}\\in T_{h,j}\\operatorname{\\mathcal{B}}$ such that $d_{h,j}\\operatorname{ev}\\cdot\\tilde{v}=v_0$. On the other hand, since \n$D_{h,j}: T_{h,j}\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}} \\to \\operatorname{\\mathcal{E}}_{h,j}$ is onto, we can find $v\\in T_{h,j}\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}}$ with $D_{h,j}v=D_{h,j}\\tilde{v}$, that is, \n$\\tilde{v}-v\\in\\ker D_{h,j}= T_{h,j}\\operatorname{\\mathcal{M}}$. On the other hand, since $d_{h,j}\\operatorname{ev}\\cdot v \\in TW^+(\\bar{\\Gamma})$ for all $v\\in T_{h,j}\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}}$ by definition, we have $d_{h,j}\\operatorname{ev}\\cdot(\\tilde{v}-v) = d_{h,j}\\operatorname{ev}\\cdot\\tilde{v} = v_0$ and the claim follows. $\\qed$ \\\\\n\nWe have seen that, instead of requiring transversality for the Cauchy-Riemann operator in the Banach space bundle over $\\operatorname{\\mathcal{B}}$ and geometric transversality for the evaluation map, it suffices to require transversality for the Cauchy-Riemann operator in the Banach space bundle over the smaller Banach manifold $\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}}$. Along the same lines as for proposition 2.1 in [F2] it can be shown that the linearized Cauchy-Riemann operator is of the form \n\\begin{eqnarray*}\n &D_{h,j}:& H^{1,p,d}_{\\operatorname{const}}(\\dot{S},\\operatorname{\\mathbb{C}})\\oplus H^{1,p}_{\\bar{\\Gamma}}(h^*\\xi)\\oplus T_j\\operatorname{\\mathcal{M}}_{0,n} \\\\\n &&\\to L^{p,d}(\\Lambda^{0,1}\\dot{S}\\otimes_{j,i}\\operatorname{\\mathbb{C}}) \\oplus L^p(\\Lambda^{0,1}\\dot{S}\\otimes_{j,\\underline{J}_{\\xi}}h^*\\xi),\\\\\n && D_{h,j} \\cdot (v_1,v_2,y) = (\\bar{\\partial} v_1+ D_j y, D_h^{\\xi} v_2),\n\\end{eqnarray*}\nwhere $\\bar{\\partial}: H^{1,p,d}_{\\operatorname{const}}(\\dot{S},\\operatorname{\\mathbb{C}}) \\to L^{p,d}(\\Lambda^{0,1}\\dot{S}\\otimes_{j,i}\\operatorname{\\mathbb{C}})$ is the standard \nCauchy-Riemann operator, $D_h^{\\xi}: H^{1,p}(h^*\\xi) \\to L^p(\\Lambda^{0,1}\\dot{S}\\otimes_{j,\\underline{J}_{\\xi}}h^*\\xi)$ describes the linearization of $\\bar{\\partial}_J$ in the direction of \n$\\xi \\subset TT^*Q$ and $D_j: T_j\\operatorname{\\mathcal{M}}_{0,n} \\to L^{p,d}(T^*\\dot{S}\\otimes_{j,i}\\operatorname{\\mathbb{C}})$ describes the variation of $\\bar{\\partial}_J$ with $j\\in\\operatorname{\\mathcal{M}}_{0,n}$. \\\\\n\nIn [F2] we have shown that for branched covers of orbit cylinders the cokernels of the linearizations of the Cauchy-Riemann operator have the same dimension for every branched cover and hence fit together to give a smooth vector bundle over the nonregular moduli space of branched covers, so that we can prove transversality without waiting for the completion of the polyfold project of Hofer, Wysocki and Zehnder. The following proposition, proved in complete analogy, outlines that this still holds true for branched covers of trivial half-cylinders. \\\\\n\\\\\n{\\bf Proposition 2.7:} {\\it The cokernels of the linearizations of the Cauchy-Riemann operator fit together to give a smooth finite-dimensional vector bundle over the moduli space of branched covers of the half-cylinder.} \\\\\n\\\\\n{\\it Proof:} As in [F2] this result relies on the transversality of the standard Cauchy-Riemann operator and the super-rigidity of the trivial half-cylinder \n\\begin{equation*} \\operatorname{coker}\\bar{\\partial}=\\{0\\}\\;\\; \\textrm{and}\\;\\; \\ker D_h^{\\xi} =\\{0\\}, \\end{equation*}\nwhere the second statement is now just a linearized version of lemma 7.2 in [CL] which states that, as for orbit cylinders in the symplectizations, the branched covers of the trivial half-cylinder are characterized by the fact that they carry no energy in the sense that the action of Reeb orbits above agrees with the lenghts of the closed geodesics below. $\\qed$ \\\\\n\nIt remains to study the extension $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ of the cokernel bundle $\\operatorname{Coker}\\bar{\\partial}_{\\underline{J}}$ to the compactified moduli space. For this recall that the components of the codimension-one-boundary of the nonregular moduli space $\\overline{\\operatorname{\\mathcal{M}}}=\\overline{\\operatorname{\\mathcal{M}}}_{\\bar{\\Gamma}}$ of branched covers of the half-cylinder are either of the form $\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2$, where $\\overline{\\operatorname{\\mathcal{M}}}_1=\\overline{\\operatorname{\\mathcal{M}}_1(\\Gamma_1^+,\\Gamma_1^-)\/\\operatorname{\\mathbb{R}}}$, $\\overline{\\operatorname{\\mathcal{M}}}_2=\\overline{\\operatorname{\\mathcal{M}}_2(\\Gamma_2,\\bar{\\Gamma}_2)}$ are nonregular compactified moduli spaces of branched covers of the orbit cylinder or of the trivial half-cylinder, respectively, or of the form $\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1$, where $\\overline{\\operatorname{\\mathcal{M}}}_0=\\overline{\\operatorname{\\mathcal{M}}_0(\\Gamma,\\bar{\\Gamma}_0)}$ is again a nonregular compactified moduli space of branched covers of the trivial half-cylinder while $S^1$ refers to the concatenation or splitting locus, which agrees with the locus where the single branch point is leaving the branched covering through the boundary. Note that for $\\bar{\\Gamma}=(\\bar{\\gamma}^{n_1},...,\\bar{\\gamma}^{n_{s^-}})$ the ordered set $\\bar{\\Gamma}_0$ is either of the form \n\\begin{eqnarray*} \n\\bar{\\Gamma}_0&=&(\\bar{\\gamma}^{n_1},...,\\bar{\\gamma}^{n_{k-1}},\\bar{\\gamma}^{n_k^1},\\bar{\\gamma}^{n_k^2},\\bar{\\gamma}^{n_{k+1}},...,\\bar{\\gamma}^{n_{s^-}}) \n\\;\\;\\textrm{or}\\\\ \\bar{\\Gamma}_0&=&(\\bar{\\gamma}^{n_1},...,\\bar{\\gamma}^{n_{k-1}},\\bar{\\gamma}^{n_k+n_{k+1}},\\bar{\\gamma}^{n_{k+2}},...,\\bar{\\gamma}^{n_{s^-}}),\n\\end{eqnarray*}\ncorresponding to concatenating $\\bar{\\gamma}^{n_k^1}$ and $\\bar{\\gamma}^{n_k^1}$ to get $\\bar{\\gamma}^{n_k}$ ($n_k^1+n_k^2=n_k$) or the splitting of $\\bar{\\gamma}^{n_k+n_{k+1}}$ to get $\\bar{\\gamma}^{n_k}$ and $\\bar{\\gamma}^{n_{k+1}}$. Restricting to the concatenation case, recall that the chosen special point on the simple closed Reeb orbit determines a special point on the underlying simple geodesic and that we may assume that every holomorphic curve comes equipped with asymptotic markers in the sense of [EGH] not only on the cylindrical ends but also on the boundary circles. In particular, for the concatenation and splitting processes we may assume that all multiply-covered geodesics come equipped with a parametrization by $S^1$. Denoting by $t_1, t_2\\in S^1$ the points on $\\bar{\\gamma}^{n_k^1}$, $\\bar{\\gamma}^{n_k^1}$, where we want to concatenate the two multiply-covered geodesics to get the multiply-covered geodesic $\\bar{\\gamma}^{n_k^1+n_k^2}$, we see that the coordinates must satisfy $n_k^1 t_1=n_k^2 t_2$ in order to represent the same point on the underlying simple geodesic, so that the configuration space agrees with $S^1$ by setting $t_1=n_k^2 t$, $t_2=n_k^1 t$ for $t\\in S^1$. \\\\\n\nWhile it directly follows from [F2] that over the boundary components $\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2\\subset\\overline{\\operatorname{\\mathcal{M}}}$ the extended cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ is of the form \n\\begin{equation*} \\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}|_{\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2} = \\pi_1^*\\overline{\\operatorname{Coker}}^1\\bar{\\partial}_{\\underline{J}}\\oplus \\pi_2^*\\overline{\\operatorname{Coker}}^2\\bar{\\partial}_{\\underline{J}}, \\end{equation*}\nwhere $\\overline{\\operatorname{Coker}}^1\\bar{\\partial}_{\\underline{J}}$, $\\overline{\\operatorname{Coker}}^2\\bar{\\partial}_{\\underline{J}}$ denote the (extended) cokernel bundles over $\\overline{\\operatorname{\\mathcal{M}}}_1$, $\\overline{\\operatorname{\\mathcal{M}}}_2$, respectively, it remains to study the cokernel bundle over the boundary components $\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1$. \\\\\n\\\\\n{\\bf Proposition 2.8:} {\\it Over the boundary components $\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1\\subset \\overline{\\operatorname{\\mathcal{M}}}$ the extended cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ is also of product form,}\n\\begin{equation*} \\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}|_{\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1} \\;=\\; \\pi_1^*\\overline{\\operatorname{Coker}}^0\\bar{\\partial}_{\\underline{J}}\\oplus \\pi_2^*\\Delta, \\end{equation*}\n{\\it where $\\overline{\\operatorname{Coker}}^0\\bar{\\partial}_{\\underline{J}}$ denotes the (extended) cokernel bundle over the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_0$ and $\\Delta$ is a vector bundle over $S^1$ which is determined by the tangent spaces to the stable manifolds of the multiply-covered closed geodesics involved into the concatenation or splitting process.}\\\\ \n\\\\\n{\\it Proof:} Still restricting to the concatenation case, let $\\dot{S}_0=\\dot{S}_{01}\\cup\\dot{S}_{02}$ denote the disconnected Riemann surface of genus zero with $s^+$ punctures and $s^-+1$ boundary circles $C_1,...,C_k^1,C_k^2,...,C_{s^-}$, where we assume that $\\partial\\dot{S}_{01}=C_1\\cup...\\cup C_k^1$ and $\\partial\\dot{S}_{02}= C_k^2,...,C_{s^-}$. As before we know that the tangent spaces to the corresponding Banach manifolds of maps $\\operatorname{\\mathcal{B}}^0$, $\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}$ at a branched covering $(h_0,j_0): (\\dot{S}_0,\\partial\\dot{S}_0)\\to(\\operatorname{\\mathbb{R}}^+_0\\times S^1,\\{0\\}\\times S^1)$ are given by \n\\begin{eqnarray*}\n T_{h_0,j_0}\\operatorname{\\mathcal{B}}^0 &=& H^{1,p,d}_{\\operatorname{const}}(\\dot{S}_0,\\operatorname{\\mathbb{C}})\\oplus H^{1,p}(h_0^*\\xi)\\oplus T_{j_0}\\operatorname{\\mathcal{M}}_{0,n},\\\\\n T_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0} &=& H^{1,p,d}_{\\operatorname{const}}(\\dot{S}_0,\\operatorname{\\mathbb{C}})\\oplus H^{1,p}_{\\bar{\\Gamma}_0}(h_0^*\\xi)\\oplus T_{j_0}\\operatorname{\\mathcal{M}}_{0,n} \\\\\n &=&\\{v\\in T_{h_0,j_0}\\operatorname{\\mathcal{B}}^0: v|_{\\partial\\dot{S}_0}\\in TW^+(\\bar{\\Gamma}_0)\\} \n\\end{eqnarray*} \nwhile the fibre of the corresponding Banach space bundle is given by\n\\begin{equation*} \n \\operatorname{\\mathcal{E}}^0_{h_0,j_0} = L^{p,d}(\\Lambda^{0,1}\\dot{S}_0\\otimes_{j_0,i}\\operatorname{\\mathbb{C}}) \\oplus L^p(\\Lambda^{0,1}\\dot{S}_0\\otimes_{j_0,\\underline{J}_{\\xi}}h_0^*\\xi).\n\\end{equation*}\nFor $(h_0,j_0,t)\\in\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1$ we further introduce the Banach manifold of maps $\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}}\\subset\\operatorname{\\mathcal{B}}^*\\subset\\operatorname{\\mathcal{B}}^0$ which should consist of all branched covers of the trivial half-cylinder in $\\operatorname{\\mathcal{B}}^0$ for which the boundary circles $C_k^1, C_k^2\\cong S^1$ are concatenated at $(t_1,t_2)=(n_k^2 t, n_k^1 t)\\in C_k^1\\times C_k^2$, to give the singular Riemann surface $\\dot{S}_*$ with $s^-$ boundary circles $C_1,...,C_k^1\\cup_t C_k^2,...,C_{s^-}$ and we have\n\\begin{eqnarray*} \n T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*&=&\\{v\\in T_{h_0,j_0}\\operatorname{\\mathcal{B}}^0: v_k^1(n_k^2 t)=v_k^2(n_k^1 t)\\;\\textrm{for}\\; v_k^{1,2}:=v|_{C_k^{1,2}}\\} \\\\\n T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}} &=&\\{v\\in T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*: v|_{\\partial\\dot{S}_*}\\in TW^+(\\bar{\\Gamma}_0)\\}. \n\\end{eqnarray*}\n\nThe proof of the general gluing theorem in [MDSa] suggests that over $(h_0,j_0,t)\\in\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1\\subset\\overline{\\operatorname{\\mathcal{M}}}$ the extended cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ has fibre \n\\begin{equation*} \n (\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}})_{h_0,j_0,t} = \\operatorname{coker} D_{h_0,j_0,t},\\;\\; D_{h_0,j_0,t}: T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}}\\to\\operatorname{\\mathcal{E}}^0_{h_0,j_0}.\n\\end{equation*}\nBefore we describe the relation to the cokernel bundle $\\overline{\\operatorname{Coker}}^0\\bar{\\partial}_{\\underline{J}}$ over the first factor $\\overline{\\operatorname{\\mathcal{M}}}_0$ with fibre\n\\begin{equation*} \n (\\overline{\\operatorname{Coker}}^0\\bar{\\partial}_{\\underline{J}})_{h_0,j_0} = \\operatorname{coker} D_{h_0,j_0},\\;\\; D_{h_0,j_0}: T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}\\to\\operatorname{\\mathcal{E}}^0_{h_0,j_0}.\n\\end{equation*}\nobserve that we still have \n\\begin{eqnarray*} \n\\operatorname{coker} D_{h_0,j_0} = \\operatorname{coker} D^{\\xi}_{h_0},&& D^{\\xi}_{h_0}: T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}\\to\\operatorname{\\mathcal{E}}^{0,\\xi}_{h_0,j_0},\\\\\n\\operatorname{coker} D_{h_0,j_0,t} = \\operatorname{coker} D^{\\xi}_{h_0,t},&& D^{\\xi}_{h_0,t}: T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}}\\to\\operatorname{\\mathcal{E}}^{0,\\xi}_{h_0,j_0},\n\\end{eqnarray*}\nand $\\ker D^{\\xi}_{h_0}=\\ker D^{\\xi}_{h_0,t}=\\{0\\}$, where $T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}\\subset T_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}$, $T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}}\\subset T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}}$ and $\\operatorname{\\mathcal{E}}^{0,\\xi}_{h_0,j_0}\\subset \\operatorname{\\mathcal{E}}^0_{h_0,j_0}$ are the subspaces corresponding to normal deformations. \\\\\n\nNow observing that \n\\begin{equation*} \n TW^+(\\bar{\\gamma}^{n_k^1})\\oplus TW^+(\\bar{\\gamma}^{n_k^2}) \\subset \\{v\\in TW^+(\\bar{\\gamma}^{n_k}): v_k^1(n_k^2 t) = v_k^2(n_k^1 t)\\}\n\\end{equation*}\nwe get from \n\\begin{eqnarray*} \n T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0} =\\{v\\in T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0: v|_{\\partial\\dot{S}_0}\\in TW^+(\\bar{\\Gamma}_0)\\},\\\\\n T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}} =\\{v\\in T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*: v|_{\\partial\\dot{S}_0}\\in TW^+(\\bar{\\Gamma})\\}\\\\\n\\end{eqnarray*}\nthat $T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}\\subset T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}}$ with quotient space\n\\begin{equation*} \n \\frac{T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}}{T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}}} \\;=\\; \n \\frac{TW^+(\\bar{\\gamma}^{n_k^1})\\oplus TW^+(\\bar{\\gamma}^{n_k^2})}{\\{v\\in TW^+(\\bar{\\gamma}^{n_k}): v_k^1(n_k^2 t) = v_k^2(n_k^1 t)\\}}.\n\\end{equation*}\nOn the other hand, since $\\ker D^{\\xi}_{h_0}=\\ker D^{\\xi}_{h_0,t}=\\{0\\}$ we also find for the quotient space that\n\\begin{equation*} \n \\frac{T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}}{T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}}} \\;=\\; \n \\frac{\\operatorname{im} D^{\\xi}_{h_0}}{\\operatorname{im} D^{\\xi}_{h_0,t}} \\;=\\; \\frac{\\operatorname{coker} D^{\\xi}_{h_0,t}}{\\operatorname{coker} D^{\\xi}_{h_0}},\n\\end{equation*}\nwhere the last equality follows from the fact that $D^{\\xi}_{h_0}$ and $D^{\\xi}_{h_0,t}$ both map to the same Banach space $\\operatorname{\\mathcal{E}}^{0,\\xi}_{h_0}$. \\\\\n\nDefining an obstruction bundle $\\Delta$ over $S^1$ by setting \n\\begin{equation*}\n \\Delta_t = \\frac{TW^+(\\bar{\\gamma}^{n_k^1})\\oplus TW^+(\\bar{\\gamma}^{n_k^2})}{\\{v\\in TW^+(\\bar{\\gamma}^{n_k}): v_k^1(n_k^2 t) = v_k^2(n_k^1 t)\\}}\n\\end{equation*}\nand putting everything together we hence found that \n\\begin{equation*} \n (\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}})_{h_0,j_0,t} \\cong (\\overline{\\operatorname{Coker}}^0\\bar{\\partial}_{\\underline{J}})_{h_0,j_0} \\oplus \\Delta_t,\n\\end{equation*}\nas desired. $\\qed$ \\\\\n \nWith this we can prove the desired statement about $\\operatorname{\\mathbf{g}}^0_{\\bar{\\gamma}}$. \\\\\n\\\\\n{\\bf Corollary 2.9:} {\\it We have $\\operatorname{\\mathbf{g}}^0_{\\bar{\\gamma}}=0$.} \\\\\n\\\\\n{\\it Proof:} It follows that the obstruction bundle over the one-dimensional configuration space has rank\n\\begin{equation*} \\operatorname{rank} \\Delta = \\operatorname{Morse}(\\bar{\\gamma}^{n_k})-\\operatorname{Morse}(\\bar{\\gamma}^{n_k^1})-\\operatorname{Morse}(\\bar{\\gamma}^{n_k^2})+\\dim Q-1 \\geq 0,\n\\end{equation*}\nwhere the latter inequality can be verified as in [F2] using the multiple cover index formulas in [Lo]. When by index reasons the configuration is expected to be discrete we get a rank-one obstruction bundle over the boundary of the branched cover, which by orientability reasons must indeed be trivial. $\\qed$ \\\\\n\nOn the other hand, we want to emphasize that the proof of $\\operatorname{\\mathbf{g}}^0_{\\bar{\\gamma}}=0$ is much simpler than the proof of $\\operatorname{\\mathbf{h}}^0_{\\gamma}=0$ in [F2], which has to involve obstruction bundles of arbitrary large rank and uses induction. Besides that our proof in [F2] also holds for Reeb orbits in general contact manifolds, this does not come as surprise. Going back to the symplectic field theory of unit cotangent bundles $S^*Q$, it is already mentioned in [CL] that the SFT differential $\\operatorname{\\mathbf{D}}^0_{\\operatorname{SFT}}=\\overrightarrow{\\operatorname{\\mathbf{H}}^0}:\\AA^0[[\\hbar]]\\to\\AA^0[[\\hbar]]$ involving all moduli spaces of holomorphic curves in $\\operatorname{\\mathbb{R}}\\times S^*Q$ is much larger than the string differential $\\operatorname{\\mathbf{D}}^0_{\\operatorname{string}}=\\partial+\\Delta+\\hbar\\nabla:\\operatorname{\\mathfrak{C}}^0[[\\hbar]]\\to\\operatorname{\\mathfrak{C}}^0[[\\hbar]]$, which just involves the singular boundary operator and the string bracket and cobracket operations. \\\\\n\n\\subsection{Additional marked points and gravitational descendants}\nWe now want to understand the system of commuting operators defined for Reeb orbits by studying moduli spaces of branched covers over the cylinder over $\\gamma$ in terms of operations defined for the underlying closed geodesic $\\bar{\\gamma}$. To this end we have to extend the picture of [CL] used for computing the symplectic field theory of Reeb orbits to include additional marked points on the moduli spaces, integration of differential forms and gravitational descendants. \\\\\n\nReintroducing the sequence of formal variables $t_j$, $j\\in\\operatorname{\\mathbb{N}}$, we now consider the graded Weyl algebras $\\operatorname{\\mathfrak{W}}_{\\gamma}$, $\\operatorname{\\mathfrak{W}}_{\\bar{\\gamma}}$ of power series in $\\hbar$, the $p$-variables corresponding to multiples of $\\gamma$, $\\bar{\\gamma}$ and $t$-variables with coefficients which are polynomials in the $q$-variables corresponding to multiples of $\\gamma$, $\\bar{\\gamma}$. In the same way we can introduce the graded commutative algebras $\\AA_{\\gamma}$, $\\operatorname{\\mathfrak{C}}_{\\bar{\\gamma}}$ of power series in $\\hbar$, the $t$-variables with coefficients which are polynomials in the $q$-variables corresponding to multiples of $\\gamma$, $\\bar{\\gamma}$. For the expansion $\\operatorname{\\mathbf{H}}_{\\gamma}=\\operatorname{\\mathbf{H}}^0_{\\gamma}+\\sum_j t_j \\operatorname{\\mathbf{H}}^1_{\\gamma,j}+o(t^2)$ of the Hamiltonian from before, we are hence looking for an extended potential $\\operatorname{\\mathbf{L}}_{\\gamma,\\bar{\\gamma}}$ as well as extended string Hamiltonian $\\operatorname{\\mathbf{G}}_{\\bar{\\gamma}}$, \n\\begin{eqnarray*}\n \\operatorname{\\mathbf{L}}_{\\gamma,\\bar{\\gamma}}&=&\\operatorname{\\mathbf{L}}^0_{\\gamma,\\bar{\\gamma}}+\\sum_j t_j \\operatorname{\\mathbf{L}}^1_{\\gamma,\\bar{\\gamma},j}+o(t^2), \\\\\n \\operatorname{\\mathbf{G}}_{\\bar{\\gamma}}&=&\\operatorname{\\mathbf{G}}^0_{\\bar{\\gamma}}+\\sum_j t_j \\operatorname{\\mathbf{G}}^1_{\\bar{\\gamma},j}+o(t^2),\n\\end{eqnarray*}\nsuch that $\\overrightarrow{\\operatorname{\\mathbf{L}}_{\\gamma,\\bar{\\gamma}}}: (\\AA_{\\gamma}[[\\hbar]],\\overrightarrow{\\operatorname{\\mathbf{H}}_{\\gamma}}) \\to (\\operatorname{\\mathfrak{C}}_{\\bar{\\gamma}}[[\\hbar]],\\overrightarrow{\\operatorname{\\mathbf{G}}_{\\bar{\\gamma}}})$ is an isomorphism of $\\operatorname{BV}_{\\infty}$-algebras. \nFor this we have to prove the extended master equation\n\\begin{equation*} \n e^{\\operatorname{\\mathbf{L}}_{\\gamma,\\bar{\\gamma}}}\\overleftarrow{\\operatorname{\\mathbf{H}}_{\\gamma}} - \\overrightarrow{\\operatorname{\\mathbf{G}}_{\\bar{\\gamma}}}e^{\\operatorname{\\mathbf{L}}_{\\gamma,\\bar{\\gamma}}} = 0, \n\\end{equation*}\nwhile the isomorphism property again follows using the natural filtration given by the $t$-variables. \\\\\n\nSince we are only interested in the system of commuting operators $\\operatorname{\\mathbf{H}}^1_{\\gamma,j}$, $j\\in\\operatorname{\\mathbb{N}}$, which is defined by counting branched covers of orbit cylinders with at most one additional marked point, we again will only discuss the required compactness statements in the case of one additional marked point. Furthermore we will still just restrict to the rational case. In other words we will prove the following proposition, which is just a reformulation of our theorem from above. \\\\\n\\\\\n{\\bf Proposition 2.10:} {\\it The system of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ is isomorphic to a system of Poisson-commuting functions $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\bar{\\gamma}}=\\operatorname{\\mathfrak{P}}^0_{\\gamma}$, where for every $j\\in\\operatorname{\\mathbb{N}}$ the descendant Hamiltonian $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ given by} \n\\begin{equation*} \n \\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j} \\;=\\; \\sum \\epsilon(\\vec{n})\\frac{q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}}{(j+2)!} \n\\end{equation*}\n{\\it where the sum runs over all ordered monomials $q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}$ with $n_1+...+n_{j+2} = 0$ \\textbf{and which are of degree $2(m+j-3)$}. Further $\\epsilon(\\vec{n})\\in\\{-1,0,+1\\}$ is fixed by a choice of coherent orientations in symplectic field theory and is zero if and only if one of the orbits $\\gamma^{n_1},...,\\gamma^{n_{j+2}}$ is bad.} \\\\\n\n{\\it Proof:} While the proof seems to require the definition of gravitational descendants for moduli spaces of holomorphic curves not only with punctures but also with boundary, instead of defining them recall that we have shown in the previous subsection 2.2 that the gravitational descendants can be replaced by imposing branching conditions over the special marked point on the orbit cylinder. More precisely, recall the lemma in subsection 2.2 states that we can indeed write each of the Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ as a weighted sum, \n\\begin{equation*} \n \\operatorname{\\mathbf{h}}^1_{\\gamma,j} \\;=\\; \\frac{1}{j!}\\;\\cdot\\;\\operatorname{\\mathbf{h}}^1_{\\gamma,(j)} \\;+\\;\\sum_{|\\mu| 0, \\quad B(r) \\geq B_{min} > 0, \\quad J(r) \\geq R_{min} > 0,\n\\]\nwith $R_{min}$ standing for the radius of the wormhole throat.\nNote that $r$ is a globally defined coordinate, so that the\nhorizon absence and ``flaring-out''\nconditions are satisfied automatically. \n\nIn what follows we aim to construct an asymptotically flat wormhole \non both ends, which is the case provided that\n\\[\nA(r) \\to 1 \\; , \\;\\;\\;\\; B(r) \\to 1 \\; ,\n\\;\\;\\;\\; J(r) = |r| + O(1) \\;\\;\\;\\; \\mbox{as} \\;\\;\\; r \\to \\pm \\infty \\; .\n\\]\nWe choose a specific form of the metric functions \n$A(r)$, $B(r)$ and $J(r)$ which\ndescribe a wormhole in Sec.~\\ref{sec:wormhole_example}.\nEven though $B(r)$ in eq.~\\eqref{eq:backgr_metric}\ncan be set equal to 1 by coordinate transformation,\nwe keep it arbitrary for generality.\nIn our setting the scalar field $\\pi$ is static and\ndepends on the radial coordinate only, $\\pi = \\pi(r)$ and\nhence $X = -B(r) \\pi'^2 \/2$, where prime denotes the\nderivative with respect to $r$.\n\nIn what follows we make use of background equations of motion following\nfrom action~\\eqref{eq:lagrangian}, especially \nwhen we reconstruct\nthe specific form of Lagrangian functions of beyond Horndeski theory which admits a wormhole solution in Sec.~\\ref{sec:wormhole_example}. We keep the notations used in Ref.~\\cite{wormhole1} for the equations of motion:\n\\[\n\\label{eq: background_eqs}\n {\\cal E}_A = 0, \\quad {\\cal E}_B = 0, \\quad {\\cal E}_J = 0,\n \\quad {\\cal E}_{\\pi} = 0 \\; ,\n \\]\n where $ {\\cal E}_A$ is obtained by varying the action with respect to\n $A$, etc. We give the explicit forms of\n${\\cal E}_A, {\\cal E}_B, {\\cal E}_J$ and ${\\cal E}_{\\pi}$\nin Appendix A. \n\nIn the following section we develop the linearized theory aiming to formulate a complete set of stability conditions for high energy modes\nabout \na static, spherically symmetric background~\\eqref{eq:backgr_metric}. \n\n\n\n\\section{Linearized theory}\n\\label{sec:linearized_th}\nAs it was stated above one of our priorities is to ensure stability of a wormhole solution against small perturbations. In what follows we\nconsider the\nlinearized metric\n\\begin{equation}\n\\label{eq:metric}\ng_{\\mu\\nu} = \\bar{g}_{\\mu\\nu} + h_{\\mu\\nu},\n\\end{equation}\nwhere the background $\\bar{g}_{\\mu\\nu}$ is given \nin eq.~\\eqref{eq:backgr_metric} and $h_{\\mu\\nu}$ stand for small metric perturbations, while $\\delta\\pi$ denotes perturbation about a background scalar field $\\bar{\\pi}$. \n\nTo study the behaviour of perturbations about a static and spherically-symmetric background we follow a standard approach and make use of the Regge-Wheeler formalism~\\cite{ReggeWheeler}:\nperturbations are classified into odd-parity and even-parity sectors \nbased on their behaviour under two-dimensional reflection.\nWith further expansion of perturbations into series of the spherical harmonics \n$Y_{\\ell m}(\\theta,\\phi)$ the modes with\ndifferent parity, $\\ell$ and $m$ do not mix at the linear level, so \nit is legitimate to consider them separately.\nIn the next two subsections\nwe consider odd-parity and even-parity sectors one by one. \n\n\\subsection{Odd-parity sector}\n\\label{sec:odd_sector}\nEven though ghost-like instabilities typically arise in the even-parity \nsector, we give a brief review of the stability constraints in the odd-parity sector for completeness. The contents of this subsection\nare a short summary of Sec.3.2 in Ref.~\\cite{wormhole1}. \n\nThe scalar field $\\pi$ does not obtain odd-parity perturbations, so the only source of perturbation modes in this sector is metric $g_{\\mu\\nu}$.\nThe most general form of metric \nperturbations in the odd-parity sector read~\\cite{ReggeWheeler}:\n\\[\n\\label{odd_parity}\n\\mbox{Parity~odd}\\quad \\begin{cases}\n\\begin{aligned}\n & h_{tt}=0,~~~h_{tr}=0,~~~h_{rr}=0,\\\\\n & h_{ta}=\\sum_{\\ell, m}h_{0,\\ell m}(t,r)E_{ab}\\partial^{b}Y_{\\ell m}(\\theta,\\varphi),\\\\\n & h_{ra}=\\sum_{\\ell, m}h_{1,\\ell m}(t,r)E_{ab}\\partial^{b}Y_{\\ell m}(\\theta,\\varphi),\\\\\n & h_{ab}=\\frac{1}{2}\\sum_{\\ell, m}h_{2,\\ell m}(t,r)\\left[E_{a}^{~c}\\nabla_{c}\\nabla_{b}Y_{\\ell m}(\\theta,\\varphi)+E_{b}^{~c}\\nabla_{c}\\nabla_{a}Y_{\\ell m}(\\theta,\\varphi)\\right],\n\\end{aligned}\n\\end{cases}\n\\]\nwhere $a,b = \\theta,\\varphi$, $E_{ab} = \\sqrt{\\det \\gamma}\\: \\epsilon_{ab}$, with $\\gamma_{ab} = \\mbox{diag}(1, \\;\\sin^2\\theta)$;\n$\\epsilon_{ab}$ is\ntotally antisymmetric symbol ($\\epsilon_{\\theta\\varphi} = 1$) and $\\nabla_a$ is covariant derivative on a 2-sphere. Note that perturbations with\n$\\ell=0$ do not exist and modes with $\\ell=1$ are pure gauges~\\cite{Armendariz,Kobayashi:odd}. So below in this section we consider perturbations with $\\ell \\geq 2$. \n\n\nWe make use of the gauge freedom and set $h_{2,\\ell}=0$ (Regge-Wheeler gauge), so that the quadratic action in the odd-parity sector is derived in terms of $h_{0,\\ell m}$ and $h_{1,\\ell m}$. It was explicitly shown in Refs.~\\cite{Kobayashi:odd,wormhole1} that $h_{0,\\ell m}$ is a non-dynamical DOF and one can integrate it out from the quadratic action. The resulting action for the only dynamical DOF reads \n(see Ref.~\\cite{wormhole1}\nfor a detailed derivation):\n\\[\n\\label{eq:action_odd_final}\n\\begin{aligned}\nS^{(2)}_{odd} = \\int \\mbox{d}t\\:\\mbox{d}r\\:\\sqrt{\\frac{A}{B}} J^2\n\\frac{\\ell(\\ell+1)}{2(\\ell-1)(\\ell+2)}\\cdot \\frac{B}{A}\\left[ \\frac{\\mathcal{H}^2}{A \\mathcal{G}} \\dot{Q}^2 - \\frac{B \\mathcal{H}^2}{\\mathcal{F}} (Q')^2 -\\frac{l(l+1)}{J^2}\\cdot \\mathcal{H} Q^2 - V(r) Q^2 \\right] \\; ,\n\\end{aligned}\n\\]\nwhere an overdot stands for a time derivative, $A$, $B$ and $J$ are metric functions~\\eqref{eq:backgr_metric}, $Q$ is an auxiliary field and\n\\footnote{We have introduced different notations for $\\mathcal{H}$ and $\\mathcal{G}$ in action~\\eqref{eq:action_odd_final} despite their equality for the quadratic subclass of beyond Horndeski theory~\\eqref{eq:lagrangian} as these coefficients become different as soon as one adds the cubic subclass to the set-up~\\cite{Kobayashi:odd,wormhole1}. } \n\\begin{equation}\n\\label{eq:cal_FGH}\n{\\cal F}=2 G_4 ,\n\\qquad\n{\\cal H}={\\cal G}= 2 \\left[G_4 - 2X G_{4X} + 4 X^2 F_4 \\right].\n\\end{equation} \nOnce $Q$ is known both $h_{0,\\ell m}$ and $h_{1,\\ell m}$ can be restored.\nNote that the third term in eq.~\\eqref{eq:action_odd_final} is the \nangular part of the Laplace operator, which governs stability in the angular direction. The \"potential\" $V(r)$ in eq.~\\eqref{eq:action_odd_final} reads\n\\begin{equation}\n\\label{eq:Vr}\n\\begin{aligned}\nV(r) &= \\frac{B {\\cal H}^2 }{{2\\cal F}}\n\\left[\\frac{{\\cal F}'}{{\\cal F}} \\left(2\\frac{{\\cal H}'}{{\\cal H}} -\\frac{A'}{A}+\\frac{B'}{B}+ 4\\frac{J'}{J}\\right) - \\frac{{\\cal H}'}{{\\cal H}} \\left( -\\frac{A'}{A}+3 \\frac{B'}{B}+4 \\frac{J'}{J}\\right)\\right.\\\\\n&\\left. - \\left(\\frac{A'^2}{A^2} -\\frac{A'}{A} \\frac{B'}{B} - \\frac{A''}{A} +\\frac{B''}{B} +4 \\frac{B'}{B} \\frac{J'}{J}-4 \\frac{J'^2}{J^2}+ 4 \\frac{J''}{J} \\right) - \\frac{4}{J^2 B}\\cdot \\frac{\\cal F}{\\cal H} \\right],\n\\end{aligned}\n\\end{equation}\nand governs the possible \"slow\" tachyonic instabilities, which we do not discuss in this paper. However, corresponding constraints following from the potential $V(r)$ may not be particularly restrictive~\\cite{Trincherini}.\n\n\nThe quadratic action~\\eqref{eq:action_odd_final} gives the following set of stability conditions which ensure the absence of both ghost and gradient instabilities in the odd-parity sector:\n\\begin{subequations}\n\\label{eq:stability_odd}\n\\begin{align}\n\\label{eq:stability_G}\n\\mbox{No ghosts:}\\quad &\\mathcal{G} > 0, \\\\\n\\label{eq:stability_F}\n\\mbox{No radial gradient instabilities:}\\quad &\\mathcal{F} > 0, \\\\\n\\label{eq:stability_H}\n\\mbox{No angular gradient instabilities:}\\quad &\\mathcal{H} > 0.\n\\end{align}\n\\end{subequations}\nTo have both radial and angular modes propagating at (sub)luminal speed one has to ensure that corresponding sound speeds squared are not greater than the speed of light\n\\begin{equation}\n\\label{eq:speed_odd}\nc_r^2 = \\frac{\\mathcal{G}}{\\mathcal{F}} \\leq 1, \\quad c_{\\theta}^2 = \\frac{\\mathcal{G}}{\\mathcal{H}} \\leq 1,\n\\end{equation}\nwhich is the case provided that\n\\begin{equation}\n\\label{eq:sublum_odd}\n\\mathcal{F} \\ge \\mathcal{G} > 0, \\qquad \n\\mathcal{H} \\ge \\mathcal{G} > 0 \\; .\n\\end{equation}\n\n\n\\subsection{Even-parity sector}\n\\label{sec:even_parity}\n\nWe now move on to the even-parity sector, which is the main source of stability constraints for a spherically-symmetric solution.\nWe adopt the following general form of parametrization for\nmetric perturbations:\n\\[\n\\label{eq:even_parity}\n\\mbox{Parity~even}\\quad \\begin{cases}\n\\begin{aligned}\nh_{tt}=&A(r)\\sum_{\\ell, m}H_{0,\\ell m}(t,r)Y_{\\ell m}(\\theta,\\varphi), \\\\\nh_{tr}=&\\sum_{\\ell, m}H_{1,\\ell m}(t,r)Y_{\\ell m}(\\theta,\\varphi),\\\\\nh_{rr}=&\\frac{1}{B(r)}\\sum_{\\ell, m}H_{2,\\ell m}(t,r)Y_{\\ell m}(\\theta,\\varphi),\\\\\nh_{ta}=&\\sum_{\\ell, m}\\beta_{\\ell m}(t,r)\\partial_{a}Y_{\\ell m}(\\theta,\\varphi), \\\\\nh_{ra}=&\\sum_{\\ell, m}\\alpha_{\\ell m}(t,r)\\partial_{a}Y_{\\ell m}(\\theta,\\varphi), \\\\\nh_{ab}=&\\sum_{\\ell, m} K_{\\ell m}(t,r) g_{ab} Y_{\\ell m}(\\theta,\\varphi)+\\sum_{\\ell, m} G_{\\ell m}(t,r) \\nabla_a \\nabla_b Y_{\\ell m}(\\theta,\\varphi)\\,.\n\\end{aligned}\n\\end{cases}\n\\]\nThe scalar field $\\pi$ also acquires non-vanishing perturbations \nin the parity even sector:\n\\[\n\\label{chi}\n\\pi (t,r,\\theta,\\varphi) = \\pi(r) + \\sum_{\\ell, m}\\chi_{\\ell m}(t,r)Y_{\\ell m}(\\theta,\\varphi),\n\\]\nwhere $\\pi(r)$ is the spherically symmetric background field. As we have\nmentioned above modes with different $\\ell$ and $m$ does not get mixed at the linear level, so to simplify the expressions we usually drop both subscripts in what follows.\n\nThe next natural step is to either work with gauge-invariant variables~\\cite{Gerlach:1979rw} or to fix the gauge. Below we choose the latter option. Under the infinitesimal coordinate change $x^{\\mu} \\to x^{\\mu} + \\xi^{\\mu}$ with $\\xi^{\\mu}$ parametrized as\n\\begin{equation}\n\\label{eq:xi}\n\\xi^{\\mu} = \\Big(T_{\\ell m}(t,r), R_{\\ell m}(t,r), \\Theta_{\\ell m}(t,r) \\partial_{\\theta}, \\dfrac{\\Theta_{\\ell m}(t,r) \\partial_{\\varphi}}{\\sin^2\\theta}\\Big)\\, Y_{\\ell m}(\\theta,\\varphi )\n\\end{equation}\nthe metric perturbations in eq.~\\eqref{eq:even_parity} transform as follows\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:gauge_laws}\nH_0 &\\rightarrow H_0 + \\dfrac{2}{A} \\dot{T} - \\dfrac{A'}{A} B R, \\\\\nH_1 &\\rightarrow H_1 + \\dot{R} + T' - \\dfrac{A'}{A} T, \\\\\nH_2 &\\rightarrow H_2 + 2 B R' + B' R, \\\\\n\\beta &\\rightarrow \\beta + T + \\dot{\\Theta}, \\\\\n\\alpha &\\rightarrow \\alpha + R + \\Theta' - 2 \\dfrac{J'}{J} \\Theta,\\\\\nK &\\rightarrow K + 2 B \\dfrac{J'}{J} R, \\\\\nG &\\rightarrow G + 2 \\Theta, \\\\\n\\chi &\\rightarrow \\chi + B \\pi' R,\n\\end{aligned}\n\\end{equation}\nwhere we have dropped the arguments of functions for clarity. \nFixing the gauge amounts to choosing specific functions $T$, $R$ and $\\Theta$ in eq.~\\eqref{eq:gauge_laws} so that \nsome of the perturbations vanish. \nFor instance, \nthe original Regge--Wheeler gauge, used e.g. in Refs.~\\cite{ReggeWheeler,Armendariz}, amounts to setting $\\alpha=\\beta=G=0$. \nAnother possible gauge choice was adopted \nin Refs.~\\cite{Kobayashi:even,wormhole1} where $\\beta=K=G=0$ (let us call it a \"spherical gauge\" for brevity). Alternatively, Ref.~\\cite{Trincherini} used the Regge--Wheeler--unitary gauge with $\\beta=G=\\chi=0$. \n\n\nIn our previous work in Ref.~\\cite{wormhole1} we have already \nderived the \naction for perturbations in beyond Horndeski theory adopting \nthe spherical gauge $\\beta=K=G=0$ and found that for modes \nwith large $\\ell$ the coefficients in the action inevitably become singular at some point. This singularity is unavoidable as soon as one requires the background to be free of ghost instabilities at all points. We provide a detailed explanation in Sec.~\\ref{sec:old_gauge}.\n\nThere is another subtlety with the gauge choice where $K=0$. \nNamely, according to the \ntransformation laws~\\eqref{eq:gauge_laws} \nit is impossible to make $K$ vanishing at all points if\nthe metric function $J(r)$ is not a monotonous function, i.e. $J'(r)=0$ at some point. And this is exactly what happens in a wormhole throat, \nsince function $J(r)$ describes the profile of a wormhole \nand $J'(r)=0$ at $r=R_{min}$~\\eqref{eq:metric_bounded}. Hence, \nthe spherical gauge in Ref.~\\cite{wormhole1} is \nnot entirely convenient for checking linear stability \nof the wormhole solutions near the throat.\n\nIn this paper we aim to formulate a complete set of stability conditions \nfor modes with high momentum, which does not involve any singularities and, hence, is suitable for stability analysis throughout the whole space, where the wormhole is present. \n\n\n\nIn Sec.~\\ref{sec:old_gauge} we give a schematic derivation of the action for even-parity perturbations \nfor beyond Horndeski Lagrangian~\\eqref{eq:lagrangian} \nin the spherical gauge (see Ref.~\\cite{wormhole1} for a detailed procedure)\nand discuss its drawbacks in the context of stability analysis for spherically-symmetric solutions like wormholes. In Sec.~\\ref{sec:new_gauge} we\ncalculate the quadratic action in Regge--Wheeler--unitary gauge and find out if we can avoid singularities in the coefficients of quadratic action analogous to those in the old gauge. In result, we derive the whole set of stability conditions for modes with high $\\ell$. \n\n\n\\subsubsection{Existing results in the spherical gauge ($\\beta=K=G=0$)}\n\\label{sec:old_gauge}\nThe quadratic action for even-parity perturbations~\\eqref{eq:even_parity}\nwith $\\ell \\geq 2$ \n\\footnote{Both $\\ell = 0$ and $\\ell=1$ are somewhat special cases, which, however, do not give anything particularly new for stability analysis as compared to $\\ell \\geq 2$~\\cite{Kobayashi:even,wormhole1}.}\nand gauge conditions $\\beta=K=G=0$\nreads:\n\\begin{equation}\n\\label{eq:action_even}\n\\begin{aligned}\n&S_{even}^{(2)} = \\int \\mbox{d}t\\:\\mbox{d}r \\left(H_0 \\left[ a_1 \\chi''+ a_2\n\\chi'+a_3 H_2'+j^2 a_4 \\alpha'+\\left( a_5+j^2 a_6 \\right) \\chi\\right.\\right. \\\\\n&\\left.+\\left( a_7+j^2 a_8 \\right)H_2+j^2a_9 \\alpha \\right]\n+j^2 b_1 H_1^2+H_1 \\left[b_2 {\\dot {\\chi}}'+b_3 {\\dot {\\chi}}+b_4 {\\dot H_2}+j^2b_5 {\\dot \\alpha}\\right] \\\\\n&+ c_1{\\dot H_2} {\\dot {\\chi}} + H_2 \\left[c_2 \\chi'+\\left( c_3+j^2 c_4\\right)\\chi + j^2 c_5 \\alpha \\right] + c_6 H_2^2+j^2d_1 {\\dot \\alpha}^2\n\\\\\n&\\left.+j^2 d_2 \\alpha\\chi'+j^2 d_3 \\alpha\\chi+j^2 d_4 \\alpha^2\n+ e_1{\\dot {\\chi}}^2+e_2 \\chi'^2+\\left( e_3+j^2 e_4 \\right) \\chi^2\\right),\n\\end{aligned}\n\\end{equation}\nwhere the subscripts $\\ell$, $m$ are again omitted, $j^2=\\ell(\\ell+1)$, and\nwe have integrated over $\\theta$ and $\\phi$. The\nexplicit expressions for coefficients $a_i$, $b_i$, $c_i$, $d_i$\nand $e_i$ with $\\sqrt{-g}$ included are given in Appendix B.\n\nIt follows from action~\\eqref{eq:action_even} that $H_0$ is a Lagrange multiplier and $H_1$ is also a non-dynamical DOF, which give the following constraints, respectively:\n\\begin{equation}\n\\label{eq:H0old}\n\\begin{aligned}\na_1 \\chi''+a_2 \\chi'+a_3 H_2'+\nj^2 a_4 \\alpha'+\\left( a_5+j^2 a_6 \\right) \\chi\n+\\left( a_7+j^2 a_8 \\right)H_2+j^2a_9 \\alpha = 0,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\label{eq:H1old}\n H_1 = - \\frac{1}{2 j^2 b_1} \\left( b_2 {\\dot {\\chi}}'+b_3 {\\dot {\\chi}}+b_4 {\\dot H_2}+j^2b_5 {\\dot \\alpha}\\right).\n\\end{equation}\nIntroducing a new variable $\\psi$ as\n\\begin{equation}\n\\label{eq:H2old}\nH_2 = \\psi - \\frac{1}{a_3}\\left( a_1 \\chi' + j^2 a_4 \\alpha\\right),\n\\end{equation}\nand so that upon substitution of $H_2$ into eq.~\\eqref{eq:H0old}\nboth $\\chi''$ and $\\alpha'$ get cancelled out automatically, the resulting equation becomes algebraic for $\\alpha$ and one can express it in terms of $\\psi$ and $\\chi$ \n\\begin{equation}\n\\label{eq:alphaold}\n\\alpha = \\frac{a_3^2 \\psi' + a_3(a_7+j^2a_8) \\psi + [a_3 (a_2 - a_1') -j^2 a_1 a_8] \\chi' + a_3 (a_5 + j^2 a_6) \\chi}{j^2 \\left[ a_3 a_4' - a_3' a_4 - a_3 a_9 + a_4 (a_7 + j^2 a_8\\right]} \\;.\n\\end{equation}\nThen it becomes possible to express both $H_2$ in eq.~\\eqref{eq:H2old} and $H_1$ in eq.~\\eqref{eq:H1old} in terms \nof $\\psi$ and $\\chi$ only. Hence, the quadratic action~\\eqref{eq:action_even} can be cast in terms of two dynamical DOFs:\n\\[\n\\label{eq:even_action_final_old}\nS_{even}^{(2)} = \\int \\mbox{d}t\\:\\mbox{d}r \\sqrt{\\frac{A}{B}} J^2 \\left(\n\\frac12 \\mathcal{K}_{ij} \\dot{v}^i \\dot{v}^j - \\frac12 \\mathcal{G}_{ij} v^{i\\prime} v^{j\\prime} - \\mathcal{Q}_{ij} v^i v^{j\\prime} - \\frac12 \\mathcal{M}_{ij} v^i v^j \\right),\n\\]\nwhere $i=1,2$ and $v^1=\\psi$, $v^2=\\chi$. We note that\nterms which are higher order in derivatives, like\n$\\dot{\\psi}' \\dot{\\chi}$, disappear upon integrating by parts.\n\nThe no-ghost condition amounts to requiring that $\\mathcal{K}_{ij}$ in eq.~\\eqref{eq:even_action_final_old} is positive definite:\n\\[\n\\label{eq:no_ghost_old}\n\\mathcal{K}_{11}>0, \\qquad \\det(\\mathcal{K}) > 0,\n\\]\nwhere\n\\begin{equation}\n\\label{eq:K11old}\n{\\cal K}_{11}=\\frac{8 B {\\left( 2{\\cal H} J J' +\n\\Xi \\pi' \\right)}^2 \\left[ \\ell (\\ell+1){\\cal P}_1-{\\cal F}\\right]}\n{\\ell (\\ell+1) A^2 \\mathcal{H}^2 \\mathbf{\\Theta}^2},\n\\end{equation}\n\\begin{equation}\n\\label{eq:detKold}\n\\det ({\\cal K})=\\frac{16 B J'^2 (\\ell-1) (\\ell+2){\\left( 2{\\cal H} J J' + \\Xi \\pi'\\right)}^2 \\left[{\\cal F}(2{\\cal P}_1-{\\cal F})\\right]}\n{\\ell (\\ell+1) A^3 J^2 \\pi'^2 \\mathcal{H}^2 \\mathbf{\\Theta}^2},\n\\end{equation}\nwith $\\mathcal{F}$ and $\\mathcal{H}$ are given by \\eqref{eq:cal_FGH},\n\\[\n \\begin{aligned}\n\\label{eq:Xi}\n\\Xi = 2G_{4\\pi}J^2 + 4G_{4X}BJJ'\\pi'\n -2G_{4\\pi X}BJ^2\\pi'^2\n \n - 4G_{4XX}B^2JJ'\\pi'^3 \n + 16F_{4}B^2JJ'\\pi'^3 - 4F_{4X}B^3JJ'\\pi'^5, \\\\\n \\end{aligned}\n\\]\nand\n\\begin{subequations}\n\\begin{align}\n\\label{eq:Theta}\n\\mathbf{\\Theta} &= 2 \\ell (\\ell+1) \\: J\\left(\\mathcal{H} - 2F_4 B^2 \\pi'^4\\right) +\\mathcal{P}_2,\\\\\n\\label{eq:P1}\n\\mathcal{P}_1 &= \\frac{\\sqrt{B}}{\\sqrt{A}} \\cdot\n\\frac{\\mbox{d}}{\\mbox{d}r}\\left[\\frac{\\sqrt{A}}{\\sqrt{B}}\n\\frac{J^2 \\mathcal{H}\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)}{2{\\cal H} J J' + \\Xi \\pi'}\\right],\\\\\n\\mathcal{P}_2 &= \\frac{B(A' J - 2 A J')}{A}\n\\left(2{\\cal H} J J' + \\Xi \\pi'\\right)\n\\; .\n\\end{align}\n\\end{subequations}\nHere and in what follows we highlight $\\mathbf{\\Theta}$ in bold to emphasize that it involves $\\ell(\\ell+1)$. It follows immediately form eqs.~\\eqref{eq:K11old}-\\eqref{eq:detKold}\nthat both no-ghost conditions~\\eqref{eq:no_ghost_old} are satisfied \nprovided\n\\[\n\\label{eq:stability_even_ghost}\n2{\\cal P}_1-{\\cal F} > 0.\n\\]\n\n\nTo have no radial gradient instabilities one also has to require positive definiteness for matrix $\\mathcal{G}_{ij}$ in eq.~\\eqref{eq:even_action_final_old}:\n\\[\n\\label{eq:no_gradient_old}\n\\mathcal{G}_{11}>0, \\qquad \\det{\\mathcal{G}}>0.\n\\]\nHere\n\\begin{equation}\n\\label{eq:G11old}\n\\mathcal{G}_{11}=\\frac{4 B^{2} \\left[ \\mathcal{G} (\\ell+2)(\\ell-1) (2 \\mathcal{H} J J'+ \\Xi \\pi')^2+\\ell(\\ell+1)(2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B )\\right]}\n{ \\ell (\\ell+1) A \\mathcal{H}^2 \\mathbf{\\Theta} ^2},\n\\end{equation}\n\\begin{equation}\n\\label{eq:detGold}\n\\det(\\mathcal{G})=\\frac{16 B^3 J'^2 \\mathcal{G} (\\ell-1) (\\ell+2) (2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B )}\n{ \\ell (\\ell+1) A J^2 \\pi'^2 \\mathcal{H}^2 \\mathbf{\\Theta}^2},\n\\end{equation}\nwith\n\\begin{align}\n\\label{eq:Gamma}\n& \\Gamma = \\Gamma_1 + \\frac{A'}{A} \\Gamma_2,\n\\\\\n& \\nonumber\\Gamma_{1} = 4\\left(G_{4\\pi} + 2XG_{4\\pi X} + \\frac{B\\pi'J'}{J}(G_{4X} + 2XG_{4XX}) \\right) - \\frac{16J'}{J}B\\pi'X(2F_{4} + XF_{4X}), \\nonumber\\\\\n&\\Gamma_{2} = 2B\\pi'\\left(G_{4X} - B\\pi'^2G_{4XX}\\right)\n- 8B\\pi'X(2F_{4} + XF_{4X}),\n\\nonumber\n\\end{align}\nand\n\\begin{align}\n\\label{eq:KSI}\n& \\Sigma = XF_{X} + 2F_{XX}X^2 +2\\left(\\frac{1 - BJ'^2}{J^2} - \\frac{BJ'}{J} \\frac{A'}{A}\\right)X(G_{4X} + 2XG_{4XX})\n\\nonumber\\\\& \n - \\frac{4BJ'}{J}\\left( \\frac{J'}{J } + \\frac{A'}{A}\\right)X^2(3G_{4XX} + 2XG_{4XXX}) + 2B\\pi'\\left(\\frac{4J'}{J } + \\frac{ A'}{A} \\right)X( \\frac{3}{2} G_{4\\pi X} + XG_{4 \\pi XX})\n \\nonumber\\\\& \n + \\frac{B^3J'(A'J + AJ')}{AJ^2}\\pi'^4(12F_{4} - 9F_{4X}B\\pi'^2 + F_{4XX}B^2\\pi'^4).\n\\end{align}\nAccording to eqs.~\\eqref{eq:G11old} and~\\eqref{eq:detGold} both conditions~\\eqref{eq:no_gradient_old} hold provided that\n\\[\n\\label{eq:stability_even_radial}\n2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B > 0.\n\\]\n\nThe corresponding sound speeds squared for perturbations propagating along the radial direction are given by the eigenvalues of matrix $(AB)^{-1}(\\mathcal{K})^{-1}\\mathcal{G}$:\n\\begin{equation}\n\\label{eq:speed_even_old}\nc_{s1}^2 = \\frac{\\mathcal{G}}{\\mathcal{F}},\n\\qquad\nc_{s2}^2 = \\frac{(2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B )}\n{\\left(2{\\cal H} J J' + \\Xi \\pi'\\right)^2 (2{\\cal P}_1-{\\cal F})} \\; ,\n\\end{equation}\nwhere $c_{s1}^2$ coincides with the radial speed squared $c^2_r$ for odd-parity modes in eq.~\\eqref{eq:speed_odd}, which enables us\nto interpret it as a propagation speed in the radial direction of two tensor modes. \n\nPositive definiteness of matrices $\\mathcal{M}_{ij}$ and \n$\\mathcal{Q}_{ij}$ in the action~\\eqref{eq:even_action_final_old} also provide stability conditions, which ensure that angular gradient instabilities and \"slow\"\ntachyonic modes are absent. This set of conditions was not thoroughly addressed in previous works in Refs.~\\cite{Kobayashi:even,wormhole1,Trincherini} and we also omit discussing them for now. \nWe explicitly give the matrix $\\mathcal{M}_{ij}$ for high angular momentum modes\nas well as corresponding stability conditions for angular gradient instabilities in Regge-Wheeler-unitary \ngauge in Sec.~\\ref{sec:new_gauge}.\n\nLet us now show that the coefficients in the action~\\eqref{eq:even_action_final_old} become singular for modes with \nhigh momentum $\\ell$ if the no-ghost condition~\\eqref{eq:stability_even_ghost} is satisfied at all points.\nThe no-ghost condition~\\eqref{eq:stability_even_ghost} together with eq.~\\eqref{eq:P1} can be rewritten as follows:\n\\[\n\\label{eq:no_go1}\n\\frac{\\sqrt{B}}{\\sqrt{A}} \\xi' >\\frac{\\mathcal{F}}{2} \\quad \\mbox{with}\n\\quad \\xi = \\frac{\\sqrt{A}}{\\sqrt{B}}\n\\frac{J^2 \\mathcal{H}\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)}{2{\\cal H} J J' + \\Xi \\pi'}.\n\\]\nAs $\\mathcal{F} > 0$ due to stability constraints in the odd-parity sector~\\eqref{eq:stability_odd} it follows from eq.~\\eqref{eq:no_go1} that $\\xi'$ has to be positive.\nThen $\\xi$ is a monotonously growing function, so $\\xi=0$ at some point(s)\n\\footnote{All other options where $\\xi'>0$ but $\\xi$ does not cross zero involve either fine-tuning or special cases like $\\mathcal{F} \\to 0$ as $r \\to -\\infty$, which signals potential strong coupling in the odd-parity sector, see eq.~\\eqref{eq:action_odd_final} (see e.g. Ref.~\\cite{wormhole1} for a discussion).}.\nThe latter happens provided the numerator of $\\xi$ crosses zero,\nwhich is possible since the \ncombination $\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)$ can take zero value, while $\\mathcal{H} >0$ due to stability in the odd-parity sector~\\eqref{eq:stability_odd}. \nLet us note here that \nin Horndeski theories $F_4 = 0$, so $\\xi$ cannot cross zero in a healthy way and, hence, the no-ghost condition~\\eqref{eq:stability_even_ghost} cannot be met at all points in this case. This is what is usually referred to as a no-go theorem in Horndeski theories, see Refs.~\\cite{Rubakov:2016zah,Olegi} for details.\n\nWhat is important for us here is that the same combination \n$\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)$ gives the leading contribution into $\\mathbf{\\Theta}$ in eq.~\\eqref{eq:Theta} for modes with $\\ell \\gg 1$. \nThis in turn means that as soon as\nthe no-ghost condition~\\eqref{eq:stability_even_ghost} is satisfied\nby having $\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)=0$ \nat some point(s), $\\mathbf{\\Theta} \\to 0$ for high momenta modes. Thus,\nthe components of both matrices \n$\\mathcal{K}_{ij}$ and $\\mathcal{G}_{ij}$\nhit infinity for high momenta modes since they involve $\\mathbf{\\Theta}^2$ in denominators, see eqs.~\\eqref{eq:K11old},~\\eqref{eq:detKold},~\\eqref{eq:G11old} and ~\\eqref{eq:detGold}. \nThis is the problem\nof singular coefficients in the quadratic action upon $\\mathbf{\\Theta}$-crossing. \n\nThe appearance of $\\mathbf{\\Theta}$ in denominator seems to result from the gauge choice, and this choice, in particular, dictates the way non-dynamical DOFs are integrated out in action~\\eqref{eq:action_even}. Namely, the $\\ell^2$ part of $\\mathbf{\\Theta}$ comes from \nthe combination (see Appendix B for definitions of $a_i$)\n$$j^2 a_4 a_8 = j^2 \\frac A2 \\cdot \\mathcal{H} \\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right) \\;,$$\nwhich arises in the denominator of eq.~\\eqref{eq:alphaold} and originates from the field redefinition~\\eqref{eq:H2old}. \n\nThe fact that $\\mathbf{\\Theta}$-crossing\nis directly related to ensuring stability at the linearized level makes the old gauge unsuitable for construction of a completely stable wormhole solution.\n\nIn the following section we rederive the quadratic action for\nbeyond Horndeski Lagrangian~\\eqref{eq:lagrangian} \nin a Regge-Wheeler-unitary gauge and pay special attention to potential divergencies in the coefficients of the action due to $\\mathbf{\\Theta}$ crossing zero. So we keep track of $\\mathbf{\\Theta}$ and ensure the coefficients in the quadratic action do not hit singularities, so that we end up with \na regular quadratic action.\n\n\n\n\n\\subsubsection{Original calculations in the Regge-Wheeler-unitary gauge ($\\beta=G=\\chi=0$)}\n\\label{sec:new_gauge}\n\nThe quadratic action for even-parity modes in the Regge-Wheeler-unitary gauge reads:\n\\begin{equation}\n\\label{eq:action_even_RW}\n\\begin{aligned}\n&S_{even}^{(2)} = \\int \\mbox{d}t\\:\\mbox{d}r \n\\left(H_0 \\left[ a_3 H_2' + \\left( a_7 + j^2 a_8 \\right)H_2 + j^2 a_4 \\alpha' + j^2 a_9 \\alpha + a_{13} K'' + a_{14} K'\\right.\\right. \\\\\n&\\left.\\left. \n+ (a_{15} + j^2 a_{16}) K \\right]\n+j^2 b_1 H_1^2 + H_1 \\left[b_4 {\\dot H_2}+j^2b_5 {\\dot \\alpha} + b_{8}\\dot{K}' + b_{9}\\dot{K} \\right] + c_{8} \\dot{H}_2 \\dot{K} + c_6 H_2^2 \\right. \\\\\n&\\left. + H_2 \\left[ j^2 c_5 \\alpha + c_{11} K' + (c_{12} +j^2 c_{13})K \\right] + j^2d_1 {\\dot \\alpha}^2 +j^2 d_4 \\alpha^2 + j^2 d_{7} \\alpha K' + p_{8}\\dot{K}^2 + p_{9}K'^2\n\\right),\n\\end{aligned}\n\\end{equation}\nwhere the notations are similar to those in action~\\eqref{eq:action_even}\nand the coefficients $a_i$, $b_i$, $c_i$, $d_i$ and $p_i$ are given in Appendix B. In full analogy with the spherical gauge case above variation of the action~\\eqref{eq:action_even_RW} w.r.t. $H_0$ and $H_1$ gives two constraint equations, respectively:\n\\begin{equation}\n\\label{eq:H0_RW}\na_3 H_2' +\\left( a_7+j^2 a_8 \\right)H_2 + j^2 a_4 \\alpha'\n+ j^2 a_9 \\alpha + a_{13}K'' + a_{14}K' +(a_{15}+j^2 a_{16})K = 0,\n\\end{equation}\n\\begin{equation}\n\\label{eq:H1_RW}\nH_1 = - \\frac{1}{2 j^2 b_1} \\left( b_4 {\\dot H_2}+j^2b_5 {\\dot \\alpha} \n + b_8 \\dot{K}' +b_9 \\dot{K} \\right).\n\\end{equation}\nThis time the constraint~\\eqref{eq:H0_RW} contains not only the first derivatives of $H_2$, $\\alpha$ and $K$ but also $K''$. To simplify things\nlet us introduce an axillary field $\\psi$ as follows (cf. eq.~\\eqref{eq:H2old})\n\\begin{equation}\n\\label{eq:alpha_shifted}\n\\alpha = \\psi - \\dfrac{a_{13}}{j^2 a_4} K'. \n\\end{equation}\nLet us note here that $a_{13}= J^2 \\cdot a_4$ (see Appendix B), so the redefinition above is regular.\nUpon substitution of $\\alpha$ and $\\alpha'$ from eq.~\\eqref{eq:alpha_shifted} into the constraint~\\eqref{eq:H0_RW}, one can see that not only $K''$ get canceled out but also the resulting factor in front of $K'$ is automatically vanishing (see Appendix B for the explicit form of $a_i$):\n\\[\na_{14} - a_{13}' +\\frac{a_{13}}{a_4}(a_4' - a_9) = 0.\n\\]\nHence, the resulting equation is algebraic for $K$ and reads\n\\begin{equation}\n\\label{eq:K}\nK = - \\dfrac{1}{a_{15} +j^2 a_{16}} \\left[ a_3 H_2' +(a_7+j^2 a_8) H_2 +j^2 a_4 \\psi' +j^2 a_9 \\psi \\right].\n\\end{equation}\nNow by substituting $\\alpha$ from eq.~\\eqref{eq:alpha_shifted} and $K$ from eq.~\\eqref{eq:K} into the second constraint~\\eqref{eq:H1_RW} one can express $H_1$ in terms of $H_2$ and $\\psi$. Therefore, by making use of both constraint equations~\\eqref{eq:H0_RW}-\\eqref{eq:H1_RW} and field redefinition~\\eqref{eq:alpha_shifted} one can cast the quadratic action~\\eqref{eq:action_even_RW} in terms of $H_2$ and $\\psi$:\n\\begin{equation}\n\\label{eq:unconstrained_action_RW}\n\\begin{aligned}\nS_{even}^{(2)} = &\\int \\mbox{d}t\\:\\mbox{d}r \\left(\ng_{4(i)}\\dot{\\psi}'\\:^2 + g_{5(i)}\\psi''\\:^2 \n+f_{4(i)}\\dot{H_2}'^2 + f_{5(i)} H_2''^2\n+ h_{7(i)} \\dot{H_2}' \\dot{\\psi}' + h_{8(i)} H_2'' \\psi''\n\\right.\\\\ \n&\\left.\n+g_{1(i)}\\dot{\\psi}^2 + g_{2(i)}\\psi'\\:^2 + g_{3(i)}\\psi^2\n + f_{1(i)}\\dot{H_2}^2 + f_{2(i)}H_2'^2 + f_{3(i)}H_2^2 \n+ h_{5(i)}\\dot{H_2} \\dot{\\psi}' \n\\right.\\\\ \n&\\left.\n+ h_{6(i)} H_2' \\psi'' + h_{1(i)}\\dot{H_2} \\dot{\\psi} + h_{2(i)} H_2' \\psi' + h_{3(i)} H_2 \\psi' + h_{4(i)} H_2 \\psi\n\\right),\n\\end{aligned}\n\\end{equation}\nwhere $g_{j(i)}$, $f_{j(i)}$ and $h_{j(i)}$ are some combinations of the initial coefficients $a_i$, $b_i$, $c_i$, $d_i$ and $p_i$ in eq.~\\eqref{eq:action_even_RW} and we give explicitly only those attributed to terms with four derivatives (see the first line of eq.~\\eqref{eq:unconstrained_action_RW}): \n\\begin{equation}\n\\label{eq:action_coef1}\n\\begin{aligned}\n& g_{4(i)} = \\frac{2\\ell(\\ell+1)B^{3\/2} J^2 }{(\\ell+2)(\\ell-1)A^{1\/2}}\\frac{\\mathcal{H}^2}{\\mathcal{F}}, \\qquad \\qquad\ng_{5(i)}= - g_{4(i)} \\cdot A B \\frac{\\mathcal{G}}{\\mathcal{F}}, \\\\\n& f_{4(i)} = \\frac{B^{3\/2} J^2}{2 \\ell(\\ell+1)(\\ell+2)(\\ell-1) A^{1\/2}} \\frac{(2\\mathcal{H} J J' + \\Xi \\pi')^2}{\\mathcal{F}}, \\quad\nf_{5(i)} = - f_{4(i)} \\cdot A B \\frac{\\mathcal{G}}{\\mathcal{F}}, \\\\\n& h_7 = - \\frac{2 B^{3\/2} J^2}{(\\ell+2)(\\ell-1) A^{1\/2}} \\frac{\\mathcal{H}(2\\mathcal{H} J J' + \\Xi \\pi')}{\\mathcal{F}}, \\qquad \\qquad\nh_8 = - h_7 \\cdot A B \\frac{\\mathcal{G}}{\\mathcal{F}},\n\\end{aligned}\n\\end{equation}\nwhere we made use of Appendix B to substitute $a_i$, $b_i$, etc. The rest of coefficients in eq.~\\eqref{eq:unconstrained_action_RW} are irrelevant at this point, however, we should note that all of them are regular.\nNote that according to\nthe unconstrained action~\\eqref{eq:unconstrained_action_RW} \nboth $\\psi$ and $H_2$ are naively described by the fourth order equations. Let us now show that in fact both DOFs are described by the second order equations upon making suitable field redefinitions.\n\nIt immediately follows from eq.~\\eqref{eq:action_coef1} that the following relations hold\n\\begin{equation}\n\\begin{aligned}\n&4\\; g_{4(i)} \\cdot f_{4(i)} - h_7^2 = 0, \\\\\n&4\\;g_{5(i)} \\cdot f_{5(i)} - h_8^2 = 0,\n\\end{aligned}\n\\end{equation}\nwhich means that the corresponding terms $\\dot{\\psi}'^2$, $\\dot{H_2}'^2$, $\\dot{H_2}'\\dot{\\psi}'$ and $\\psi''^2$, $H_2''^2$, $\\psi''H_2''$ get combined into perfect squares in eq.~\\eqref{eq:unconstrained_action_RW}. Hence, upon a field redefinition\n\\begin{equation}\n\\label{eq:tildePsi}\n\\begin{aligned}\n& \\psi = \\tilde{\\psi} + \\frac{(2\\mathcal{H} J J' + \\Xi \\pi')}{2\\ell(\\ell+1) \\; \\mathcal{H}} \\tilde{H_2}, \\\\\n& H_2 = \\tilde{H_2}.\n\\end{aligned}\n\\end{equation}\nthe terms $\\dot{\\tilde{H_2}}'^2$, $\\tilde{H_2}''^2$, $\\dot{\\tilde{H_2}}' \\dot{\\psi}'$ and $\\tilde{H_2}'' \\psi''$ vanish and the action~\\eqref{eq:unconstrained_action_RW} reduces to the following:\n\\begin{equation}\n\\label{eq:unconstrained_action_tildePsi}\n\\begin{aligned}\nS_{even}^{(2)} = &\\int \\mbox{d}t\\:\\mbox{d}r \\left(\ng_4\\; \\dot{\\tilde\\psi}'\\:^2 + g_5 \\;\\tilde\\psi''\\:^2 + g_1\\; \\dot{\\tilde\\psi}^2 + g_{2} \\;\\tilde\\psi'\\:^2 + g_{3} \\;\\tilde\\psi^2\n+ f_1 \\;\\dot{\\tilde{H_2}}^2 + f_2 \\;\\tilde{H_2}'^2 \\right.\\\\ \n&\\left.+ f_{3} \\;\\tilde{H_2}^2 \n+ h_5\\; \\dot{\\tilde{H_2}} \\dot{\\tilde\\psi}' + h_6\\; \\tilde{H_2}' \\tilde\\psi'' + h_1\\; \\dot{\\tilde{H_2}} \\dot{\\tilde\\psi} + h_2\\; \\tilde{H_2}' \\tilde\\psi' + h_{3}\\; \\tilde{H_2} \\tilde\\psi'+ h_{4}\\; \\tilde{H_2} \\tilde\\psi\n\\right),\n\\end{aligned}\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:coef_tildePsi}\n\\begin{aligned}\n& g_4 = \\frac{2 \\ell(\\ell+1) B^{3\/2} J^2 }{(\\ell^2+\\ell-2) A^{1\/2}} \\frac{\\mathcal{H}^2}{\\mathcal{F}}, \n\\qquad \\qquad\ng_5 = - \\frac{2 \\ell(\\ell+1) A^{1\/2} B^{5\/2} J^2}{\\ell^2 + \\ell -2} \\frac{\\mathcal{H}^3}{\\mathcal{F}^2}, \\\\\n& \nh_5 = - \\frac{B^{1\/2} J}{(\\ell^2+\\ell-2) A^{1\/2}} \\frac{\\mathcal{H} \\mathbf{\\Theta}}{\\mathcal{F}},\n\\qquad \\qquad\nh_6 = \\frac{A^{1\/2}B^{3\/2} J}{(\\ell^2+\\ell-2)} \\frac{\\mathcal{H}^2 \\mathbf{\\Theta}}{\\mathcal{F}^2},\\\\\n&f_1 = \\frac{1}{8\\ell(\\ell+1)(\\ell^2+\\ell-2)(AB)^{1\/2}} \\frac{\\mathbf{\\Theta}^2}{ \\mathcal{F}},\n\\qquad \\qquad\nf_2 = \\frac{ (AB)^{1\/2}}{8\\ell(\\ell+1)(\\ell^2+\\ell-2)}\\frac{\\mathcal{H} \\mathbf{\\Theta}^2}{ \\mathcal{F}^2},\n\\end{aligned}\n\\end{equation}\nwhile the explicit form of the rest of coefficients \n$g_j$, $f_j$ and $h_j$ is irrelevant here (they do not involve $\\mathbf{\\Theta}$ and were checked to be perfectly regular).\nNote that there are still terms with higher order derivatives \nof $\\tilde{\\psi}$ in the\naction~\\eqref{eq:unconstrained_action_tildePsi}. \nHowever, we see again\nthat terms $g_4\\; \\dot{\\tilde\\psi}'\\:^2 $, $h_5\\; \\dot{\\tilde{H_2}}\\dot{\\tilde\\psi}'$, $f_1 \\dot{\\tilde{H_2}}^2$ and $g_5 {\\tilde\\psi}''\\:^2$,\n$h_6\\; \\tilde{H_2}'{\\tilde\\psi}''$, $f_2\\;\\tilde{H_2}'^2$ get combined into perfect squares since\n\\begin{equation}\n\\label{eq:pefect_squares2}\n\\begin{aligned}\n&4\\; g_{4} \\cdot f_{1} - h_5^2 = 0, \\\\\n&4\\;g_{5} \\cdot f_{2} - h_6^2 = 0,\n\\end{aligned}\n\\end{equation}\nhence, it is possible to redefine $\\tilde{H_2}$ as\n\\begin{equation}\n\\label{eq:tildePsi2}\n\\begin{aligned}\n& \\tilde{\\psi} = \\bar{\\psi}, \\\\\n& \\tilde{H_2} = \\bar{H_2} + 4 \\ell(\\ell+1) B J \\dfrac{\\;{\\cal H} }{\\mathbf{\\Theta}} \\; \\bar\\psi'.\n\\end{aligned}\n\\end{equation}\nThen the terms with higher derivatives vanish and the resulting action gives the second order differential equations of motion for both $\\bar{\\psi}$ and $\\bar{H_2}$ in full analogy with the case of Sec.~\\ref{sec:old_gauge}. \n\nThere is, however, a problem with redefinition~\\eqref{eq:tildePsi2}: it diverges in the vicinity of $\\mathbf{\\Theta} = 0$, which means that the final action for $\\bar{\\psi}$ and $\\bar{H_2}$\nalso involves singular coefficients and this is exactly what we tried to avoid. One possible way to proceed is to refrain from the redefinition~\\eqref{eq:tildePsi2} and use the action~\\eqref{eq:unconstrained_action_tildePsi} for further stability analysis. The advantage of the latter approach is that all coefficients in the action are regular everywhere\nincluding $\\mathbf{\\Theta}=0$, but at the same time there are still higher derivative terms \nof $\\tilde{\\psi}$, which naively signal that the corresponding system of equations of motion\nmight admit more than four solutions for $\\tilde{\\psi}$ and $\\tilde{H_2}$, i.e. there are additional pathological DOFs. \nOne should note, however, that the redefinition~\\eqref{eq:tildePsi2} is perfectly fine away from $\\mathbf{\\Theta}=0$ and it demonstrates that in fact the action~\\eqref{eq:unconstrained_action_tildePsi} gives two second order differential equations, which normally admit four solutions. So in the following section let us\nprove that even in the vicinity of $\\mathbf{\\Theta}=0$ the action~\\eqref{eq:unconstrained_action_tildePsi} describes a healthy system, giving four regular solutions ${\\left(\n \\begin{array}{c}\n \\tilde{\\psi} \\\\\n \\tilde{H}_2 \\\\\n \\end{array}\n\\right)}$.\nThis legitimizes redefinition~\\eqref{eq:tildePsi2} at all points including $\\mathbf{\\Theta}=0$ since we explicitly show that potential divergencies in the final quadratic action for $\\bar{\\psi}$\nand $\\bar{H_2}$ are not inherited from the original action~\\eqref{eq:unconstrained_action_tildePsi} and\ndo not have any physics behind them but result from our field redefinition. \n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{$\\mathbf{\\Theta}$-crossing}\n\\label{sec:TT_crossing}\n\nLet us explicitly solve the linearized equations for $\\tilde{H_2}$ and $\\tilde{\\psi}$ following from the action~\\eqref{eq:unconstrained_action_tildePsi} in the vicinity of $\\mathbf{\\Theta}=0$.\nThe general form of these linearized equations reads:\n\\begin{subequations}\n\\label{eq:deltaPsiH2}\n\\begin{align}\n\\label{eq:deltaPsi}\n\\delta\\tilde{\\psi}: \\;\\; &\n 2 g_5 \\tilde{\\psi}'''' + 4 g_5' \\tilde{\\psi}''' + (2 g_5 '' - 2 g_2)\\tilde{\\psi}'' - 2 g_2' \\tilde{\\psi}' + 2 g_3 \\tilde{\\psi} + h_6 \\tilde{H_2}''' + (2 h_6' - h_2) \\tilde{H_2}'' \\nonumber\\\\\n& + (h_6'' - h_2' - h_3) \\tilde{H_2}' + (h_4 - h_3')\\tilde{H_2} + 2 g_4 \\ddot{\\tilde{\\psi}}'' + 2 g_4' \\ddot{\\tilde{\\psi}}' \n- 2 g_1 \\ddot{\\tilde{\\psi}} + h_5 \\ddot{\\tilde{H_2}}' + (h_5' - h_1) \\ddot{\\tilde{H_2}} = 0, \\\\\\nonumber\n\\\\\n\\label{eq:deltaH2}\n\\delta\\tilde{H_2}: \\;\\; &\n2 f_2 \\tilde{H_2}'' + 2 f_2 \\tilde{H_2}' - 2 f_3 \\tilde{H_2} + h_6 \\tilde{\\psi}''' + (h_6'+h_2)\\tilde{\\psi}''\n+ (h_2' - h_3)\\tilde{\\psi}' - h_4 \\tilde{\\psi} \n+ 2 f_1 \\ddot{\\tilde{H_2}} + h_5 \\ddot{\\tilde{\\psi}}' \n+h_1 \\ddot{\\tilde{\\psi}} = 0.\n\\end{align}\n\\end{subequations}\nSince we consider a static, spherically-symmetric background \nfor both $\\tilde{\\psi}(t,r)$ and \n$\\tilde{H_2}(t,r)$ we use constant energy Ansatz:\n\\begin{equation}\n\\tilde{\\psi}(t,r) = \\tilde{\\psi}(r) e^{i\\omega t}, \\qquad\n\\tilde{H_2}(t,r) = \\tilde{H_2}(r) e^{i\\omega t},\n\\end{equation}\nwhere $\\omega$ denotes frequency. Until the end of this section we generally drop the argument of both $\\tilde{\\psi}(r)$ and $\\tilde{H_2}(r)$ for brevity. \n\nThe system~\\eqref{eq:deltaPsiH2} consists of the fourth and third order differential equations, which can be combined into the system of two third order differential equations. To see this one has to take a \ncoordinate derivative of eq.~\\eqref{eq:deltaH2} and then take a linear combination\nof the resulting equation and eq.~\\eqref{eq:deltaPsi} with a factor\n\\begin{equation}\n\\label{eq:constC}\n\\mathcal{C} = \\dfrac{1}{4 \\ell(\\ell+1) B J } \\dfrac{\\mathbf{\\Theta}}{\\cal{H}},\n\\end{equation}\nso that both \n$(2 g_5 \\mathcal{C} + h_6)\\tilde{\\psi}''''$ and $(h_6 \\mathcal{C} + 2 f_2)\\tilde{H_2}'''$ vanish (see eq.~\\eqref{eq:action_coef1}).\nThen the resulting system reads as follows:\n\\begin{subequations}\n\\label{eq:deltaPsiH2new}\n\\begin{align}\n& [4 g_5' \\mathcal{C} + 2 h_6' + h_2]\\;\\tilde{\\psi}''' + [2g_5'' \\mathcal{C} - 2g_2 \\mathcal{C} +h_6'' +2 h_2' - h_3 - \\omega^2 (2 g_4 \\mathcal{C} +h_5) ]\\; \\tilde{\\psi}'' \n+ [ h_2''-2 g_2' \\mathcal{C} - h_3' - h_4 \n\\nonumber\\\\\\nonumber\n&- \\omega^2(2 g_4' \\mathcal{C} + h_5' + h_1)]\\; \\tilde{\\psi}' + [2g_3 \\mathcal{C} - h_4' - \\omega^2(2 g_1 \\mathcal{C} + h_1')] \\tilde{\\psi}\n+ [2 h_6' \\mathcal{C} - h_2 \\mathcal{C} + 4 f_2'] \\tilde{H_2}'' \n+ [h_6'' \\mathcal{C} - h_2' \\mathcal{C} \n\\nonumber\\\\\n& \n- h_3 \\mathcal{C} + 2f_2'' -2 f_3 - \\omega^2 (h_5 \\mathcal{C} + 2 f_1)] \\tilde{H_2}' - [h_3' \\mathcal{C} - h_4 \\mathcal{C} + 2 f_3' - \\omega^2 (\\mathcal{C}(h_5' - h_1) + 2 f_1')] \\tilde{H_2} = 0,\\\\\\nonumber\n\\\\\n& 2 f_2 \\tilde{H_2}'' + 2 f_2 \\tilde{H_2}' - 2 f_3 \\tilde{H_2} + h_6 \\tilde{\\psi}''' + (h_6'+h_2)\\tilde{\\psi}''\n+ (h_2' - h_3)\\tilde{\\psi}' - h_4 \\tilde{\\psi} - \\omega^2 ( 2 f_1 \\tilde{H_2} + h_5 \\tilde{\\psi}' \n+h_1\\tilde{\\psi}) = 0,\n\\end{align}\n\\end{subequations}\nwhere the leading terms with derivative \nare $\\tilde{\\psi}'''$ and $\\tilde{H_2}''$.\nLet us now prove that the system~\\eqref{eq:deltaPsiH2new} admits only four regular solutions in the vicinity of $\\mathbf{\\Theta} = 0$.\n\nWe arrange the coordinate system so that $\\mathbf{\\Theta}$ crosses zero at $r=0$ so we write\n\\begin{equation}\n\\label{eq:linear_theta}\n\\mathbf{\\Theta} = \\gamma \\cdot r + \\dots,\n\\end{equation}\nwhere $\\gamma$ is a regular constant and dots denote higher order \nin $r$. So from now on we focus on the behaviour \nof the system~\\eqref{eq:deltaPsiH2new} in the vicinity of $r=0$. \nThe coefficients $h_j$, $g_j$ and $f_j$ \nin eqs.~\\eqref{eq:deltaPsiH2new} are regular at all points and can be expanded into power-series around $r=0$: \n\\begin{equation}\n\\begin{aligned}\n\\label{eq:ansatz_hgf}\n& h_j = h_{j,0} + h_{j,1} \\cdot r + h_{j,2}\\cdot r^2 + \\dots, \\quad (j=\\overline{1,4})\\\\\n& h_5 = h_{5,1}\\cdot r + h_{5,2}\\cdot r^2 + \\dots,\\\\\n& h_6 = h_{6,1}\\cdot r + h_{6,2}\\cdot r^2 + \\dots,\\\\\n& g_j = g_{j,0} + g_{j,1}\\cdot r + g_{j,2}\\cdot r^2 + \\dots, \\quad (j=\\overline{1,5})\\\\\n& f_3 = f_{3,0} + f_{3,1}\\cdot r + f_{3,2}\\cdot r^2 + \\dots,\n\\end{aligned}\n\\end{equation}\nwhere $h_{j,k}$, $g_{j,k}$ and $f_{j,k}$ are constants and for \n$f_1$ and $f_2$ we make use of eqs.~\\eqref{eq:pefect_squares2} to express them as follows: \n\\begin{equation}\n\\label{eq:f1f2}\nf_1 = \\frac{h_5^2}{4\\; g_4}, \\qquad f_2 = \\frac{h_6^2}{4\\; g_5}.\n\\end{equation}\nLet us note the expansions for $h_5$ and $h_6$ start from linear terms according to their definitions in eqs.~\\eqref{eq:coef_tildePsi}.\n\nSo upon substitution of \nthe expansions~\\eqref{eq:linear_theta}-\\eqref{eq:f1f2} into \neqs.~\\eqref{eq:deltaPsiH2new}\nwe find four independent solutions, which to the leading order read:\n\\begin{subequations}\n\\label{eq:regular_solutions}\n\\begin{align}\n&\\left(\n \\begin{array}{c}\n \\tilde{\\psi} \\\\\n \\tilde{H}_2 \\\\\n \\end{array}\n\\right)\n=\nc_1\\exp(i\\omega t)\\left(\n \\begin{array}{c}\n 1 + O(r^4)\\\\\n \\dfrac{-h_{1,0}\\cdot\\omega^2-h_{4,0}}{2 f_{3,0}}+O(r) \\\\\n \\end{array}\n \\right),\\\\\n&\\left(\n \\begin{array}{c}\n \\tilde{\\psi} \\\\\n \\tilde{H}_2 \\\\\n \\end{array}\n\\right)\n=\nc_2\\exp(i\\omega t)\\left(\n \\begin{array}{c}\n r +O(r^4) \\\\\n \\dfrac{h_{2,1}-h_{3,0}}{2 f_{3,0}} + O(r)\\\\\n \\end{array}\n \\right),\\\\\n&\\left(\n \\begin{array}{c}\n \\tilde{\\psi} \\\\\n \\tilde{H}_2 \\\\\n \\end{array}\n\\right)\n=\nc_3\\exp(i\\omega t)\\left(\n \\begin{array}{c}\n r^2 + O(r^4)\\\\\n \\dfrac{h_{2,0}+h_{6,1}}{f_{3,0}} + O(r)\\\\\n \\end{array}\n \\right),\\\\\n&\\left(\n \\begin{array}{c}\n \\tilde{\\psi} \\\\\n \\tilde{H}_2 \\\\\n \\end{array}\n\\right)\n=\nc_4\\exp(i\\omega t)\\left(\n \\begin{array}{c}\n r^3 + O(r^4)\\\\\n \\dfrac{(6h_{2,0}+12h_{6,1})g_{5,0}}{2g_{5,0}f_{3,0}-h_{6,1}^2} r + O(r^2)\\\\\n \\end{array}\n \\right),\n\\end{align}\n\\end{subequations}\nwhere $c_i$ are constants.\nThe solutions~\\eqref{eq:regular_solutions} \nare indeed regular at $r=0$, i.e. when $\\mathbf{\\Theta}$ crosses zero. \nLet us note that as soon as \n$\\tilde{\\psi}$ and $\\tilde{H_2}$ are found the rest of\nmetric perturbations, namely, $K$, $\\alpha$, $H_1$ and $H_0$ can be restored from \neqs.~\\eqref{eq:tildePsi},~\\eqref{eq:H0_RW}-\\eqref{eq:alpha_shifted} \nand they turn out to be regular as well since denominators in the corresponding relations are non-zero.\n\nBy proving that solutions~\\eqref{eq:regular_solutions} behave \nregularly around $\\mathbf{\\Theta}$-crossing \nwe have shown that $\\mathbf{\\Theta}=0$ is not a special\npoint for the action~\\eqref{eq:unconstrained_action_tildePsi}. Hence, \nthe potential singularities at $\\mathbf{\\Theta}=0$ \nin the resulting action for $\\bar{\\psi}$ and\n$\\bar{H_2}$ after redefinition~\\eqref{eq:tildePsi2} should be \nattributed\nto the singularity of the redefinition upon $\\mathbf{\\Theta}$-crossing.\nKeeping this in mind, \nin the following section we explicitly adopt redefinition~\\eqref{eq:tildePsi2}\nin order to have the quadratic action for perturbations in a conventional form and \nderive a set of stability conditions for two DOFs $\\bar{\\psi}$ and $\\bar{H_2}$.\n\n\n\n\n\n\n\n\n\\subsubsection{Stability constraints in Regge-Wheeler-unitary gauge}\n\\label{sec:new_stability_constraints}\n\nLet us now make use of the redefinition~\\eqref{eq:tildePsi2} and cast \nthe action~\\eqref{eq:unconstrained_action_tildePsi} in a conventional form similar to that in the spherical gauge (cf.~\\eqref{eq:even_action_final_old}): \n\\begin{equation}\n\\label{eq:action_KGQM_new}\nS_{even}^{(2)} = \\int \\mbox{d}t\\:\\mbox{d}r \n\\left(\n\\frac12 \\bar{\\mathcal{K}}_{ij} \\dot{v}^i \\dot{v}^j - \\frac12 \\bar{\\mathcal{G}}_{ij} v^{i\\prime} v^{j\\prime} - \\bar{\\mathcal{Q}}_{ij} v^i v^{j\\prime} - \\frac12 \\bar{\\mathcal{M}}_{ij} v^i v^j\n\\right),\n\\end{equation}\nwhere $i=1,2$ with $v^1 = \\bar{H_2}$, $v^2 = \\bar{\\psi}$ and both DOFs are described by the second order differential equations.\nThe explicit expressions for $\\bar{\\mathcal{K}}_{11}$, $\\det\\bar{\\mathcal{K}}$, $\\bar{\\mathcal{G}}_{11}$ and $\\det\\bar{\\mathcal{G}}$ \nare as follows:\n{\\small\n\\begin{equation}\n\\label{eq:k11detk_RW}\n\\bar{\\mathcal{K}}_{11} = \n\\dfrac{ 1}{4 \\ell(\\ell+1)(\\ell +2)(\\ell-1) \\sqrt{AB}}\n\\dfrac{\\mathbf{\\Theta}^2}{\\mathcal{F}},\n\\quad\n\\det\\bar{\\mathcal{K}} = \\dfrac{\\ell(\\ell+1) B}{4(\\ell+2)(\\ell-1)A}\n\\dfrac{(2\\mathcal{H} J J' + \\Xi \\pi')^2 (2 \\mathcal{P}_1 - \\mathcal{F})}{\\mathcal{F}},\n\\end{equation}\n\\begin{equation}\n\\label{eq:g11detg_RW}\n\\bar{\\mathcal{G}}_{11} = \\dfrac{\\sqrt{AB} }{4 \\ell(\\ell+1)(\\ell +2)(\\ell-1)} \\dfrac{\\mathcal{H}\\mathbf{\\Theta}^2}{\\mathcal{F}^2}, \\quad\n\\det\\bar{\\mathcal{G}}=\\dfrac{\\ell (\\ell+1) A B^3}\n{ 4(\\ell+2) (\\ell-1)}\n\\dfrac{\\mathcal{G} (2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B )}{ \\mathcal{F}^2 },\n\\end{equation}}\nwhile the rest of components in both matrices are given in Appendix C.\nLet us note that $\\det\\bar{\\mathcal{K}}$ differ from its counterpart in the old gauge~\\eqref{eq:detKold} by a certain non-negative factor:\n\\begin{equation}\n\\label{eq:relationKK}\n\\det\\bar{\\mathcal{K}} = \\dfrac{\\ell^2(\\ell+1)^2 AB \\pi'^2}{64(\\ell-1)^2(\\ell+2)^2 J'^2} \\dfrac{\\mathcal{H}^2 \\mathbf{\\Theta}^2}{\\mathcal{F}^2} \\; \\det{\\mathcal{K}},\n\\end{equation}\nand the same holds for $\\det\\bar{\\mathcal{G}}$ and $\\det{\\mathcal{G}}$~\\eqref{eq:detGold}. The relation~\\eqref{eq:relationKK} serves as a cross check for our calculations in the Regge-Wheeler-unitary gauge. \nAccording to eqs.~\\eqref{eq:k11detk_RW} and~\\eqref{eq:g11detg_RW}\nno-ghost and no radial gradient instabilities constraints are still given by eqs.~\\eqref{eq:no_ghost_old} and~\\eqref{eq:no_gradient_old}, and sound speeds squared for radial perturbations are also unchanged and are given by eqs.~\\eqref{eq:speed_even_old}.\n\nAs for potentially divergent coefficients in the action~\\eqref{eq:action_KGQM_new} upon $\\mathbf{\\Theta}$-crossing\nwe see that both\n$\\det\\bar{\\mathcal{K}}$, $\\det\\bar{\\mathcal{G}}$, $\\bar{\\mathcal{K}}_{11}$ and $\\bar{\\mathcal{G}}_{11}$\nare non-singular \nat $\\mathbf{\\Theta}=0$ while $\\bar{\\mathcal{K}}_{22}$ and $\\bar{\\mathcal{G}}_{22}$ hit singularities (see Appendix C). \nLet us recall that $\\mathbf{\\Theta}$ inevitably crosses zero at some point(s)\nprovided that the no-ghost constraint is satisfied (see Sec.~\\ref{sec:old_gauge}).\nSo in the end we have not avoided \nsingular coefficients in the quadratic \naction~\\eqref{eq:action_KGQM_new}, but unlike the case of \naction~\\eqref{eq:even_action_final_old} in the spherical gauge we have explicitly checked that these divergencies \nappear due to our technical redefinition~\\eqref{eq:tildePsi2}\nand have no physical nature. \n\nThe significant difference between the new \nmatrices $\\bar{\\mathcal{K}}_{ij}$, $\\bar{\\mathcal{G}}_{ij}$ \nand ${\\mathcal{K}}_{ij}$, ${\\mathcal{G}}_{ij}$\nin the old gauge is that the determinants of former ones do not involve $J'$ as a common factor, cf. eqs.~\\eqref{eq:k11detk_RW},~\\eqref{eq:g11detg_RW} and eqs.~\\eqref{eq:detKold},~\\eqref{eq:detGold}. \nThis fact becomes important in stability analysis for a wormhole\nsolution, whose characteristic tunnel-like profile is encoded in the metric function\n$J(r)$, and $J'(r)=0$ at the narrowest point of the throat. The latter \nmade both $\\det\\mathcal{K}$ and $\\det\\mathcal{G}$ automatically vanishing, which made the \nstability analysis in the spherical gauge not entirely conclusive close to the center of a wormhole's throat. \n\n\nLet us now turn to matrices $\\bar{\\mathcal{Q}}_{ij}$ and $\\bar{\\mathcal{M}}_{ij}$ in action~\\eqref{eq:action_KGQM_new}. We adopt a convention where matrix $\\bar{\\mathcal{Q}}_{ij}$ has the only non-vanishing element $\\bar{\\mathcal{Q}}_{12}$, while other elements are included in matrix $\\bar{\\mathcal{M}}_{ij}$ upon integration by parts. \nIn full analogy with the odd-parity sector in Sec.~\\ref{sec:odd_sector}\nmatrices $\\bar{\\mathcal{Q}}_{ij}$ and \n$\\bar{\\mathcal{M}}_{ij}$ generally give \ntwo types of constraints for the linearized theory: one for angular gradient instabilities and the other one for \"slow\" tachyonic instabilities. In what follows we concentrate on angular gradient instabilities, so it is sufficient for us to consider only those parts \nof both $\\bar{\\mathcal{Q}}_{ij}$ and $\\bar{\\mathcal{M}}_{ij}$ which are \nproportional to $\\ell(\\ell+1)$. \nWe have found that in the leading order $\\bar{\\mathcal{Q}}_{12}$ does\nnot depend on $\\ell$\nso it does not provide any\nconstraints in the context of angular gradient instabilities, while \nthe leading order in angular momentum of both $\\bar{\\mathcal{M}}^{(\\ell^2)}_{11}$ and $\\det\\bar{\\mathcal{M}}^{(\\ell^2)}$ read:\n\\begin{equation}\n\\label{eq:m11_RW}\n\\bar{\\mathcal{M}}^{(\\ell^2)}_{11} = \\ell^2 \\dfrac{\\sqrt{A}}{\\sqrt{B}}\\dfrac{(\\mathcal{H} - 2 F_4 B^2 \\pi'^4)^2}{\\mathcal{F}},\n\\end{equation}\n\n{\\small\n\\begin{equation}\n\\label{eq:detM_RW}\n\\det\\bar{\\mathcal{M}}^{(\\ell^2)} = - \\dfrac{\\ell^2 \\left( \\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)^2 \\mathcal{H}^2}{16 \\mathcal{F}^2 A J^2} \n\\left(\n8 A^2 \\dfrac{\\mathcal{F} \\mathcal{G}}{\\mathcal{H}^2} + B[\\mathcal{F}\\mathcal{P}_4 - (A'J - 2 A J')]^2 + 4 A^{3\/2}B J^4 \\mathcal{F} \\frac{d}{dr}\\left[ \\frac{\\sqrt{B}}{\\sqrt{A} J^3\n} \\mathcal{P}_4 \\right]\n\\right),\n\\end{equation}\n}\nwith\n\\begin{equation}\n\\mathcal{P}_4 = \\dfrac{\\mathcal{H} A' J + 2 \\mathcal{G} A J' + \\Gamma A J \\pi'}{\\mathcal{H} \\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)},\n\\end{equation}\nand the rest of matrix components are given in Appendix C.\nWe have introduced a superscript $\\ell^2$ to emphasize that this is only\na part of matrix $\\bar{\\mathcal{M}}$ that is proportional to $\\ell(\\ell+1)$.\n\nWe see that in full analogy with $\\bar{\\mathcal{K}}_{ij}$ and $\\bar{\\mathcal{G}}_{ij}$ matrix $\\bar{\\mathcal{M}}^{(\\ell^2)} _{ij}$ involves \nthe key combination $\\left( \\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)$ which comes from $\\mathbf{\\Theta}$ (see eq.~\\eqref{eq:Theta}) upon taking the leading order in $\\ell$ and has to cross zero in order to avoid ghost instabilities, see eq.~\\eqref{eq:no_go1}. Hence, the\nmatrix $\\bar{\\mathcal{M}}^{(\\ell^2)} _{ij}$ behaves in a similar way to \n$\\bar{\\mathcal{K}}_{ij}$ and $\\bar{\\mathcal{G}}_{ij}$\nupon $\\mathbf{\\Theta}$-crossing: $\\det\\bar{\\mathcal{M}}^{(\\ell^2)} $ is regular while\n$\\bar{\\mathcal{M}}^{(\\ell^2)}_{11}$ crosses zero. \n\nAs before to ensure that angular gradient instabilities are absent \nit is sufficient to require positive definiteness of matrix $\\bar{\\mathcal{M}}$:\n\\begin{equation}\n\\label{eq:positiveM}\n\\bar{\\mathcal{M}}^{(\\ell^2)} _{11} > 0, \\qquad \\det\\bar{\\mathcal{M}}^{(\\ell^2)} >0.\n\\end{equation}\nAccording to eq.~\\eqref{eq:m11_RW} $\\bar{\\mathcal{M}}^{(\\ell^2)} _{11}$ is automatically non-negative, while making the determinant positive\nrequires the following: \n\\begin{equation}\n\\label{eq:no_ang_grad}\n\\frac{d}{dr}\\left[ \\frac{\\sqrt{B}}{\\sqrt{A} J^3\n} \\mathcal{P}_4 \\right]< - \\frac{1}{4 A^{3\/2} J^4 }\n\\left(8 \\frac{A^2}{B} \\dfrac{ \\mathcal{G}}{\\mathcal{H}^2} \n+ \\frac{[\\mathcal{F}\\mathcal{P}_4 - (A'J - 2 A J')]^2}{\\mathcal{F}} \\right).\n\\end{equation}\nThus, parity even modes have no angular gradient instabilities \nprovided that inequality~\\eqref{eq:no_ang_grad} is satisfied. The corresponding sound speeds squared propagating along the angular direction $c^2_{a1,2}$ are given as eigenvalues \nof matrix $(A^{-1}J^2)(\\bar{\\mathcal{K}})^{-1}\\bar{\\mathcal{M}}^{(\\ell^2)} $. We omit their explicit expressions here as they are quite cumbersome and not really illuminating.\n\nWe stress again that our stability analysis for the even-parity modes in this section remains incomplete as opposed to the odd-parity modes, see Sec.~\\ref{sec:odd_sector}. Namely, we have not addressed \n\"slow\" tachyonic instabilities, which become significant if we consider all modes, including those with low momentum. As we mentioned above the corresponding stability conditions for tachyons are associated with the remaining parts of matrices $\\bar{\\mathcal{Q}}_{ij}$ and \n$\\bar{\\mathcal{M}}_{ij}$ which are lower order in $\\ell$. \nFormulating these conditions in a concise form is challenging and\nwe leave it for future development.\n\nTo sum up, the quadratic action~\\eqref{eq:action_KGQM_new} gives the following set of stability conditions which ensure the absence of both ghost and gradient instabilities in the even-parity sector\n(see eqs.~\\eqref{eq:stability_even_ghost},~\\eqref{eq:stability_even_radial} and~\\eqref{eq:no_ang_grad}):\n\\begin{subequations}\n\\label{eq:stability_even}\n\\begin{align}\n\\label{eq:stability_K_RW}\n\\mbox{No ghosts:}\\quad &2\\frac{\\sqrt{B}}{\\sqrt{A}} \\cdot\n\\frac{\\mbox{d}}{\\mbox{d}r}\\left[\\frac{\\sqrt{A}}{\\sqrt{B}}\n\\frac{J^2 \\mathcal{H}\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)}{2{\\cal H} J J' + \\Xi \\pi'}\\right]-\\mathcal{F} > 0, \\\\\n\\label{eq:stability_G_RW}\n\\mbox{No radial gradient instabilities:}\\quad &2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B > 0, \\\\\n\\label{eq:stability_M_RW}\n\\mbox{No angular gradient instabilities:}\\quad & \\frac{d}{dr}\\left[ \\frac{\\sqrt{B}}{\\sqrt{A} J^3\n} \\mathcal{P}_4 \\right]< - \\frac{1}{4 A^{3\/2} J^4 }\n\\left(8 \\frac{A^2}{B} \\dfrac{ \\mathcal{G}}{\\mathcal{H}^2} \n+ \\frac{[\\mathcal{F}\\mathcal{P}_4 - (A'J - 2 A J')]^2}{\\mathcal{F}} \\right).\n\\end{align}\n\\end{subequations}\nSo the complete set of stability conditions for high momenta modes in parity odd and parity even sectors is given by eqs.~\\eqref{eq:stability_odd} and~\\eqref{eq:stability_even}.\n\n\n\\section{Wormhole beyond Horndeski: an example}\n\\label{sec:wormhole_example}\n\nIn this section we give a specific example of beyond Horndeski Lagrangian~\\eqref{eq:lagrangian} which admits a static, spherically-symmetric wormhole solution. Our main requirement to the solution is that it has to be stable against ghosts and gradient instabilities in both parity odd and parity even sectors.\nIn full analogy with our previous wormhole solution in Ref.~\\cite{wormhole1} we adopt a \"reconstruction\" procedure: we\nchoose the background metric~\\eqref{eq:backgr_metric} of a wormhole form\nand concoct the Lagrangian functions so that background equations of motion~\\eqref{eq: background_eqs} and stability conditions in eqs.~\\eqref{eq:stability_odd} and~\\eqref{eq:stability_even}\nare satisfied. We also ensure that\nthe modes in the odd sector propagate at safely subluminal speed~\\eqref{eq:sublum_odd}, \nand the same is done for the speeds of the even parity modes propagating in the radial direction~\\eqref{eq:speed_even_old}. \n\nWe begin with choosing a specific form of metric functions in~\\eqref{eq:backgr_metric}:\n\\[\n\\label{eq:AJ}\nA = 1, \\qquad J = \\tau\\log\\left[1+2\\cosh\\left(\\frac{r}{\\tau}\\right)\\right],\n\\]\nwhere the parameter $\\tau \\sim R_{min}$\nregulates the size of the wormhole\nthroat at $r=0$. Our choice for the last metric function $B(r)$ is somewhat less straightforward. There is a linear combination of the \nbackground equations of motion~\\eqref{eq: background_eqs} which involves metric functions explicitly, while all the Lagrangian functions get combined into $\\mathcal{F}$ and $\\mathcal{H}$ (see eq.~\\eqref{eq:cal_FGH}):\n\\begin{equation}\n\\label{eq:dY}\nA\\cdot\\mathcal{E}_J + J^2\\cdot\\mathcal{E}_A = -\\frac{\\mathcal{F}}{J^2}+\\frac{BA'^2}{4A^2}+\\frac{B'J'}{2J}-\\frac{A'(JB'+2BJ')}{4AJ}-\\frac{BA''}{2A}+B\\left(\\frac{J'^2}{J^2}+\\frac{J''}{J}\\right)=0,\n\\end{equation}\nwhere $\\mathcal{E}_J$ and $\\mathcal{E}_A$ are given in Appendix A and we have already set $\\mathcal{H} = 1$ for simplicity (this is a deliberate choice in our model below although not an obligatory one). Stability conditions for the odd parity sector~\\eqref{eq:stability_odd} require $\\mathcal{F} > 0$ so\naccording to eq.~\\eqref{eq:dY} we cannot freely choose all\nthree metric functions. Having fixed $A(r)$ and $J(r)$ in eq.~\\eqref{eq:AJ}\nwe take $B(r)$ as follows:\n\\begin{equation}\n\\label{eq:B}\nB(r) = 1+ \\mbox{sech}\\left(\\frac{r}{\\tau}\\right),\n\\end{equation} \nwhere $\\tau$ is the same parameter as in eq.~\\eqref{eq:AJ}. This choice for $B$ ensures that eq.~\\eqref{eq:dY} holds for $\\mathcal{F} > \\mathcal{H}$ with already fixed $\\mathcal{H}=1$ above. \n\n\nFinally, we choose static, spherically-symmetric scalar field $\\pi(r)$, which supports the throat of a wormhole, as follows:\n\\begin{equation}\n\\label{eq:pi}\n\\pi(r)=\\tanh\\left(\\frac{r}{\\tau}\\right)-1.\n\\end{equation} \nThis completes our arrangement of the background setting.\nOur choice for $\\pi(r)$, $A$, $B$ and $J$ above has the following asymptotical behaviour as $r \\to\\pm\\infty$:\n\\begin{equation}\n\\label{eq:asymp_background}\n\\pi(r) \\to 0, \\qquad A(r) = 1, \\qquad B(r) \\to 1, \\qquad J(r) \\to r,\n\\end{equation} \nwhich means that the space-time\nin our set up tends to an empty Minkowski space far away from the wormhole. \n\nLet us note that we choose the metric functions $A$, $B$ and $J$ above differently as compared to the original set of $A=B=1$ and $J=\\sqrt{\\tau^2 + r^2}$ in Ref.~\\cite{wormhole1}. This is due to the fact that the corresponding equations of motion and stability conditions in the latter case become oversimplified due to random cancellations so that we have less parameters in the model to control stability at the linearized level.\n \n\nTo find the Lagrangian functions for the devised background, \nwe choose the\nfollowing Ansatz:\n\\begin{subequations}\n\\label{ansatz}\n\\begin{align}\n\\label{F}\n& F(\\pi, X) = f_0(\\pi) + f_1(\\pi)\\cdot X + f_2(\\pi)\\cdot X^2, \\\\\n\\label{G4}\n& G_4(\\pi, X) = \\frac12 + g_{40}(\\pi) + g_{41}(\\pi) \\cdot X + g_{42}(\\pi) \\cdot X^2,\\\\\n\\label{F4}\n& F_4(\\pi, X) = f_{40}(\\pi) + f_{41}(\\pi) \\cdot X,\n\\end{align}\n\\end{subequations}\nin full analogy with Ref.~\\cite{wormhole1}, but here we have an additional term $g_{42}(\\pi)$ which will be used to satisfy the new stability condition in the even parity sector~\\eqref{eq:no_ang_grad}.\n\nThe following steps \nare almost identical to those in Ref.~\\cite{wormhole1}, however, all\nthe formulae become significantly more involved due to non-trivial $J(r)$ and $B(r)$ in eqs.~\\eqref{eq:AJ} and~\\eqref{eq:B}. \nIn Sec.\\ref{sec:wormhole_reconstruction} we schematically describe the way we find functions $f_i(\\pi)$, $g_{4i}(\\pi)$ and $f_{4j}(\\pi)$ by making use of stability conditions~\\eqref{eq:stability_odd},~\\eqref{eq:stability_even} and background equations, but we skip some of the most lengthy intermediate steps of calculations and do not give the resulting Lagrangian functions analytically to avoid bulky expressions.\nWe present all necessary plots in Sec.~\\eqref{sec:wormhole_results} though, which show the graphs of Lagrangian functions we come up with and explicitly prove that the stability conditions are indeed satisfied. The reader that is interested in the results only\nmay safely skip the following technical section and go straight to Sec.~\\eqref{sec:wormhole_results}.\n\n\n\\subsection{Beyond Horndeski Lagrangian functions: reconstruction}\n\\label{sec:wormhole_reconstruction}\n\nWe make use of eq.~\\eqref{ansatz} and express $\\mathcal{H}$, $\\mathcal{F}$, \n$\\Xi$, $\\Sigma$ and $\\Gamma$ in terms of $f_i(\\pi)$, $g_{4i}(\\pi)$ and $f_{4j}(\\pi)$:\n\\begin{subequations}\n\\begin{align}\n\\label{eq:D:H}\n& {\\cal H} = {\\cal G} = 1 + 2 g_{40}(\\pi) + B \\pi'^2 \\cdot g_{41}(\\pi) +\n \\frac12 {B^2} \\pi'^4 \\cdot [4\\; f_{40}(\\pi) - 3\\; g_{42}(\\pi)] -\n B^3 \\pi'^6 \\cdot f_{41}(\\pi) ,\\\\\n %\n\\label{eq:D:F}\n& {\\cal F} = 1 + 2 g_{40}(\\pi) - B \\pi'^2 \\cdot g_{41}(\\pi) +\n \\frac12 B^2 \\pi'^4 \\cdot g_{42}(\\pi), \\\\\n %\n\\label{eq:D:Xi}\n& \\Xi = 4 B J J' \\pi' \\left[-3 B^2 \\pi'^4 \\cdot f_{41}(\\pi) \n+ 4 B \\pi'^2 \\cdot f_{40}(\\pi)\n- 3 B \\pi'^2 \\cdot g_{42}(\\pi) \n+ g_{41}(\\pi) \\right]\\\\\\nonumber\n&\\qquad + J^2 \\left[ \\frac52 B^2 \\pi'^4 \\cdot g_{42}'(\\pi) \n- 3 B \\pi'^2 \\cdot g_{41}'(\\pi) + 2 \\cdot g_{40}'(\\pi)\\right],\\\\\n\\label{eq:D:Sigma}\n&\\Sigma=- \\frac12 B^2 \\pi'^2\\left( 6 B \\left(\\frac{A'}{A} + \\frac{J'}{J }\\right)\\frac{J'}{J } \\pi'^2 (5 B f_{41}(\\pi) \\pi'^2-4 f_{40}(\\pi)+3 g_{42}(\\pi)) \\right.\\\\\\nonumber&\\left.\n+ \\left(\\frac{A'}{A} + 4 \\frac{J'}{J }\\right)\n\\pi'(3 g_{41}'(\\pi)-5 B \\pi'^2 g_{42}'(\\pi)) - 2 g_{41}(\\pi) \\left(\\frac{A'}{A} + \\frac{J'}{J }\\right) \\frac{J'}{J} \\right) \n\\\\\\nonumber\n& + \\frac{1}{2 J^2}B \\pi'^2 \\left( 6 B g_{42}(\\pi) \\pi'^2 + 2 g_{41}(\\pi)\\right) \n -\\frac{3}{2} B^2 f_{2}(\\pi) \\pi'^4 - \\frac{1}{2}B \\pi'^2 f_{1}(\\pi),\\\\\n\\label{eq:D:Gamma}\n& \\Gamma = \\left(2 B \\pi' \\frac{A'}{A} + 4 B \\pi' \\frac{ J'}{ J}\\right)\n\\left[{4 B \\pi'^2} \\cdot f_{40}(\\pi) \n- {3 B ^2 \\pi'^4 \\cdot f_{41}(\\pi) } + { g_{41}(\\pi)} \n- {3 B \\pi'^2 \\cdot g_{42}(\\pi) } \\right] \\\\\\nonumber\n& \\qquad + 4 g_{40}'(\\pi) - 6 B \\pi'^2 \\cdot g_{41}'(\\pi) +5 B ^2 \\pi'^4 \\cdot g_{42}'(\\pi),\n\\end{align}\n\\end{subequations}\nwhere we do not explicitly substitute the coordinate dependence of $\\pi$\nfrom eq.~\\eqref{eq:pi} to keep the equations more concise, but we bear in mind that in fact we work with functions of $r$.\n\nFirst, we recall that we have already fixed $\\mathcal{H} = 1$ (and $\\mathcal{G}=1$, see eq.~\\eqref{eq:cal_FGH}) for simplicity when we chose $B$ in eq.~\\eqref{eq:B}, so now we can make use of eq.~\\eqref{eq:D:H} and express $g_{40}(\\pi)$ in terms of $f_{40}(\\pi)$, $f_{41}(\\pi)$, $g_{41}(\\pi)$ and $g_{42}(\\pi)$:\n\\begin{equation}\n\\label{eq:g40}\ng_{40}(\\pi) = - \\frac12 \\left[ B \\pi'^2 \\cdot g_{41}(\\pi) +\n \\frac12 {B^2} \\pi'^4 \\cdot [4\\; f_{40}(\\pi) - 3\\; g_{42}(\\pi)] -\n B^3 \\pi'^6 \\cdot f_{41}(\\pi) \\right].\n\\end{equation}\nThen, with $A$, $B$ and $J$ set in eqs.~\\eqref{eq:AJ},~\\eqref{eq:B} \nwe can write eq.~\\eqref{eq:dY} substituting $\\mathcal{F}$ \nfrom~\\eqref{eq:D:F} and $g_{40}(\\pi)$ from eq.~\\eqref{eq:g40}\nand express $g_{41}$ in terms of $g_{42}(\\pi)$, $f_{40}(\\pi)$ and\n$f_{41}(\\pi)$\nas follows:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:g41}\n& g_{41}(\\pi) = \\frac{1}{J^2 } \\left(\\frac12 B^2 \\pi'^4 \\cdot f_{41}(\\pi) - {B} \\pi'^2\n(f_{40}(\\pi) - g_{42}(\\pi)) \\right) \\\\\n&- \\frac{1}{8 \\pi'^2 }\n\\left(\\frac{A'^2}{A^2} -\n \\frac{[2 B A' J' + J (A' B' + 2 B A)]}{A B J} +\n \\frac{2[ B' J J' + 2 B J'^2 + 2 B J J''-2]}{B J^2}\\right).\n \\end{aligned}\n\\end{equation}\nNote that we chose $B(r)$ in eq.~\\eqref{eq:B} so that eq.~\\eqref{eq:dY} holds for $\\mathcal{F} > \\mathcal{H}$, i.e in our set up $\\mathcal{F} > 1$. Therefore, we have ensured that the stability conditions in the odd-parity sector~\\eqref{eq:stability_odd} are satisfied everywhere and both radial and angular speed~\\eqref{eq:speed_odd} is luminal at most.\n\nLet us turn to the no-ghost condition in the even parity sector~\\eqref{eq:stability_K_RW}:\n\\[\n\\label{eq:D:no_go}\n\\frac{\\sqrt{B}}{\\sqrt{A}}\\cdot \\frac{\\mbox{d}}{\\mbox{d}r}\\left[ \\frac{\\sqrt{A}}{\\sqrt{B}}\n\\frac{J^2 \\mathcal{H}\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)}{2{\\cal H} J J' + \\Xi \\pi'} \\right] > \\frac{{\\cal F}}{2}.\n\\]\nAs we have discussed in Sec.~\\ref{sec:old_gauge} to satisfy this no-ghost constraint for all $r$\nit is necessary to make \n$\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)$ cross zero at some point(s). Since we have fixed $\\mathcal{H}=1$ above, we can choose\n\\[\n\\label{eq:D:F4_choice}\n2 F_4(\\pi,X) B^2 \\pi'^4 \\equiv 2 \\left(f_{40}(\\pi) + f_{41}(\\pi)\\cdot X\\right) B^2 \\pi'^4 =\nw\\cdot \\mbox{sech}\\left(\\frac{r}{\\tau}+u\\right),\n\\]\nso that $\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)$ crosses zero twice. \nWe use eq.~\\eqref{eq:D:F4_choice} to express $f_{40}(\\pi)$ through $f_{41}(\\pi)$. In our numerical results below we take specific values of parameters $u$, $w$ and $\\tau$ involved in eq.~\\eqref{eq:D:F4_choice}, namely, \n\\[\nu = 0, \\quad w=2, \\quad \\tau =100,\n\\]\nwhere $\\tau$ still defines the size or the wormhole throat. \nWe have introduced $u$ in eq.~\\eqref{eq:D:F4_choice} to emphasize that it is possible to take $u \\neq 0$, which proves that there is no fine-tuning and it is not obligatory to have $F_4'=0$ right at the throat $r=0$. We still take $u=0$ in our calculations for the sake of simplicity.\n\nNext we turn to the denominator of the no-ghost constraint~\\eqref{eq:D:no_go}, where $\\Xi$ involves yet undefined $g_{42}(\\pi)$ and $f_{41}(\\pi)$, see eq.~\\eqref{eq:D:Xi}. The simplest option that satisfies the stability condition~\\eqref{eq:D:no_go} everywhere is to take $\\Xi = 0$, so that we can substitute $g_{40}(\\pi)$, $g_{41}(\\pi)$ and $f_{40}(\\pi)$ from eqs.~\\eqref{eq:g40},~\\eqref{eq:g41},~\\eqref{eq:D:F4_choice} and find $f_{41}(\\pi)$ in terms of $g_{42}(\\pi)$ from the\nthe resulting equation. \n\nThe constraint that governs angular gradient instabilities in the even\nparity sector~\\eqref{eq:stability_M_RW} involves the only yet unconstrained coefficient\n$\\Gamma$. To satisfy inequality~\\eqref{eq:stability_M_RW} everywhere it turned out to be sufficient to take $\\Gamma = 0$, so we make use of eq.~\\eqref{eq:D:Gamma} and\nexpress $g_{42}(\\pi)$ as it is the only function that has not been defined above. \nThus, at this step we have a closed system for $g_{40}(\\pi)$, $g_{41}(\\pi)$, $g_{42}(\\pi)$, $f_{40}(\\pi)$ and $f_{41}(\\pi)$ so this \ncompletes the definition of $G_4(\\pi,X)$\nand $F_4(\\pi,X)$ in eqs.~\\eqref{G4} and~\\eqref{F4}.\n\nIn full analogy with Ref.~\\cite{wormhole1} \nwe find $f_0(\\pi)$ and $f_1(\\pi)$\nfrom solving two background equations of \nmotion~\\eqref{eq: background_eqs}, e.g. $\\mathcal{E}_A=0$ and $\\mathcal{E}_B=0$ (recall that we have already used a linear combination of $\\mathcal{E}_A$ and $\\mathcal{E}_J$ in eq.~\\eqref{eq:dY}, while $\\mathcal{E}_{\\pi}$ is linearly dependent on the other three equations). As before the final function $f_2(\\pi)$ is used to satisfy the stability condition against\nradial gradient instabilities~\\eqref{eq:stability_G_RW} in parity even sector and simultaneously ensure that the speed $c_{s2}^2 \\leq 1$ in eq.~\\eqref{eq:speed_even_old} ($c^2_{s1}$ is already set to be subluminal above). Both requirements can be written in a compact form:\n\\[\n\\label{D:constraint_f2}\n0 < 2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B \\leq (2 {\\cal H} J J' + \\Xi \\pi' )^2 (2 {\\cal P}_1 -{\\cal F}),\n\\]\nwhere $\\Sigma$ involves $f_2(\\pi)$, see eq.~\\eqref{eq:D:Sigma}. \nThe combination on the right hand side coincides with the no-ghost condition~\\eqref{eq:stability_K_RW} and we have already insured its positivity, so we choose $\\Sigma$ appropriately to have $c_{s2}^2 \\leq 1$:\n\\[\n\\label{eq:sigma_choice}\nc^2_{s2} = \\frac{(2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B)}{(2 {\\cal H} J J' + \\Xi \\pi' )^2 (2 {\\cal P}_1 -{\\cal F})} = \n\\frac{\\cosh\\left(\\frac{r}{\\tau}\\right)}{1+\\cosh\\left(\\frac{r}{\\tau}\\right)},\n\\]\nand express $f_2(\\pi)$ by making use of eq.~\\eqref{eq:D:Sigma} and recalling that we fixed $\\Xi=0$ in our model. This completes the reconstruction of the function $F(\\pi,X)$.\n\n\n\\subsection{Beyond Horndeski Lagrangian functions: results}\n\\label{sec:wormhole_results}\n\nThe reconstructed functions $f_i(\\pi)$, $g_{4i}(\\pi)$ and $f_{4j}(\\pi)$\nentering eqs.~\\eqref{ansatz} are given in Fig.~\\ref{fig:f-g-f4} and have the following asymptotic behaviour as $r \\to \\pm \\infty$:\n\\[\n\\label{eq:fg4f4_asymp}\nf_{0}\\propto e^{-\\frac{r}{\\tau}},\\quad \nf_{1} \\propto e^{\\frac{3r}{\\tau}}, \\quad\nf_{2}\\propto e^{\\frac{7r}{\\tau}}, \\quad\ng_{40} \\propto e^{-\\frac{r}{\\tau}}, \\quad\ng_{41} \\propto e^{\\frac{3r}{\\tau}}, \\quad \ng_{42} \\propto e^{\\frac{7r}{\\tau}}, \\quad\nf_{40} \\propto e^{\\frac{7r}{\\tau}},\\quad \nf_{41} \\propto e^{\\frac{11 r}{\\tau}} \\; ,\n\\] \nor, equivalently,\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:fg4f4XXX_asymp}\n&f_{0} =\\frac{3}{\\tau^2}e^{-\\frac{r}{\\tau}},\\quad \nf_{1} \\cdot X = -\\frac{6}{\\tau^2}e^{-\\frac{r}{\\tau}}, \\quad\nf_{2} \\cdot X^2 =\\frac{1}{\\tau^2}e^{-\\frac{r}{\\tau}}, \\quad\ng_{40} = -\\frac{13}{8}e^{-\\frac{r}{\\tau}}, \\quad\ng_{41}\\cdot X = \\frac{9}{4}e^{-\\frac{r}{\\tau}}, \\\\\n&\\qquad\\qquad \ng_{42} \\cdot X^2 = -\\frac{5}{8}e^{-\\frac{r}{\\tau}}, \\quad\nf_{40} = \\frac{15}{8}e^{-\\frac{r}{\\tau}},\\quad \nf_{41} \\cdot X = -\\frac{11}{8}e^{-\\frac{r}{\\tau}} \\; ,\n\\end{aligned}\n\\end{equation}\nwhich shows that all terms equally contribute to the Lagrangian functions. Thus, the Lagrangian functions $F(\\pi,X)$, $G_4(\\pi,X)$\nand $F_4(\\pi,X)$\nhave the following asymptotics far away from the throat (recall that $\\pi(r)$ in eq.~\\eqref{eq:pi})\n\\[\n\\label{eq:FG4F4asymp}\n\\begin{aligned}\n&F(\\pi, X) = q_1\\cdot(-\\pi)^{1\/2} + q_2\\cdot (-\\pi)^{-3\/2}\\cdot X + q_3\\cdot (-\\pi)^{-7\/2}\\cdot X^2, \\\\\n&G_4(\\pi, X) = \\frac12 + q_4\\cdot(-\\pi)^{1\/2} + q_5\\cdot (-\\pi)^{-3\/2}\\cdot X + q_6 \\cdot (-\\pi)^{-7\/2}\\cdot X^2, \\\\\n&F_4(\\pi, X) = q_7 \\cdot (-\\pi)^{-7\/2} + q_8 \\cdot (-\\pi)^{-11\/2} \\cdot X,\n\\end{aligned}\n\\]\nwhere $q_i$, $i = \\overline{1,8}$ are constants, which are irrelevant at this point. \nWe see that all functions in eq.~\\eqref{eq:fg4f4XXX_asymp} decay \nexponentially with growing $r$, which means that, \nwhile our theory is still of beyond Horndeski type away from the \nwormhole, the space-time quite rapidly becomes empty Minkowski space\nwith pure Einstein gravity. \n\n\n\n\n\n\\begin{figure}[H]\\begin{center}\\hspace{-1cm}\n{\\includegraphics[width=0.5\\textwidth]{f-2.pdf}}\\hspace{2.8cm}\\hspace{-3cm}\n{\\includegraphics[width=0.5\\textwidth] {g4-2.pdf}}\n\n\\vspace{-0.2cm}\n{\\includegraphics[width=0.5\\textwidth]\n{f4-2.pdf}}\\hspace{1cm}\n\\caption{\\footnotesize{The Lagrangian functions $f_0(r)$, $f_1(r) \\cdot X$, $f_2(r)\\cdot X^2$, $g_{40}(r)$, $g_{41}(r)\\cdot X$, \n$g_{42}(r)\\cdot X^2$,\n $f_{40}(r)$ and $f_{41}(r) \\cdot X $, with the following choice of the parameters: $u=0$, $w=2$ and $\\tau = 100$. This choice\n guarantees that the size\n of the wormhole throat safely exceeds the Planck length.}} \\label{fig:f-g-f4}\n\\end{center}\\end{figure}\n\nLet us demonstrate that our wormhole solution\n~\\eqref{eq:AJ},~\\eqref{eq:B}-\\eqref{eq:pi} \nsatisfies our set of stability conditions. We have arranged the Lagrangian functions so that $\\mathcal{H}=\\mathcal{G}=1$ and \n$\\mathcal{F} > 1$ which complies with the constraints in the\nodd-parity sector~\\eqref{eq:stability_odd},~\\eqref{eq:sublum_odd} \nand gives at most luminal sound speed squared, i.e. $c^2_{\\theta}=1$ and $c^2_r < 1$~\\eqref{eq:speed_odd}. As for the parity even modes we show in Fig.~\\ref{fig:KGM} that for our set of Lagrangian functions \n$\\bar{\\mathcal{K}}_{11}$, \n$\\det\\bar{\\mathcal{K}}$, $\\bar{\\mathcal{G}}_{11}$, $\\det\\bar{\\mathcal{G}}$, $\\bar{\\mathcal{M}}^{(\\ell^2)} _{11}$, $\\det\\bar{\\mathcal{M}}^{(\\ell^2)} $ are positive everywhere. Hence, the constraints\n~\\eqref{eq:stability_even} \nare satisfied at all points.\n\n\\begin{figure}[H]\\begin{center}\n{\\includegraphics[width=0.49\\textwidth]\n{kgm22.pdf}}\n\\hspace{0.01cm}\n{\\includegraphics[width=0.49\\textwidth]\n{kgmdet.pdf}}\n\\caption{\\footnotesize{Functions $\\bar{\\mathcal{K}}_{11}$, \n$\\det\\bar{\\mathcal{K}}$, $\\bar{\\mathcal{G}}_{11}$, $\\det\\bar{\\mathcal{G}}$, $\\bar{\\mathcal{M}}^{(\\ell^2)} _{11}$ and $\\det\\bar{\\mathcal{M}}^{(\\ell^2)} $, which govern the stability of the \nparity even sector (we choose $\\ell = 10$ here for definiteness). We have introduced a normalizing numerical factor for both $\\bar{\\mathcal{M}}^{(\\ell^2)} _{11}$ and $\\det\\bar{\\mathcal{M}}^{(\\ell^2)}$, which enabled us to put them on the same plots with matrices $\\bar{\\mathcal{K}}$ and $\\bar{\\mathcal{G}}$.\n}}\n\\label{fig:KGM}\n\\end{center}\\end{figure}\n\n\n\nAs for the sound speeds squared in the even-parity sector~\\eqref{eq:speed_even_old} \n$c^2_{s1} = c^2_r < 1$, while $c^2_{s2}$ was fixed in eq.~\\eqref{eq:sigma_choice} and obviously does not exceed unity. \nThe situation, however, is no so bright for the angular sound speeds squared $c^2_{a1,2}$, which are given by eigenvalues of matrix \n$(A^{-1}J^2)(\\bar{\\mathcal{K}})^{-1}\\bar{\\mathcal{M}}^{(\\ell^2)}$. Even though we do not give these speeds explicitly we can do a quick check if their product is greater than unity, which would signal superluminal propagation in the angular direction. Indeed, the product of $c^2_{a1}$ and $c^2_{a2}$\nis given by\n\\[\nc^2_{a1} \\cdot c^2_{a2} = \\frac{J^4}{A^2}\\frac{\\det\\bar{\\mathcal{M}}^{(\\ell^2)}}{\\det\\bar{\\mathcal{K}}},\n\\]\nwhere $\\det\\bar{\\mathcal{K}}$ and $\\det\\bar{\\mathcal{M}}^{(\\ell^2)}$ are given in eqs.~\\eqref{eq:k11detk_RW} and~\\eqref{eq:detM_RW}, respectively.\nOur numerical check has shown that for our choice of Lagrangian functions this ratio is greater than one, meaning that either of speeds\n$c^2_{a1}$ or $c^2_{a2}$ (or both) is greater than the speed of light. This is another troublesome \ndrawback of our solution and we hope to deal with it in\nfuture studies.\n\n\\vspace{-0.2cm}\n\\section{Conclusion}\n\\label{sec:outlook}\nTo sum up, this work aimed to support the point that scalar-tensor theories of modified gravity like beyond Horndeski theories are promising candidates admitting stable Lorentzian wormholes. \nWe have formulated a set of stability conditions which eliminate \nhigh energy pathologies like ghosts and gradient instabilities.\nAlthough imperfect we have put forward a wormhole solution \nthat complies with this set of stability constraints.\nUnfortunately, we still cannot say anything definite about the tachyon\nsector, since dealing with mass matrices is technically demanding\nin this class of field theories. It remains a challenging and intriguing\nquestion whether it is possible to have a completely stable wormhole in theories like generalised Horndeski family, let alone make it realistic and observationally traceable. \n\n\\vspace{-0.2cm}\n \\section*{Acknowledgements}{}\n\nWhen this work was close to completion, Valery Rubakov untimely deceased. All the findings in this paper are the result of our fruitful and long-lasting collaboration with Professor Rubakov. Unfortunately, he did not take part in preparing this manuscript.\n\\\\ \n\\\\\nThe work of S.M. and V.V. \non Sec.~\\ref{sec:linearized_th} of this paper has been\nsupported by Russian Science Foundation grant\n19-12-00393, while the part of work on Sec.~\\ref{sec:wormhole_example}\nhas been supported by the Foundation for the Advancement of Theoretical Physics and Mathematics \"BASIS\".\n\n\n\n\\section*{Appendix A}\n\\label{app:A}\nIn this Appendix we give Einstein and scalar field equations of motion for action~\\eqref{eq:lagrangian} in the background~\\eqref{eq:backgr_metric}:\n\\[\n\\mathcal{E}_A = 0, \\qquad \\mathcal{E}_B = 0,\n\\qquad \\mathcal{E}_J = 0, \\qquad \\mathcal{E}_{\\pi} = 0,\n\\]\nwhere\n{\\small\n\\begin{eqnarray}\n\\hspace{-1cm}\n&& \\label{CalEa} {\\cal E}_A = F + \\frac{2}{J}\\left(\\frac{1 - BJ'^2}{J} - (2BJ'' + J'B')\\right)G_{4}\n +\\frac{4BJ'}{J}\\left( \\frac{J'}{J } + \\frac{ X'}{X} \n + \\frac{2BJ'' + J'B'}{BJ'}\\right)XG_{4X} \n\\nonumber\\\\&&\n + \\frac{8BJ'}{J}XX'G_{4XX} -\n B\\pi'\\left(\\frac{4J'}{J } + \\frac{ X'}{X} \\right)G_{4\\pi}+ 2B\\pi'\\left(\\frac{4J'}{J } - \\frac{ X'}{X} \\right)XG_{4\\pi X}+ 4XG_{4\\pi\\pi}\n\\nonumber\\\\&&\n -\\frac{2B^2\\pi'^3}{J^2}(5B'JJ' + B(J'^2 + 2JJ''))\\pi' F_{4}\n -\\frac{16 B^3 J' \\pi'^3}{J} \\pi'' F_4 \n + \\frac{2B^3J'\\pi'^5}{J} \\left(B'\\pi' + 2B\\pi''\\right)F_{4X}\n \\nonumber\\\\&&\n - \\frac{4B^3J'\\pi'^5}{J}F_{4\\pi} , \\\\\\nonumber\n\\end{eqnarray}}\n{\\small\n\\begin{eqnarray}\n\\nonumber\n&&\\label{CalEb} {\\cal E}_B = F - 2XF_{X} + \\frac{2}{J}\\left(\\frac{1 - BJ'^2}{J } - BJ'\\frac{A'}{A} \\right)G_{4}\n -\\frac{4}{J}\\left(\\frac{1 - 2BJ'^2}{J } - 2BJ'\\frac{A'}{A}\\right)XG_{4X}\n- \\left(\\frac{4J'}{J } + \\frac{ A'}{A} \\right)B\\pi'G_{4\\pi}\n \\nonumber\\\\&& \n + 8\\frac{BJ'}{J}\\left( \\frac{J'}{J } + \\frac{ A'}{A}\\right)X^2G_{4XX} \n - 2\\left(\\frac{4J'}{J } + \\frac{ A'}{A} \\right)B\\pi'XG_{4\\pi X} \n -\\frac{10B^3J'(A'J + AJ')\\pi'^4}{AJ^2} F_{4}\n \\nonumber\\\\&&\n +\\frac{2B^4J'(A'J + AJ')\\pi'^6}{AJ^2} F_{4X},\\\\\\nonumber\n\\end{eqnarray}}\n{\\small\n\\begin{eqnarray}\n&&\\label{CalEc} {\\cal E}_J = F \n -\\left(\\frac{1}{J}\\frac{\\sqrt{B}}{\\sqrt{A}}\\left(J\\frac{\\sqrt{B}}{\\sqrt{A}}A'\\right)' + \\frac{2BJ'' + J'B'}{J} \\right)G_{4} - B\\pi'\\left(\\frac{2J'}{J } + \\frac{ A'}{A } + \\frac{2BJ'' + J'B'}{BJ'} +\n 2\\frac{\\pi'' - \\frac{J''}{J'}\\pi'}{\\pi'}\\right)G_{4\\pi} \n\\nonumber\\\\&&\n + BX\\left(-\\frac{A'^2}{A^2} + \\frac{2}{J}\\frac{2BJ'' + J'B'}{B} + \\frac{A'}{A}\\frac{2BJ'' + J'B'}{BJ'} + 2\\frac{A'J' + J(A'' - \\frac{J''}{J'}A')}{JA}\\right)G_{4X} \n\\nonumber\\\\&&\n + BX'\\left(\\frac{2J'}{J } + \\frac{ A'}{A} \\right)G_{4X} + 4XG_{4\\pi \\pi} +\n 2B\\pi'\\left(\\frac{2J'}{J } + \\frac{ A'}{A } - \\frac{ X'}{X}\\right)XG_{4\\pi X} + 2B\\left(\\frac{2J'}{J } + \\frac{ A'}{A} \\right)XX'G_{4XX}\n\\nonumber\\\\&&\n-\\frac{B^2\\pi'^4}{2}\\left(-\\frac{A'^2B}{A^2} + \\frac{A'(5B'J + 2BJ')}{AJ} + 2\\frac{A''BJ + 5AB'J' + 2ABJ''}{AJ}\\right) F_{4}\n\\nonumber\\\\&&\n - \\frac{B^3(A'J + 2AJ')}{AJ} \\pi'^3 \\left( 4 \\pi'' F_{4} + \\pi'^2 F_{4\\pi} -\n \\frac{\\pi'^2(B'\\pi' + 2B\\pi'')}{2} F_{4X} \\right),\n \n\\end{eqnarray}}\nand\n\\begin{eqnarray}\n&&\\label{CalEphi} {\\cal E}_{\\pi} = \\frac{1}{J^2}\\sqrt{\\frac{B}{A}} \\cdot\n\\frac{\\mbox{d}}{\\mbox{d}r}\\left[J^2 J' \\sqrt{AB} \\cdot\n\\mathcal{J}_H\\right] - \\mathcal{S}_H + \\mathcal{JS}_{BH},\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n&&\\mathcal{J}_{H} = \\frac{\\pi'}{J'}F_{X} + 2\\frac{\\pi'}{J'}\\left(\\frac{1 - BJ'^2}{J^2 } - \\frac{BJ'}{J}\\frac{A'}{A}\\right)G_{4X} \n - \\frac{4B\\pi'}{J}\\left( \\frac{J'}{J } + \\frac{ A'}{A}\\right)XG_{4XX}\n \\nonumber \\\\&&\n - 2\\left( \\frac{4}{J } + \\frac{ A'}{AJ'}\\right)XG_{4\\pi X},\\\\\\nonumber\n \\\\\n &&\n\\mathcal{S}_{H} = -F_{\\pi} -\n \\left[\\frac{B}{2}\\left( \\frac{A'}{A} \\right)^2 + \\frac{2}{J}\\left(\\frac{1 - BJ'^2}{J } - (2BJ'' + J'B')\\right)\\right]G_{4\\pi} \n + B\\left[\\frac{X'}{X}\\left(\\frac{4J'}{J } + \\frac{ A'}{A} \\right) \n\\right.\\nonumber \\\\&&\\left.\n + \\frac{4J'}{J}\\left( \\frac{J'}{J } + \\frac{ A'}{A}\\right)\\right]XG_{4\\pi X}\n\n\n+\n \\left[\\frac{B}{2}\\left(2\\frac{A'' - \\frac{J''}{J'}A'}{A} + \\frac{A'}{A}\\left(\\frac{4J'}{J } + \\frac{2BJ'' + J'B'}{BJ'}\\right)\\right)\\right]G_{4\\pi}\n\\end{eqnarray}\n\\begin{eqnarray}\n &&\n\\mathcal{JS}_{BH} = -4B^2\\pi'^2\\left[\\frac{-A'^2BJ'}{A^2J} \\pi'+ \\frac{J'(2A''BJ + 5AB'J' + 4ABJ'')}{AJ^2}\\pi'\n\\right.\n\\nonumber\\\\ &&\n\\left.\n + \\frac{A'(5B'JJ' + 3BJ'^2 + 2BJJ'')}{AJ^2} \\pi' +\n \\frac{6BJ'(A'J + AJ')}{AJ^2}\\pi''\\right]F_{4}\n\\nonumber\\\\&&\n -\\frac{6B^3J'(A'J + AJ')\\pi'^4}{AJ^2}F_{4\\pi} +\n B^3\\pi'^4\\left[-\\frac{A'^2BJ'}{A^2J}\\pi' + \\frac{J'(2A''BJ + 11AB'J' + 4ABJ'')}{AJ^2}\\pi'\n\\right.\n\\nonumber\\\\&&\n \\left.\n + \\frac{A'(11B'JJ' + 3BJ'^2 + 2BJJ'')}{AJ^2}\n \\pi' + \\frac{18BJ'(A'J + AJ')}{AJ^2}\\pi''\\right]F_{4X}\n \\nonumber\\\\&&\n + \\frac{2B^4J'(A'J + AJ')\\pi'^6}{AJ^2}F_{4 \\pi X} -\n \\frac{B^4J'(A'J + AJ')\\pi'^6(B'\\pi' + 2B\\pi'')}{AJ^2}F_{4XX}.\n\\end{eqnarray}\n\n\n\\section*{Appendix B}\nIn this Appendix we gather the explicit form of coefficients\n$a_i$, $b_i$, $c_i$, $d_i$, $e_i$ and $p_i$ entering the quadratic action in the spherical gauge~\\eqref{eq:action_even} and \nits counterpart in\nRegge-Wheeler-unitary gauge~\\eqref{eq:action_even_RW}. The coefficients \nare given in terms of combinations $\\mathcal{H}$, $\\mathcal{F}$, $\\mathcal{G}$, $\\Xi$, $\\Gamma$\nand $\\Sigma$ introduced in the main body of the text in eqs.~\\eqref{eq:cal_FGH},~\\eqref{eq:Xi},~\\eqref{eq:Gamma} and~\\eqref{eq:KSI} respectively:\n\\begin{eqnarray\na_1&=&\\sqrt{AB}\\,\\Xi,\\\\\na_2&=&\\frac{\\sqrt{AB}}{2\\pi'}\\left[\n2\\pi'\\Xi'-\\left(2\\pi''-\\frac{A'}{A}\\pi'\\right)\\Xi\n+2JJ'\\left(\\frac{A'}{A}-\\frac{B'}{B}\\right){\\cal H}-4{\\cal H}JJ''\n\\right.\\\\ &&\\left.\n+\\frac{2J^2}{B}\\left({\\cal E}_B-{\\cal E}_A\\right) \\right] - \n 2 \\sqrt{AB^3} \\pi'^3 \\cdot F_4 ,\\\\\na_3&=&-\\frac{\\sqrt{AB}}{2}\\left(\\pi'\\Xi+2JJ'{\\cal H}\\right),\\\\\na_4&=&\\sqrt{AB}\\,{\\cal H},\\\\\na_5&=&-\\sqrt{\\frac{A}{B}}J^2\\frac{\\partial{\\cal E}_A}{\\partial\\pi }=a_2'-a_1'',\\\\\na_6&=&-\\sqrt{\\frac{A}{B}} \\frac{1}{J\\pi'} \\left( J {\\cal H}'+J'{\\cal H}-J'{\\cal F} \\right) + \\sqrt{AB} \\pi'^2 \\left( B' \\pi' + 2 B \\pi''\\right) \\cdot F_4, \\\\\na_7&=&a_3'+\\frac{J^2}{2} \\sqrt{\\frac{A}{B}} {\\cal E}_B,\n\\end{eqnarray}\n\\begin{eqnarray}\na_8&=&-\\frac{a_4}{2B} + \\sqrt{AB^3} \\pi'^4 \\cdot F_4, \\\\\na_9&=&\\frac{\\sqrt{A}}{J}\\frac{{\\rm d}}{{\\rm d} r}\\left(\nJ\\sqrt{B}{\\cal H}\n\\right), \\\\\nb_1&=&\\frac{1}{2}\\sqrt{\\frac{B}{A}}{\\cal H},\n\\\\\nb_2&=&-2\\sqrt{\\frac{B}{A}}\\Xi,\n\\\\\nb_3&=&\\sqrt{\\frac{B}{A}}\\frac{1}{\\pi'}\n\\left[\n\\left(2\\pi''+\\frac{B'}{B}\\pi'\\right)\\Xi -2JJ'\\left(\\frac{A'}{A}-\\frac{B'}{B}\\right){\\cal H}\n+\\frac{2J^2}{B}{\\cal E}_A+4JJ''{\\cal H}\n\\right\n,\n\\\\\nb_4&=&\\sqrt{\\frac{B}{A}}\\left(\\pi'\\Xi+2JJ'{\\cal H}\\right),\n\\\\\nb_5&=&-2b_1,\\\\\nc_1&=&-\\frac{1}{\\sqrt{AB}}\\Xi - 4\\sqrt{\\frac{ B^3}{A}} J J' \\pi'^3 \\cdot F_4,\n\\\\\nc_2&=&-\\sqrt{AB} \\left( \\frac{A'}{2A} \\Xi+JJ'\\Gamma-\\frac{J^2 \\pi'}{X} \\Sigma \\right)\n,\n\\\\\nc_3&=&J^2\\sqrt{\\frac{A}{B}} \\frac{\\partial {\\cal E}_B}{\\partial \\pi},\\\\\nc_4&=&\\frac{1}{2}\\sqrt{\\frac{A}{B}} \\Gamma + \\sqrt{\\frac{ B^3}{A}} \\left(A' J + 2A J'\\right) \\pi'^3\n \\cdot F_4,\n\\\\\nc_5&=&\n-\\frac{1}{2}\\sqrt{AB}\\left(\\pi'\\Gamma+\\frac{A'}{A}{\\cal H}\n+\\frac{2J'}{J}{\\cal G}\\right),\n\\\\\nc_6&=&\\frac{J^2}{2} \\sqrt{\\frac{A}{B}} \\left( \\Sigma+\\frac{A'B \\pi'}{2J^2 A}\\Xi+\\frac{B\\pi'J'}{J}\\Gamma-\\frac{1}{2} {\\cal E}_B+\\frac{BJ'^2}{J^2} {\\cal G}+\\frac{A'BJ'}{JA}{\\cal H} \\right),\n\\\\\nd_1&=&b_1,\n\\\\\nd_2&=&\\sqrt{AB}\\,\\Gamma,\n\\\\\nd_3&=& \\frac{\\sqrt{AB}}{J^2}\\left[\n\\frac{2JJ'}{\\pi'}\\left(\\frac{A'}{A}-\\frac{B'}{B}\\right){\\cal H}\n-J^2\\left(\\frac{2J'}{J}-\\frac{A'}{A}\\right)\\frac{\\partial{\\cal H}}{\\partial\\pi}\n+\\frac{2}{B\\pi'}\\left({\\cal F}-{\\cal G}\\right)\\right.\n\\nonumber\\\\&&\\left.\n-\\frac{J^2}{2\\pi'}\\left(2\\pi''+\\frac{B'}{B}\\pi'\\right)\n\\left(\\Gamma_1+\\frac{2J'}{J}\\Gamma_2\\right)\n-\\frac{2J^2}{B\\pi'}({\\cal E}_A-{\\cal E}_B)-\\frac{4JJ''}{\\pi'}{\\cal H}\n\\right]\n,\n\\\\\nd_4&=&\\frac{\\sqrt{AB}}{J^2}\n\\left({\\cal G}-J^2{\\cal E}_B\\right),\n\\\\\ne_1&=&\\frac{1}{2\\sqrt{AB}}\\left[\n\\frac{J^2}{X}({\\cal E}_A-{\\cal E}_B)-\\frac{2}{\\pi'}\\Xi'+\\left(\\frac{A'}{A}\n-\\frac{X'}{X}\\right)\\frac{\\Xi}{\\pi'}\n+\\frac{2BJ'^2}{X}{\\cal F}-\\frac{2JJ'B}{X}{\\cal H}' \\right.\\\\&&\\left.\n-{\\cal H}\\frac{B^2J'^2}{JXA}\\frac{{\\rm d}}{{\\rm d} r}\\left(\\frac{J^2A}{B}\\right)+\\frac{2BJJ''}{X}{\\cal H}\n\\right] - \\frac{4}{B \\pi'^2} \\cdot \\frac{\\mbox{d}}{\\mbox{d} r}\n\\left[\\sqrt{\\frac{B^5}{A}} J J' \\pi'^4 \\cdot F_4\\right], \\nonumber \\\\\ne_2&=&-\\sqrt{AB}\\frac{J^2}{X}\\Sigma,\\\\\ne_3&=&J^2 \\sqrt{\\frac{A}{B}} \\frac{\\partial {\\cal E}_\\pi}{\\partial \\pi},\n\\end{eqnarray}\n\\begin{eqnarray}\ne_4&=&\\frac{\\sqrt{AB}J'^2}{8X}\\left(-\\frac{4{\\cal G}}{J^2}-\\frac{4({\\cal E}_A-{\\cal E}_B)}{BJ'^2}+\\frac{2A'{\\cal H}'}{AJ'^2}+\n\\frac{4{\\cal G}'}{JJ'}+\\frac{4}{BJ^2J'^2}\\left(1-\\frac{JJ'BA'}{A}\\right){\\cal F}\n\\nonumber\\right.\\\\ &&\\left.-\n\\frac{4{\\cal H}}{BJ^2J'^2}\\left(1-BJ'^2(1+\\frac{2A'J}{AJ'})+J(B'J'+2BJ'')\\right)-\\frac{2\\pi'}{J'^2}\\Gamma'\n\\nonumber\\right.\\\\ &&\\left.\n+\\frac{2\\pi'}{JJ'}\\left(-2+\\frac{A'J}{AJ'}\\right)\\frac{\\partial{\\cal H}}{\\partial\\pi}\n+\\frac{\\Xi\\pi'}{J^3J'}\\left(2-\\frac{A'J}{AJ'}\\right)\\left[\\frac{A'BJJ'}{A}-2+2BJ'^2\\right]\n\\nonumber\\right.\\\\ &&\\left.\n+\\frac{\\Gamma_1\\pi'}{2J'}\\left[\\frac{4}{J}+\\frac{A'^2BJ}{A^2}-\\frac{4A'}{AJ'}-\\frac{4BJ'^2}{J}\n-\\frac{2B'}{BJ'}+\\frac{2BJ''}{BJ'^2}-\\frac{4\\pi''}{\\pi'J'}\n\\right]\n-\\nonumber\\right.\\\\ &&\\left.\n-\n\\frac{\\Gamma_2\\pi'}{J'}\\left[\\frac{2A'}{AJ}+\\frac{A'^2}{A^2J'}(1-BJ'^2)-\\frac{4J'}{J^2}(1-BJ'^2)+\\frac{2B'}{BJ}+\\frac{4\\pi''}{\\pi'J}\n\\right]\\right) - e_4^{BH},\n\\\\\ne_4^{BH}&=& -\\frac{\\pi'^2}{J^2 J'} \\sqrt{\\frac{B^3}{A}}\n\\left(A'' J^2 J' - 4 A J'^3 +A' J(J'^2 - J J'')\\right) \\cdot F_4\n\\\\\\nonumber&&\n- \\frac{A'J + 2AJ'}{B J^2 J' \\pi'^2} \\cdot\n\\frac{\\mbox{d}}{\\mbox{d} r} \\left[\\sqrt{\\frac{B^5}{A}} J J' \\pi'^4 \\cdot F_4\\right],\n\\\\\na_{13}&=& J^2\\; a_4,\n\\\\\na_{14}&=& \\sqrt{{A}{B}} \\;J^2 \\left[\\mathcal{H}' + \\frac12 \\left(\\frac{B'}{B} +6\\frac{J'}{J} \\right)\\mathcal{H} \\right],\n\\\\\na_{15} &=& \\sqrt{\\frac{A}{B}}\\; \\mathcal{F},\n\\\\\na_{16} &=& -\\frac12 a_{15},\n\\\\\nb_{8} &=& - 2\\sqrt{\\frac{B}{A}} J^2 \\mathcal{H},\n\\\\\nb_{9} &=& \\sqrt{\\frac{B}{A}} J^2 \\left(\\frac{A'}{A} - 2 \\frac{J'}{J} \\right) \\mathcal{H},\n\\\\\nb_{10} &=& -b_6,\n\\\\\nb_{11} &=& \\sqrt{\\frac{B}{A}} \\frac{A'}{A} \\mathcal{H},\n\\\\\nc_{8} &=& -\\frac{J^2}{\\sqrt{{A}{B}}} (\\mathcal{H} - 2 F_4 B^2 \\pi'^4),\n\\\\\nc_{9} &=& \\frac14 \\sqrt{{A}{B}} \\left[\\Gamma \\pi' + \\left(\\frac{A'}{A} + 2 \\frac{J'}{J} \\right)\\mathcal{H} \\right],\n \\\\\nc_{10} &=& - \\frac14\\sqrt{{A}{B}} \\left[\n\\frac{J'}{J} \\left( \\Gamma + \\frac{J'}{J} \\Gamma_2\\right)\\pi' - \\frac14\\frac{A'}{A}\\left( \\Gamma_1 - 2 \\frac{J'}{J} \\Gamma_2\\right)\\pi'+\n\\right.\\\\\\nonumber&&\\left.\n\\mathcal{H} \\frac{J'}{J} \\left(\\frac{A'}{A} + 3 \\frac{J'}{J} \\right)+\n\\frac{1}{BJ^2} (\\mathcal{H} - 2 F_4 B^2 \\pi'^4) \\right],\n\\\\\nc_{11} &=& - \n\\frac12 \\sqrt{{A}{B}} J^2 \\left[\\Gamma \\pi' + \\left(\\frac{A'}{A} + 2 \\frac{J'}{J} \\right)\\mathcal{H} \\right],\n\\\\\nc_{12} &=& - \\frac12 \\sqrt{{A}{B}} J^2 \\left[\n\\frac{1}{2 J^2}\\left(\\frac{A'}{A} + \\frac{J'}{J} \\right)\\Xi\\pi' +\n\\frac34 \\frac{J'}{J} \\left( \\Gamma_1 - 2 \\frac{J'}{J} \\Gamma_2\\right)\\pi' \n\\right.\\\\\\nonumber&&\\left.\n+ \\frac{J'}{J} \\left(\\frac{A'}{A} + \\frac{J'}{J} \\right) \\mathcal{H} +\n\\frac{1}{BJ^2} (\\mathcal{H} - 2 F_4 B^2 \\pi'^4) \\right],\n\\end{eqnarray}\n\\begin{eqnarray}\nc_{13} &=& \\frac12 \\frac{\\sqrt{A}}{\\sqrt{B}} (\\mathcal{H} - 2 F_4 B^2 \\pi'^4),\n\\\\\nd_{5} &=& - \\frac{\\sqrt{{A}{B}}}{J^2} \\mathcal{H},\n\\\\\nd_{6} &=& 2\\frac{\\sqrt{{A}{B}}J'}{J^3} \\mathcal{H},\n\\\\\nd_{7} &=& a_4,\n\\\\\np_{8} &=& -\\frac{J^2}{2 \\sqrt{{A}{B}}} \\mathcal{F},\n\\\\\np_{9} &=& \\frac12 \\sqrt{{A}{B}} J^2 \\mathcal{H},\n\\end{eqnarray} \n\n\n\n\n\n\\section*{Appendix C}\nHere we put the rest of components of matrices $\\bar{\\mathcal{K}}$, $\\bar{\\mathcal{G}}$ and $\\bar{\\mathcal{M}}^{(\\ell^2)}$ entering the quadratic\naction~\\eqref{eq:action_KGQM_new} for completeness:\n\\begin{equation}\n\\bar{\\mathcal{K}}_{12} = \\bar{\\mathcal{K}}_{21} = \\frac{B^{1\/2}}{2(\\ell+2)(\\ell-1) A^{1\/2}} \\left[2(\\ell+2)(\\ell-1) (2\\mathcal{H} J J' + \\Xi \\pi') -\n\\frac{\\left[\\mathcal{H}^2 B J^2 \\right]'}{\\mathcal{H} B J} \\frac{\\mathbf{\\Theta}}{\\mathcal{F}} \\right],\n\\end{equation}\n\\begin{equation}\n\\bar{\\mathcal{K}}_{22} = \\frac{4 \\ell(\\ell+1) A^{1\/2} B^{1\/2}}{\\mathbf{\\Theta}^2} \\left[\\ell(\\ell+1) \\frac{B}{A} (2 \\mathcal{P}_1 -\\mathcal{F}) (2\\mathcal{H} J J' + \\Xi \\pi')^2\n+ (\\ell+2)(\\ell-1) \\mathcal{F}\\; \\bar{\\mathcal{K}}_{12}^2 \\right],\n\\end{equation}\n\n\n\\begin{equation}\n\\bar{\\mathcal{G}}_{12} = \\bar{\\mathcal{G}}_{21} = AB\\frac{\\mathcal{G}}{\\mathcal{F}} \\bar{\\mathcal{K}}_{12}\n\\end{equation}\n\n\\begin{equation*}\n\\bar{\\mathcal{G}}_{22} = \\frac{4\\ell(\\ell+1) A^{1\/2} B^{1\/2}}{\\mathbf{\\Theta}^2} \\frac{\\mathcal{G}}{\\mathcal{H}} \\left[ \\ell(\\ell+1) B^2 (2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B )\n+ \\frac{(\\ell+2)(\\ell-1)\\mathcal{F}^2}{AB \\;\\mathcal{G}} \\bar{\\mathcal{G}}_{12}^2\\right],\n\\end{equation*}\n\n\\begin{equation}\n\\bar{\\mathcal{M}}^{(\\ell^2)}_{12} = \\bar{\\mathcal{M}}^{(\\ell^2)}_{21} = - \\ell(\\ell+1) \\frac{ \\mathcal{H} (\\mathcal{H} - 2 F_4 B^2 \\pi'^2) } {2 \\mathcal{F} J} \\frac{A^{1\/2}}{B^{1\/2}} \\left[ 2 \\frac{\\left[\\mathcal{H}^2 B J^2 \\right]'}{\\mathcal{H}^2 J} + \\frac{B(A'J - 2 A J')}{A} - \\frac{B}{A} \\mathcal{F} \\mathcal{P}_4\n \\right],\n\\end{equation}\n\n\\begin{multline}\n \\bar{\\mathcal{M}}^{(\\ell^2)}_{22} = -\\ell(\\ell+1) \\;\\frac{ \\mathcal{H}^2}{\\mathcal{F}} \\frac{A^{1\/2} B^{1\/2}}{J^2} \\left[ 2\\frac{ \\mathcal{F} \\mathcal{G} }{\\mathcal{H}^2} + \\frac{B^{1\/2}}{A^{1\/2}} J^4\\; \\mathcal{F} \n\\left[\\frac{B^{1\/2}}{A^{1\/2}J^3} \\mathcal{P}_4\\right]' +\\frac{1}{B} \\left(\\frac{\\left[\\mathcal{H}^2 B J^2 \\right]'}{\\mathcal{H}^2 J}\\right)^2\n \\right] \\\\\n - \\frac{2}{(\\mathcal{H} - 2 F_4 B^2 \\pi'^2)} \\frac{\\left[\\mathcal{H}^2 B J^2 \\right]'}{\\mathcal{H} J^2} \\bar{\\mathcal{M}}_{12}^{(\\ell^2)}.\n\\end{multline}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nCancers evolve through waves of mutation and clonal expansion~\\cite{Nowell1976}. Darwinian selection operates on the increased variation within the tumour, favouring clones with increased fitness, according to microenvironmental and therapeutic pressures~\\cite{Fearon1990, Stratton2009, Aparicio2013, Beerenwinkel2015}. As a consequence of this evolutionary process, tumours are generally genetically heterogeneous~\\cite{Gerlinger2012, Nik-Zainal2012a} and consist of related populations of cancer cells (\\emph{clones}) with distinct genotypes, which encode the evolutionary history of each cell population~\\cite{Nik-Zainal2012a}.\nThis genetic heterogeneity is important clinically because it can confound the molecular profiling of biopsies, and increased variation may equip tumours with more avenues to escape treatment, leading to worse prognosis~\\cite{Schwarz2015}. \n\n\\paragraph{The clonal deconvolution problem}\nIdentifying clones and their proportions is a difficult task~\\cite{Beerenwinkel2015}, aggravated by the fact that cancer genomics data generally come from bulk sequencing experiments, which profile a mixture of cells from different clones. \nClones are related to each other and can be thought of as nodes in a phylogenetic tree that describes tumour development. The root of the tree corresponds to a normal, non-mutated cell; every other node is a cancer clone with a distinct complement of mutations (its \\emph{genotype}). Each clone inherits the mutations of its parent and adds more to them. This encodes a subset relationship between parent and child nodes.\n\nHowever, none of this is directly observable. Instead, the data only consist of a set of mutations and their proportions (called \\emph{allele fractions}) in a collection of tumour samples (Figure~\\ref{fig:overview}).\nThe \\emph{clonal deconvolution problem} thus asks to identify the clonal genotypes, phylogeny, and clonal fractions that best explain the observed data~\\cite{ElKebir2015}.\n\n\\paragraph{Additional challenges}\n\nThe clonal deconvolution problem is further complicated by factors such as the selection of alleles during tumour evolution and the specifics of the data obtained from sequencing experiments. \nIn particular, convergent evolution and mutational loss contradict the common assumption that mutations arise only once in the phylogeny (the infinite sites assumption) and never disappear. Tumours are subjected to internal selective pressures in their microenvironment and external pressures from therapeutic interventions. In such cases, multiple tumour clones may acquire the same mutation in convergent evolution, especially if it is a hotspot mutation or it confers resistance to the treatment. \nAt the same time, mutations can be removed by several mechanisms, including loss of heterozygosity, the deletion of the chromosome fragment carrying the mutation.\nAnother challenge is that for cost-effective sequencing options like targeted amplicon sequencing, which we will use in the case studies, the depth of sequencing is not informative of the chromosomal copy-number of the tumour. This contradicts assumptions often made by previous methods.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{overview3.pdf}\n \\caption{Overview of the general clonal deconvolution problem. Sample A shows how from a normal root node, four clones evolved according to the displayed phylogeny. In this example, genotypes consist of two loci (blue and orange), which may be mutated (red star), gained or lost. The normal genotype consists of two non-mutated alleles for each locus. Clonal fractions are represented by the diameter of the node and reported as a percentage. A second sample B from the same patient would also consist of the same tree and genotypes; clonal fractions, however, may change (as in this example). These latent parameters give rise to the observed mutation and copy-number data, shown at the bottom. The allele fraction of a mutation is the proportion of that allele in the sample. The observed copy-number is the total copy number of each clone weighted by the clonal fractions. The increase of the orange mutation's allele fraction and the decrease of its observed copy-number are due to the growth of the clone with a single mutated copy.}\n \\label{fig:overview}\n\\end{figure}\n\n\\paragraph{Previous approaches}\n\nVarious methods have been proposed in the literature~\\cite{Beerenwinkel2015} to improve on manual analyses~\\cite{Gerlinger2012,Nik-Zainal2012a}.\nTo put our approach in context, it is useful to distinguish \\emph{direct} reconstructions that directly infer clonal genotypes (like CloneHD~\\cite{Fischer2014}, Clomial~\\cite{Zare2014} and BayClone~\\cite{Sengupta2015}) from \\emph{indirect} reconstructions that obtain clusters of mutations rather than full genotypes and require additional phylogenetic analysis to obtain clonal genotypes (like PyClone~\\cite{Roth2014}, SciClone~\\cite{Miller2014}, PhyloWGS~\\cite{Deshwar2015}, and BitPhylogeny~\\cite{Yuan2015}).\n\nDirect reconstructions generally aim to infer two quantities, a matrix of mutation assignments and a matrix of clonal fractions, which come together in an admixture to form the sampling model. The mutation assignments matrix associates each mutation with zero or more classes, which can be intuitively interpreted as clonal genotypes. For models that lack a phylogeny, inference may yield biologically implausible genotypes, as shown later in the benchmarking studies (section~\\ref{sec:benchmark}).\n\nOn the other hand, indirect methods cluster mutations based on their allele fractions across multiple samples. Joint phylogenetic modelling allows these clusters to become nodes of a tree, displaying at which node each mutation first appeared. Hence, the assignment of mutation clusters to nodes of a tree is generally inflexible to episodes of convergent evolution or mutational loss. \n\n\\paragraph{Latent feature models}\n\nHere we introduce Cloe, a phylogenetic latent feature model for clonal deconvolution that belongs to the category of direct reconstruction methods. \nLatent feature models discover independent features with which to describe a set of observed objects. The set of features possessed by an object determines the parameters of its distribution~\\cite{Griffiths2005}. In our context, observed objects are mutations, and latent features are clonal genotypes. \n\nLatent feature models have been previously applied to clonal deconvolutions, but maintained the assumption that features are independent~\\cite{Zare2014,Sengupta2015}.\nIn parallel, extensions to these models have been developed to relate features hierarchically, but placed features as the leaves of the tree~\\cite{Heaukulani2014}. Moreover, these tree structure only correlated the feature assignments, making such a model unsuitable for clonal deconvolutions. \n\nThe model we propose lifts the independence assumption and relates features with a latent hierarchy. In our framework features live at every node of the tree, thus encoding a noisy subset relationship in the mutation assignments. Our model differs from the phylogenetic Indian Buffet Process as the latter relates observed objects with a latent phylogeny, rather than the features~\\cite{Miller2012}.\nOur approach is more general than previously published methods, because it relies on fewer assumptions on clones and the evolutionary model: we can readily model multiple independent primary tumours, account for loss of mutations and penalise, though still allow, convergent evolution. \n\nWe validate Cloe{} on simulated data, on a controlled biological dataset, and apply it to two published clinical datasets: longitudinal samples from three chronic lymphocytic leukaemia patients~\\cite{Schuh2012}, and from an acute myeloid leukaemia case~\\cite{Griffith2015}. Cloe{} is available as an R package at \\url{https:\/\/bitbucket.org\/fm361\/cloe}.\n\n\n\\section{The Cloe{} model}\n\nOur model follows the overview of Figure~\\ref{fig:overview}. A latent phylogenetic tree influences the clonal genotypes; these, together with clonal fractions and additional nuisance parameters, describe the distribution of the data.\n\nWe observe data for \\( J \\) mutations in \\( T \\) samples, and the data are collected in two \\( J \\times T \\) matrices: \\textbf{X} for mutant read counts, and \\textbf{D} for read depths, the number of times a particular locus of the genome is covered by sequencing reads.\n\nThe phylogenetic tree is defined by a vector \\( \\mathcal{T} \\) with \\( K > 1 \\) elements, one for each clone. We consider the normal contamination as a fixed clone. Our analysis is restricted to mutations in copy-number neutral regions: each clone, including the normal, contributes exactly two copies of each allele, of which at most one can be mutated. Clonal genotypes are defined in a binary \\( J \\times K \\) matrix \\textbf{Z}, where each column \\( z_{\\cdot k} \\) represents the genotype of clone \\( k \\). The proportions of each clone in each sample are summarised in a \\( K \\times T \\) matrix \\textbf{F} formed by \\( T \\) stochastic vectors.\n\nOur goal is to infer the phylogeny~\\( \\mathcal{T} \\), clonal genotypes~\\textbf{Z}, and clonal fractions~\\textbf{F} from the posterior distribution \\( P(\\mathcal{T}, \\textbf{Z}, \\textbf{F} \\, \\vert \\, \\textbf{X}, \\textbf{D}) \\). To do this, in the following sections we develop a probabilistic model that links observed and unobserved variables, and an inference algorithm to explore the posterior distribution.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{graphical_model_cnn_2_alt.pdf}\n \\caption{An outline of the graphical model corresponding to Cloe, omitting for simplicity overlapping plates and convergent evolution relations. Legend: \\( \\gamma \\), clonal fractions hyperparameters; \\textbf{F}, clonal fractions matrix; \\( \\mathcal{T} \\), phylogeny; \\textbf{Z}, genotypes matrix; \\textbf{X}, mutant read counts; \\textbf{D}, depths; \\( s \\), Beta-binomial overdispersion parameter.}\n \\label{fig:graphical_model_outline}\n\\end{figure}\n\n\n\\subsection{Model definition}\nFor guidance, a simplified version of our model is outlined in Figure~\\ref{fig:graphical_model_outline}, while at the end of this section Figure~\\ref{fig:graphical_model_full} presents the complete model.\n\n\n\\paragraph{Phylogeny}\nFor \\( K > 1 \\) populations, we model the phylogenetic tree as a vector \\( \\mathcal{T} \\) of length \\( K \\), where \\( \\mathcal{T}_k = l \\) means that the parent of \\( k \\) is \\( l \\). The normal clone is fixed as the first entry, the root of the tree. To ensure that the graph encoded by \\( \\mathcal{T} \\) is a tree, we let each entry only take values on the previous entries. This definition is flexible as the tree can assume any shape, even allowing phylogenies with multiple primary tumours. \\( \\mathcal{T} \\) is defined by\n\\begin{align}\n \\begin{aligned}\n\\mathcal{T}_1 & = 0, \\\\ \n\\mathcal{T}_2 & = 1, \\\\\n\\mathcal{T}_k & \\sim \\mathcal{U}\\left(\\delta, k-1 \\right) \\quad\\text{for } k \\in \\{ 3 \\dots K \\},\n \\end{aligned}\n \\label{eq:tree}\n\\end{align}\nwhere \\( \\mathcal{U}(\\delta, a) \\) is a one-deflated discrete uniform distribution with values in~\\( \\{1, \\ldots, a \\}\\). The probability of drawing a 1 is \\( \\delta \\), and the probability of drawing an integer between \\( 2 \\) and \\( a \\) is uniform:\n\\begin{align}\n\\mathcal{U}(x; \\delta, a) =\n \\begin{cases}\n \\delta & \\text{ if } x = 1 \\\\\n \\frac{1 - \\delta}{a - 1} & \\text{ if } x \\in \\left\\{ 2, 3, \\dots, a \\right\\}\n \\end{cases}\n\\end{align}\nWe penalise multiple independent primary tumours (multiple children of the normal clone) by setting \\( \\delta = (2 k)^{-1} \\).\n\n\\paragraph{Genotypes}\nGenotypes are defined in a binary \\( J \\times K \\) matrix \\(\\text{\\textbf{Z}} = (z_{jk}) \\), where \\(1\\) denotes a mutation and \\(0\\) the un-mutated (\\emph{wildtype}) state, for each mutation \\( j \\) in each clone \\( k \\). \nWe fix the genotype of the normal clone to a zero vector of length \\( J \\), implying that all mutations are somatic. More generally, the normal genotype could be modified to accommodate for known germline variants. The genotype of a clone \\( k \\) for a mutation \\( j \\) is then defined as \n\\begin{equation}\nz_{jk} \\, \\vert \\, z_{j\\mathcal{T}_k}, \\mu, \\rho \\sim \\text{Bernoulli}(p_{jk})\n\\label{eq:genotypes}\n\\end{equation}\nwhere \\( \\mu \\) is the probability of mutating if the parent does not have a mutation, and\n\\( \\rho \\) is the probability of reverting to wildtype if the parent is mutated:\n\\begin{equation}\np_{jk} =\n\\begin{cases}\n\\,\\mu & \\text{ if } z_{j\\mathcal{T}_k} = 0 \\\\\n\\,1 - \\rho & \\text{ if } z_{j\\mathcal{T}_k} = 1.\n\\end{cases}\n\\label{eq:genotypes_probability}\n\\end{equation}\n\n\n\\paragraph{Clonal fractions}\nBecause the samples may be collected latitudinally, longitudinally, and at irregular intervals, we assume that clonal fractions are independent between samples. We represent clonal fractions with a \\( K \\times T \\) matrix \\( \\mathbf{F} = (\\mathbf{f}_{\\cdot t})_{t = 1, \\ldots, T} \\) composed of a collation of stochastic column vectors \\( \\mathbf{f}_{\\cdot t} \\) describing the proportions of each clone in a sample \\(t\\). Clonal fractions for a sample \\( t \\) are modelled with a symmetric Dirichlet distribution with hyperparameter \\( \\gamma_t \\):\n\\begin{equation}\n\\mathbf{f}_{\\cdot t} \\, \\vert \\, \\gamma_t \\sim \\text{Dirichlet}(\\gamma_t)\n\\end{equation}\n\n\\paragraph{Likelihood}\nGenotypes and clonal fractions come together in an admixture, their dot product representing the expected allele fractions for each mutation in each sample. \nWe model mutant reads as successful trials from a beta-binomial distribution with overdispersion parameter \\( s \\). The probability of success is a function of the expected allele fraction \\( p_{jt} = \\frac{1}{2} \\left( \\mathbf{z}_{j \\cdot} \\cdot \\mathbf{f}_{\\cdot t} \\right) \\). To capture sequencing noise at extreme values of \\( p_{jt} \\) we replace it with a function \\( e(p_{jt}) \\) that depends on the sequencing error rate \\( \\varepsilon \\) (e.g. 0.1\\%) such that\n\\begin{equation}\ne(p_{jt}) \\; =\n\\begin{cases}\n\\; \\varepsilon & \\text{ if } p_{jt} = 0 \\\\\n\\; 1 - \\varepsilon & \\text{ if } p_{jt} = 1 \\\\\n\\; p_{jt} & \\text{ otherwise.}\n\\end{cases}\n\\label{eq:seq_noise}\n\\end{equation}\n\nThe likelihood is then specified by\n\\begin{equation}\n\\label{eq:cnn_likelihood}\nx_{jt} \\, \\vert \\, d_{jt}, \\mathbf{z}_{j \\cdot}, \\mathbf{f}_{\\cdot t}, s \\;\\sim\\; \\text{Beta-binomial}(d_{jt}, e(p_{jt}), s).\n\\end{equation}\n\n\n\\paragraph{Nuisance parameters}\nWe let the beta-binomial overdispersion parameter \\( s \\) and the Dirichlet hyperparameters \\( \\bm{\\gamma} \\) have Gamma priors, whereas the mutation and reversion probabilities \\( \\mu \\) and \\( \\rho \\) are fixed.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.6\\textwidth]{graphical_model_cnn_4.pdf}\n \\caption{The full graphical model of Cloe.}\n \\label{fig:graphical_model_full}\n\\end{figure}\n\n\n\n\\subsection{Penalising convergent evolution}\nOne of the risks of assuming the independence of features in this biological application is that the inferred genotypes may largely display convergent evolution. We can penalise such occurrences by altering the definition of genotypes (cfr. equations~\\ref{eq:genotypes} and \\ref{eq:genotypes_probability}).\n\nUnder the infinite sites assumption (ISA) every mutation occurs only once, so that if multiple clones possess a mutation \\( j \\), then the mutation must have appeared with their most recent common ancestor. In contrast, if the most recent common ancestor is not mutated, then the mutation must have appeared multiple times (convergent evolution).\nWe thus say that a mutation assignment \\( z_{jk} = 1 \\) conflicts with ISA if the most recent common ancestor of \\( \\left\\{ k' : z_{jk'} = 1, k' \\leq k \\right\\} \\) does not harbour mutation \\( j \\). \n\nWe include ISA checks into our model by using an indicator function \\( I(j, k, a) \\) that returns \\( 1 \\) if the most recent common ancestor of all clones \\( k' \\leq k \\) that harbour mutation \\( j \\) also possesses the mutation, when \\( z_{jk} = a \\).\nThat is, \\( I(j, k, a) = 1 \\) if ISA is satisfied by setting \\( z_{jk} = a \\), and 0 otherwise. \n\nThus we redefine the distribution of genotypes making them conditional on all previous genotypes and weighting assignments by a user-defined parameter \\( \\nu \\) if they comply with ISA, or by \\( 1 - \\nu \\) if they do not:\n\\begin{equation}\nP(z_{jk} = 1 \\, \\vert \\, \\mathcal{T}, \\mathbf{Z}_{j, < k}, z_{j\\mathcal{T}_k} = 0, \\mu, \\rho, \\nu) \\propto ( \\mu \\nu )^{I(j, k, 1)} ( \\mu ( 1 - \\nu ) )^{1 - I(j, k, 1)}\n\\label{eq:genotypes_isa_probability}\n\\end{equation}\nIn practice, only transitions that gain a mutation can clash with ISA, and the factor of \\( \\nu \\) immediately cancels out if the parental genotype is 1 at a given \\( j \\). An ISA-check at \\( j \\) is thus only warranted if the parent is not mutated at \\( j \\).\n\nThe graphical model corresponding to what has been described so far is shown in Figure~\\ref{fig:graphical_model_full}.\n\n\n\\begin{algorithm}\n \\begin{algorithmic}[1]\n \\For{\\( i \\) = 1, \\dots, \\#\\text{iterations}}\n \\For{\\( m \\) = 1, \\dots, \\#\\text{chains}}\n \\For{\\( j \\) = 1 \\dots \\( J \\)}\n \\Comment \\textbf{Z}\n \\State Propose new \\( \\mathbf{z}^{* (m)}_{j \\cdot} \\)\n \\State Accept with probability \\ref{eq:inference_z_mh_ratio}\n \\EndFor\n \\For{\\( k \\) = 3 \\dots \\( K \\)}\n \\Comment \\( \\mathcal{T} \\)\n \\State Compute \\( P(\\mathcal{T}^{*(m)}_k = l) \\) for \\( l \\in \\left\\{ 1 \\dots k - 1 \\right\\} \\) (eq.~\\ref{eq:inference_tree})\n \\State Sample new \\( \\mathcal{T}^{*(m)}_k \\) from \\( P(\\mathcal{T}^{*(m)}_k) \\)\n \\EndFor\n \\State Randomly swap two siblings\n \\State With probability 1\\% propose a swap between a node and its parent\n \\State Accept with probability~\\ref{eq:parent_swap}\n \\For{\\( t \\) = 1 \\dots \\( T \\)}\n \\Comment \\textbf{F}\n \\State Propose new \\( \\mathbf{f}^{*(m)}_{\\cdot t} \\) from eq.~\\ref{eq:inference_f_proposal}\n \\State Accept with probability \\ref{eq:inference_f_mh_ratio}\n \\EndFor\n \\Comment Nuisance parameters\n \\State Propose new \\( s^{*(m)} \\sim \\mathcal{N}(s^{(m)}, \\sigma_s) \\) and accept with probability~\\ref{eq:inference_s_mh_ratio}\n \\For{\\( t \\) = 1 \\dots \\( T \\)}\n \\State Propose new \\( \\gamma^{*(m)}_t \\sim \\mathcal{N}(\\gamma^{(m)}_t, \\sigma_\\gamma) \\) and accept with probability~\\ref{eq:inference_gamma_mh_ratio}\n \\EndFor\n \\EndFor\n \\If{\\( i \\) is a multiple of 100}\n \\Comment Chain swap\n \\State Propose a chain \\( j \\in \\{ 1 \\dots m - 1 \\} \\)\n \\State Accept the state swap between chains \\( j \\) and \\( j + 1 \\) with probability~\\ref{eq:inference_swap_mh_ratio}\n \\EndIf\n \\EndFor\n \\end{algorithmic}\n \\caption{MCMCMC sampling algorithm for Cloe}\n \\label{alg:inference}\n\\end{algorithm}\n\n\n\n\\subsection{Inference}\n\nWe are interested in the posterior distribution of the latent variables given the observed variables\n\\(P(\\mathcal{T}, \\mathbf{Z}, \\mathbf{F} \\, \\vert \\, \\mathbf{X}, \\mathbf{D})\\),\nwhich we explore by Metropolis-coupled Markov chain Monte Carlo (MCMCMC)~\\cite{Geyer1991}, integrating out the nuisance parameters \\( s\\) and \\( \\bm{\\gamma} \\).\nWe use Gibbs sampling to update the tree vector \\( \\mathcal{T} \\) and Metropolis-Hastings moves to update the other quantities.\n\nBecause the posterior landscape appears composed of high peaks separated by deep valleys (Supplementary Figure 1), we run five chains in parallel, with tempered posteriors. \nThe sampling strategy described hereafter is applied to each chain, and summarised in Algorithm~\\ref{alg:inference}.\n\n\\paragraph{Phylogeny}\nFor each \\( \\mathcal{T}_{k > 2} \\), we compute the conditional posterior of the parent assignment \\( \\mathcal{T}_k = l \\), for each \\( l < k \\):\n\\begin{equation}\nP( \\mathcal{T}_k = l \\, \\vert \\, \\dots) \\propto P(\\mathbf{Z} \\, \\vert \\, \\mathcal{T}_{-k}, \\mathcal{T}_k = l) \\,\nP(\\mathcal{T}_k = l)\n\\label{eq:inference_tree}\n\\end{equation}\nThe likelihood term amounts to tallying the genotype transitions from parent \\( l \\) to child \\( k \\), and reassessing how many transitions comply or clash with ISA for all clones. The prior, according to eq.~\\ref{eq:tree}, is equal to\n\\begin{equation}\n \\begin{aligned}\nP(\\mathcal{T}_k = l) & = \\delta^{\\bm{I}(l = 1)} \\left( \\frac{1 - \\delta}{k - 2} \\right)^{1 - \\bm{I}(l = 1)} \\\\\n& = \\left( \\frac{1}{2k} \\right)^{\\bm{I}(l = 1)} \\left( \\frac{2k - 1}{2k \\, (k - 2)} \\right)^{1 - \\bm{I}(l = 1)}\n \\end{aligned}\n \\label{eq:inference_tree_prior}\n\\end{equation}\n\nTo facilitate the exploration of the space of tree and genotypes configurations, we uniformly propose a pair of siblings and swap their position in the tree and in the genotypes and clonal fractions matrices. Prior to this swap, the siblings had access to one linear topology. This move allows the other linear topology to be explored, while leaving probabilities unaltered.\n\nIn addition, the swap between a node \\( k \\) and its parent \\( l \\) is proposed. A node \\( k \\) is chosen uniformly from \\( \\left\\{ 3 \\dots K \\right\\} \\). A tree \\( \\mathcal{T}^* \\) is created where \\( k \\) is the parent of \\( l \\), while any children of \\( k \\) remain children of \\( k \\); the same applies to \\( l \\). As with the sibling swap, this move requires rearranging the clone order in the genotypes and clonal fractions matrices. The parent swap affects genotype transitions from \\( \\mathcal{T}_l \\), the original parent of \\( l \\), to \\( k \\), and from \\( k \\) to \\( l \\). The proposal is accepted with probability\n\\begin{equation}\n\\min\\left( 1, \\frac{P(\\mathbf{Z}_{\\{ \\mathcal{T}_l, k, l \\}} \\, \\vert \\, \\mathcal{T}^*, \\mu, \\rho, \\nu) \\, P(\\mathcal{T}^*_{\\{ \\mathcal{T}_l, k, l \\}})}{P(\\mathbf{Z}_{\\{ \\mathcal{T}_l, k, l \\}} \\, \\vert \\, \\mathcal{T}, \\mu, \\rho, \\nu) \\, P(\\mathcal{T}_{\\{ \\mathcal{T}_l, k, l \\}})} \\right)\n\\label{eq:parent_swap}\n\\end{equation}\nWe perform this update with probability 0.01.\n\n\\paragraph{Genotypes}\nBecause mutations are independent, we update \\textbf{Z} by row, proposing a new row \\( \\mathbf{z}^*_{j \\cdot} \\) by flipping each bit of \\( \\mathbf{z}_{j \\cdot} \\) with probability \\( \\theta \\). The proposal is symmetric and the move is accepted with probability\n\\begin{equation}\n\\min\\left( 1, \\frac{P(\\mathbf{x}_{j \\cdot} \\, \\vert \\, \\mathbf{d}_{j \\cdot}, \\mathbf{z}^*_{j \\cdot}, \\mathbf{F}, s) \\, P(\\mathbf{z}^*_{j \\cdot} \\, \\vert \\, \\mathcal{T}, \\mu, \\rho, \\nu)}{P(\\mathbf{x}_{j \\cdot} \\, \\vert \\, \\mathbf{d}_{j \\cdot}, \\mathbf{z}_{j \\cdot}, \\mathbf{F}, s) \\, P(\\mathbf{z}_{j \\cdot} \\, \\vert \\, \\mathcal{T}, \\mu, \\rho, \\nu)} \\right)\n\\label{eq:inference_z_mh_ratio}\n\\end{equation}\nwhere the likelihood is only computed for mutation \\( j \\), and the prior refers to the sequence of transitions from the root genotype at \\( j \\) to the leaves, with appropriate penalties for convergent evolution.\n\n\n\\paragraph{Clonal fractions}\nBecause of the independence of the samples, the matrix \\textbf{F} is updated by column. A new vector \\( \\mathbf{f}^*_{\\cdot t} \\) is proposed from a Dirichlet distribution centred at the current value \\( \\mathbf{f}_{\\cdot t} \\):\n\\begin{equation}\nQ(\\mathbf{f}^*_{\\cdot t} \\, \\vert \\, \\mathbf{f}_{\\cdot t}) = \\text{Dirichlet}(\\psi \\, \\mathbf{f}_{\\cdot t} + \\epsilon)\n\\label{eq:inference_f_proposal}\n\\end{equation}\nwhere \\( \\psi \\) is a precision factor and \\( \\epsilon \\) a small bias to avoid sinks at 0. The proposal is accepted with probability\n\\begin{equation}\n\\min\\left( 1, \\frac{P(\\mathbf{x}_{\\cdot t} \\, \\vert \\, \\mathbf{d}_{\\cdot t}, \\mathbf{Z}, \\mathbf{f}^*_{\\cdot t}, s) \\ P(\\mathbf{f}^*_{\\cdot t} \\, \\vert \\, \\gamma_t) \\ Q(\\mathbf{f}_{\\cdot t} \\, \\vert \\, \\mathbf{f}^*_{\\cdot t})}{P(\\mathbf{x}_{\\cdot t} \\, \\vert \\, \\mathbf{d}_{\\cdot t}, \\mathbf{Z}, \\mathbf{f}_{\\cdot t}, s) \\ P(\\mathbf{f}_{\\cdot t} \\, \\vert \\, \\gamma_t) \\ Q(\\mathbf{f}^*_{\\cdot t} \\, \\vert \\, \\mathbf{f}_{\\cdot t})} \\right)\n\\label{eq:inference_f_mh_ratio}\n\\end{equation}\n\n\\paragraph{Nuisance parameters}\n\nThe remaining parameters are updated with Metropolis moves using Gaussian proposals. The Metropolis-Hastings acceptance ratios are\n\\begin{equation}\n\\min\\left( 1, \\frac{P(\\mathbf{X} \\, \\vert \\, \\mathbf{D}, \\mathbf{Z}, \\mathbf{F}, s^*) \\ P(s^*)}{P(\\mathbf{X} \\, \\vert \\, \\mathbf{D}, \\mathbf{Z}, \\mathbf{F}, s) \\ P(s)} \\right) \\quad\\text{for}~s,\n\\label{eq:inference_s_mh_ratio}\n\\end{equation}\nand\n\\begin{equation}\n\\min\\left( 1, \\frac{P(\\mathbf{f}_{\\cdot t} \\, \\vert \\, \\gamma^*_t) \\ P(\\gamma^*_t)}{P(\\mathbf{f}_{\\cdot t} \\, \\vert \\, \\gamma_t) \\ P(\\gamma_t)} \\right) \\quad\\text{for}~\\gamma_t.\n\\label{eq:inference_gamma_mh_ratio}\n\\end{equation}\n\n\n\\paragraph{Temperatures and chain swaps}\n\nRegularly at user-defined intervals, a swap between two adjacent chains is proposed as a Metropolis-Hastings move~\\cite{Geyer1991}. Let \\( M \\) denote the number of parallel chains, \\( P^{(m)} \\) denote the tempered posterior of chain \\( m \\), and \\( \\omega_m \\) denote the state of chain \\( m \\). A chain \\( m \\) is selected among the first \\( M - 1 \\) chains. The swap between chains \\( m \\) and \\( m + 1 \\) is then accepted with probability\n\\begin{equation}\n\\min\\left( 1, \\frac{P^{(m)}(\\omega_{m+1}) \\ P^{(m+1)}(\\omega_m)}{P^{(m)}(\\omega_{m}) \\ P^{(m+1)}(\\omega_{m+1})} \\right).\n\\label{eq:inference_swap_mh_ratio}\n\\end{equation}\n\nThe temperature \\( \\tau_m \\) for each chain \\( m \\) is chosen according to the following scheme:\n\\begin{equation}\n\\tau_m = (1 + \\Delta T (m - 1))^{-1}\n\\end{equation}\nwhere \\( \\Delta T > 0 \\) regulates the temperature differences between chains.\n\n\n\\paragraph{Parameter estimates}\n\nMCMCMC parameter estimates are derived solely from the first, untempered chain. After discarding a certain proportion of the initial samples as burn-in, and thinning the chain by a factor of \\( i \\), thus considering every \\( i^{th} \\) sample, we obtain a maximum \\emph{a posteriori} (MAP) estimate of the parameters by selecting the chain state of the sample with the highest posterior value.\n\n\n\n\n\n\\section{Validation and benchmarks}\n\nWe extensively validated and benchmarked Cloe{} by using simulated data and a controlled experimental set-up based on mixing cell lines. \n\n\\subsection{Simulated data}\n\nWe first tested our model on 9 simulated datasets, one for each combination of number of clones (3, 4 or 5) and depth of sequencing (means: 50\\( \\times \\), 200\\( \\times \\), 1000\\( \\times \\)).\nThe genotypes were created according to a random tree and using parameters \\( \\mu = 0.5 \\), \\( \\rho = 0.05 \\), \\( \\nu = 0.9 \\); clonal fractions were \\emph{iid} draws from a symmetric Dirichlet distribution with parameter \\( \\gamma = 2 \\). All datasets contained 100 mutations and 5 samples, with depths obtained from a Poisson distribution and mutant read counts obtained from a binomial distribution. Because of the fixed size of our model, we ran Cloe{} for 3, 4, and 5 clones on each of the 9 datasets (running parameters are reported in Table~\\ref{tab:params}).\n\nWe measured Cloe's performance in several ways: we evaluated its ability to identify the right number of clones, we assessed mixing by computing the Gelman-Rubin statistic from three consecutive runs of the algorithm, and finally we measured the reconstruction error. To perform the latter step, we calculated two metrics, the normalised genotypes error \\( Z_{err} \\) and the normalised clonal fractions error \\( F_{err} \\), both defined as the sum of the absolute differences between inferred (\\( \\mathbf{Z}^* \\)) and the real (\\( \\mathbf{Z} \\)) matrices, normalised by the real matrix dimensions (ignoring the fixed genotype in \\textbf{Z}).\nTo control for equivalent solutions with permuted clones, we used permutations \\(\\sigma\\) of the columns of \\( \\mathbf{Z}^* \\) to minimise \\( Z_{err} \\): \n\\begin{equation}\nZ_{err} = \\frac{1}{J (K - 1)} \\, \\min_{\\sigma} \\left(\\sum_{j,k} \\vert z^*_{j\\sigma(k)} - z_{jk} \\vert \\right),\n\\end{equation}\n The same permutation is then used to rearrange the rows of \\( \\mathbf{F}^* \\), which is normalised by \\( J K \\). If the real and inferred matrices have different sizes, we pad the smaller one with normal clones with zero clonal fractions. In this instance \\( K \\) refers to the real number of clones. \n\n\\paragraph{MCMC mixing and effective sample size}\nThe Gelman--Rubin statistic was calculated from the log-posterior values of the untempered chains as a proxy for the multidimensional parameters. In every case, the potential scale reduction factor is within the accepted range, less than 1.1 (Figure~\\ref{fig:simulation_psrf}). Each run was started with a different random seed.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{simulations_psrf.pdf}\n \\caption{The Gelman-Rubin statistic computed from three runs of our algorithm on the simulated datasets. Row names consist of the number of clones that was used in the run, and the characteristics of the dataset (sequencing depth and the real number of clones).}\n \\label{fig:simulation_psrf}\n\\end{figure}\n\nWe computed the effective sample size (ESS) for the first run on each dataset, focussing on the cases where the sought number of clones matched the real number of clones of the dataset. Table~\\ref{tab:ess} reports the ESS of the nuisance parameters and the log-posterior computed from 10,000 post-thinning iterations. Modulating the standard deviation of the Gaussian proposals for the nuisance parameters could decrease their autocorrelation. However, the shape of the posterior space~(Supplementary Figure 1) may prevent efficient large moves.\n\n\\begin{table}\n \\centering\n \\begin{tabular}{lrrrrrrr}\n \\hline\nDataset & \\( \\gamma_1 \\) & \\( \\gamma_2 \\) & \\( \\gamma_3 \\) & \\( \\gamma_4 \\) & \\( \\gamma_5 \\) & \\( s \\) & LP \\\\\n \\hline\n50x, 3 & 194.63 & 272.20 & 474.59 & 117.33 & 134.96 & 1696.31 & 4366.40 \\\\\n200x, 3 & 178.25 & 352.60 & 505.93 & 171.06 & 151.80 & 890.28 & 3088.45 \\\\\n1000x, 3 & 143.95 & 442.33 & 446.48 & 220.48 & 153.46 & 426.54 & 1685.87 \\\\\n50x, 4 & 648.35 & 241.09 & 586.84 & 170.57 & 342.17 & 1345.90 & 2323.20 \\\\\n200x, 4 & 735.31 & 195.13 & 683.51 & 152.63 & 346.05 & 1025.92 & 1700.03 \\\\\n1000x, 4 & 1083.17 & 302.75 & 654.13 & 140.00 & 237.40 & 406.45 & 894.08 \\\\\n50x, 5 & 395.45 & 239.42 & 209.06 & 261.33 & 369.81 & 1182.60 & 1673.17 \\\\\n200x, 5 & 803.71 & 387.01 & 394.54 & 324.07 & 644.47 & 947.42 & 2401.41 \\\\\n1000x, 5 & 990.07 & 401.38 & 373.32 & 284.74 & 657.88 & 478.26 & 1805.82 \\\\\n \\hline\n \\end{tabular}\n \\caption{Effective sample size per dataset. The effective sample size was computed on the 10,000 post-thinning iterations for the first of the three replicate runs with the correct number of clones. The dataset is denoted by the average sequencing depth and the real number of clones. LP denotes the log-posterior, as a proxy for the multidimensional parameters.}\n \\label{tab:ess}\n\\end{table}\n\n\n\\paragraph{Model selection}\nAs a model selection criterion we used the log-posterior probability of the MAP sample. We were able to recover the correct size \\( K \\) for every dataset (Figure~\\ref{fig:simulation_results}, left) in two of the three runs. In one run, a higher log-posterior probability was given to the solution with 4 clones on the dataset with 50\\( \\times \\) and 5 clones (-14464.05 compared to the 5-clone solution's -14464.58; Supplementary Figure 2). In such cases, where the log-posterior probabilities of different models are approximately equal, we prefer the solution with higher log-likelihood value.\n\n\\begin{figure}\n \\centering\n \\makebox[\\textwidth][c]{\n \\includegraphics[scale=0.52]{simulations_results_3.pdf}\n }\n \\caption{Results on the simulated datasets. Left: inferred model size for every combinations of \\( K \\) and depths. Centre and right: reconstruction errors. All datasets consisted of five samples and 100 mutations.}\n \\label{fig:simulation_results}\n\\end{figure}\n\n\n\\paragraph{Reconstruction fidelity}\nTo assess our reconstructions we considered only the MAP solution for each dataset. The reconstruction error was low, with \\( Z_{err} \\leq 0.027 \\) and \\( F_{err} \\leq 0.033 \\). The largest errors were obtained at the lowest depth (Figure~\\ref{fig:simulation_results}, centre and right), suggesting that on these random datasets Cloe{} can not only discover the correct number of clones,\nbut also infer correct genotypes and clonal fractions with \\( > 96.7\\% \\) accuracy (Supplementary Figures 3 and 4).\n\n\\paragraph{Conclusion}\nThis validation with synthetic data provided proof that the model and the inference are sound, achieving good reconstructions even with several clones and low sequencing depth (Supplementary Figure 5).\n\n\\begin{table*}\n \\begin{tabular}{lr}\n\\hline\n\\textbf{Parameter} & \\textbf{Value} \\\\\n\\hline\n\\multicolumn{2}{c}{MCMCMC} \\\\\n\\hline\niterations & 40000 \\\\\nchains & 5 \\\\\n\\( \\Delta T \\) & 0.4 \\\\\nswap interval & 50 \\\\\nburn-in & 50\\% \\\\\nthinning factor & 4 \\\\\n\\hline\n\\multicolumn{2}{c}{\\textbf{Z}} \\\\\n\\hline\n\\( \\mu \\) & 0.3 \\\\\n\\( \\rho \\) & 0.1 \\\\\n\\( \\nu \\) & 0.75 \\\\\n\\( \\theta \\) (proposal) & 0.20 \\\\\n\\( \\varepsilon \\) (likelihood) & 0.005, 0.002 \\\\\n\\hline\n\\multicolumn{2}{c}{\\textbf{F}} \\\\\n\\hline\n\\( \\psi \\) (proposal) & 200 \\\\\n\\( \\epsilon \\) (proposal) & 4 \\\\\n\\hline\n\\multicolumn{2}{c}{Nuisance parameters} \\\\\n\\hline\n\\( \\gamma \\) (prior, shape) & 2 \\\\\n\\( \\gamma \\) (prior, rate) & 1 \\\\\n\\( \\sigma_\\gamma \\) (proposal) & 0.2 \\\\\n\\( s \\) (prior, shape) & 11 \\\\\n\\( s \\) (prior, rate) & 0.10 \\\\\n\\( \\sigma_s \\) (proposal) & 16 \\\\\n\\hline\n \\end{tabular}\n \\caption{Running parameters for the simulated and validation datasets. For \\( \\varepsilon \\), the sequencing error parameter, the first value refers to the simulations, the second to the validation dataset.}\n \\label{tab:params}\n\\end{table*}\n\n\n\n\n\\subsection{Controlled experimental data}\n\nBecause synthetic data may not capture the variability seen in real biological data, we tested our method on a bespoke experiment.\nIn order to mimic heterogeneous tumour samples, we genotyped five cell lines and mixed them together at known proportions. Four single-cell-diluted cancer cell lines (HD124, HD212, HD249, and HD659, from Horizon Discovery, UK) were mixed with a normal cell line (HG00131, 1000 Genomes Project, Coriell Institute, USA). All five cell lines were subjected to whole-exome sequencing (Nextera Rapid Capture Exome, Illumina, USA).\nMutational and single nucleotide polymorphism (SNP) profiles were obtained using the standard samtools workflow \\cite{samtools}. To identify copy-number neutral regions, we generated copy-number profiles with the R package CopywriteR (version 2.2.0). We created 14 mixtures, with a median tumour cellularity of 64\\% (Table~\\ref{tab:mixtures}).\n\nBecause the cell lines were unrelated, we selected a subset of mutations (heterozygous single nucleotide variants and small indels) so as to embed the cell lines into an artificial phylogeny (Figure~\\ref{fig:data_cnn}, left). We focussed on regions that were copy-number neutral across all cell lines, and measured allele fractions by targeted sequencing (Figure~\\ref{fig:data_cnn}, right) \\cite{Forshew2012}. We excluded from the final dataset mutations whose genotypes had been erroneously inferred from exome sequencing data. The final dataset included 82 mutations, of which one displayed convergent evolution and two were reversions to wildtype, and was sequenced to a median of 17260\\( \\times \\) coverage.\nTargeted sequencing data were processed with an \\emph{in house} pipeline and the known mutations were quantified from pileups with a base quality cutoff of 30. \n\n\\begin{figure}\n \\centering\n \\makebox[\\textwidth][c]{\n \\includegraphics[scale=0.5]{data.pdf}\n }\n \\caption{The observed data for the validation dataset. On the left is the artificial phylogeny, where N denotes the normal clone, C1 to C4 are the cancer cell lines, playing in this context the role of clones. On the right are the observed mutational dynamics (allele fractions over samples) at an average depth of 17260\\( \\times \\).}\n \\label{fig:data_cnn}\n\\end{figure}\n\n\\begin{table*}\n \\centering\n \\makebox[\\textwidth][c]{\n \\begin{tabular}{rrrrrrrrrrrrrrr}\n\\hline\n\\ & M1 & M2 & M3 & M4 & M5 & M6 & M7 & M8 & M9 & M10 & M11 & M12 & M13 & M14 \\\\\n\\hline\nN & 0.26 & 0.34 & 0.22 & 0.13 & 0.37 & 0.38 & 0.72 & 0.26 & 0.13 & 0.45 & 0.65 & 0.38 & 0.53 & 0.00 \\\\\nC1 & 0.03 & 0.41 & 0.14 & 0.29 & 0.14 & 0.14 & 0.04 & 0.38 & 0.03 & 0.02 & 0.06 & 0.09 & 0.19 & 0.08 \\\\\nC2 & 0.18 & 0.00 & 0.19 & 0.25 & 0.06 & 0.04 & 0.11 & 0.05 & 0.32 & 0.18 & 0.03 & 0.28 & 0.00 & 0.30 \\\\\nC3 & 0.44 & 0.19 & 0.00 & 0.17 & 0.10 & 0.41 & 0.13 & 0.18 & 0.43 & 0.23 & 0.20 & 0.25 & 0.22 & 0.57 \\\\\nC4 & 0.10 & 0.05 & 0.44 & 0.15 & 0.33 & 0.05 & 0.00 & 0.14 & 0.10 & 0.12 & 0.07 & 0.00 & 0.06 & 0.04 \\\\\n\\hline\n \\end{tabular}\n }\n \\caption{Clonal fractions in the 14 mixtures of the validation experiment.}\n \\label{tab:mixtures}\n\\end{table*}\n\nThe large number of samples and the high depth of sequencing that we obtained afforded a sensitivity analysis, in which we varied the number of samples and the depth. Cloe{} was run on these datasets with the same parameters as for the synthetic data (Table~\\ref{tab:params}).\n\n\\paragraph{Model selection}\n\nWe ran Cloe{} for \\( K \\in \\left\\{ 3, 4, 5, 6 \\right\\} \\) and performed model selection based on the log-posterior values of the MAP estimates. In every case we were able to identify the correct number of clones (Figure~\\ref{fig:validation_k}), suggesting that either a moderate depth of sequencing or multiple samples should suffice in obtaining good estimates of the number of clones.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.75]{validation_model_selection_lp.pdf}\n \\caption{Inferred model size \\( K \\) for every combination of samples and depths in the validation dataset. The validation data consist of a mixture of five clones. Only the top solution was considered in each case.}\n \\label{fig:validation_k}\n\\end{figure}\n\n\n\\paragraph{Reconstruction fidelity}\n\nOverall, we obtained precise reconstructions for almost all depth-samples combinations. Considering for each combination only the first solution suggested by Cloe, on average 1\\% of mutation assignments were inaccurate (\\( Z_{err} \\) median 0, mean 0.013), and clonal fractions were inferred with an average error lower than 2\\% (\\( F_{err} \\) median 0.017, mean 0.019). As expected, we observed a pattern of decreasing errors as the data increase in the number of samples or in depth (Figure~\\ref{fig:validation_err5}).\n\n\\begin{figure}\n \\centering\n \\makebox[\\textwidth][c]{\n \\includegraphics[scale=0.6]{validation_norm_zf_error_k5.pdf}\n }\n \\caption{Reconstruction errors on validation data obtained running Cloe{} with \\( K = 5 \\) clones. The heatmaps show the genotypes error (left) and clonal fractions error (right) for various combinations of depth and samples.}\n \\label{fig:validation_err5}\n\\end{figure}\n\n\\paragraph{Specific low-depth cases}\nPoorer reconstructions were obtained at lower depths (\\( \\leq 60\\times \\)) for the datasets with three samples. In every case, the inferred genotypes showed a faulty separation between two expected genotypes (Supplementary Figure 6), which led to high error metrics: \\( Z_{err} \\leq 0.134 \\) and \\( F_{err} \\leq 0.065 \\). Despite the imprecise reconstruction, there is an overall good agreement with the observed data (Supplementary Figure 6 (e)).\n\nThese results could be improved by tuning the running parameters of Cloe{} for these datasets. Because the height of the posterior peaks at these levels of depth is lower than at high depth, using less tempered chains may result in higher acceptance of chain swaps, and, consequently, in a more complete exploration of the posterior space. Increasing the number of MCMCMC iterations could also prove beneficial.\n\nIt should be also noted that at low depths sampling noise may promote suboptimal parameter combinations to near-optimal. In this case, more mutations should be analysed in order to average sampling noise effects, though this may place a heavy burden on our implementation's runtime. Alternatively, one could model more data in terms of samples. If the clonal fractions are dynamic enough, meaning that most clones grow and shrink at some point in the samples, more opportunities are provided to separate clonal signals.\n\n\n\\subsection{Comparison to previous approaches}\n\\label{sec:benchmark}\n\nTo further benchmark Cloe, we compared the results of three published methods compatible with targeted sequencing on our validation dataset: BayClone \\cite{Sengupta2015} and Clomial \\cite{Zare2014}, two latent feature models, and PyClone \\cite{Roth2014}, a non-parametric model. Other methods, like PhyloWGS~\\cite{Deshwar2015} or CloneHD~\\cite{Fischer2014}, are not applicable to targeted sequencing.\n\nWe ran two tests on the validation dataset described in the last section, first using all samples and the entire depth, and then 3 samples and a depth of 100\\( \\times \\). Our method's performance on the first dataset is a perfect reconstruction of the genotypes (\\(Z_{err} = 0 \\)) and a near-perfect reconstruction of the clonal fractions (\\(F_{err} = 0.005 \\)), with a correct identification of the number of clones. With less data, there are three misassignment (\\( Z_{err} = 0.009 \\)) and the error of the clonal fractions is 0.017 (Figure~\\ref{fig:method_comparison}); again, the number of clones is correctly inferred.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.56]{method_comparison_zf_err_160404.pdf}\n \\caption{Comparison of Cloe{} against three published methods on our validation dataset, consisting of targeted sequencing for 82 mutations in 14 samples (average depth 17260\\( \\times \\)); five clones are present in the data. \\( Z_{err} \\) denotes the reconstruction error on genotypes; \\( F_{err} \\) the error on clonal fractions. The two datasets (17260\\( \\times \\) depth and 14 samples, and 100\\( \\times \\) and 3 samples) are denoted ``high'' and ``low'', respectively. The legend refers to the number of inferred clones.}\n \\label{fig:method_comparison}\n\\end{figure}\n\nBayClone \\cite{bayclone} was run with default parameters for 45000 iterations, discarding the first 5000 and thinning the chain by a factor of 4. We tested the same model sizes as for our own method, namely 3, 4, 5, and 6. Through the log-pseudo-marginal likelihood, BayClone was able to identify \\( K = 5 \\) as the best solution on the first dataset. However, the reconstruction of the genotypes was less precise (Figure~\\ref{fig:method_comparison_z}), with \\( Z_{err} = 0.25 \\) and \\( F_{err} = 0.103 \\). Here \\( Z_{err} \\), the normalised absolute difference between inferred and real genotypes matrices, ignores the ploidy of the mutation. The reconstruction was poorer on the second dataset because of the less precise data: six clones were inferred with \\( Z_{err} = 0.287 \\), and \\( F_{err} = 0.147 \\) (Supplementary Figure 7).\n\nClomial (version 1.6.0) implements an EM algorithm, and it was run with default parameters (1000 restarts, and 100 maximum EM iterations) using model sizes of 3, 4, 5, and 6. On the first dataset, model selection with BIC (and AIC) indicated \\( K = 5 \\) as the best solution, with one misassignment (\\( Z_{err} = 0.003 \\)) and an accurate reconstruction of the clonal fractions (\\( F_{err} = 0.006 \\)). With less data, model selection was unclear as AIC, BIC and the log-likelihoods were all discordant. Using the correct model size Clomial obtained a \\( Z_{err} = 0.076 \\) and \\( F_{err} = 0.044 \\).\n\nPyClone (version 0.12.9) was run for 30000 iterations with a beta-binomial density and copy-number neutral states allowing a single mutant allele out of two (AB mode). PyClone was also provided with estimates of cellularity for each of the samples. We removed the first 3000 iterations as burn-in samples and thinned the chain by a factor of 4. The output of PyClone consists of a clustering of the observed mutations, \nwhere each cluster should roughly correspond to one of the non-root nodes of Figure~\\ref{fig:data_cnn}. Phylogenetic modelling can translate these clusters into genotypes.\nOn the full dataset, PyClone produced three clusters, as shown in Figure~\\ref{fig:method_comparison_z}. Because the cluster of stem mutations was merged with one of its two children, we were unable to interpret the results phylogenetically. Hence, we could not derive genotypes nor clonal fractions. The estimate of \\( K \\) is 4 with \\( Z_{err} = 0.241 \\). On less data, two clusters were produced, leading to an estimate of \\( K \\) of 3, with \\( Z_{err} = 0.357 \\).\n\nIn summary, our benchmark shows that Cloe{} compares favourably against similar published methods (Figure~\\ref{fig:method_comparison_z}). It is expected that the accuracy of the reconstruction would be affected by the quality of the data. Indeed every model performed more poorly on less data, however Cloe{} seemed to be affected to a lesser extent (Supplementary Figure 7).\n\n\\begin{figure}\n \\centering\n \\makebox[\\textwidth][c]{\n \\includegraphics[scale=0.56]{method_comparison_z_D17260_T14_160404.pdf}\n }\n \\caption{Comparison of the genotypes inferred by the four benchmarked methods using all the data in our validation dataset. PyClone's reconstruction is a clustering of the mutations. In this representation of the genotypes we padded solutions with the normal clone (C1) for a more direct comparison with our method. The legend refers to the proportion of mutated alleles out of two.\n }\n \\label{fig:method_comparison_z}\n\\end{figure}\n\n\n\n\n\\section{Case studies}\n\nWe show the applicability of Cloe{} to clinical data in two case studies.\n\n\n\\subsection{Chronic lymphocytic leukaemia}\nThis dataset consists of five time points for each of three chronic lymphocytic leukaemia patients \\cite{Schuh2012}. The original study identified mutations by whole-genome sequencing (WGS; average depth across the mutation loci 39\\( \\times \\)) and quantified a subset of these with deep targeted sequencing (TAR; average depth 101600\\( \\times \\)). \n\nThe authors' analysis reported evolutionary trees and clonal fractions for each of the three cases. We used this information to run Cloe{} with known clonal fractions on all mutations, prioritising information from the higher depth datasets. We interpreted these results as ground truth mutation assignments for all three patients, and scored our reconstructions to these reference parameters.\n\nWe ran Cloe{} on the reported mutations with \\( K \\in \\left\\{ 3 \\dots 7 \\right\\} \\) for each case and each experiment (WGS, low-depth, and TAR, high-depth), comparing our results with the original study, with PhyloSub's results \\cite{Jiao2014}, and with CloneHD's reconstruction of case CLL003 \\cite{Fischer2014}. We also included another dataset, which consisted of the WGS dataset with data from the higher depth TAR dataset for mutations in common.\n\nCase CLL003 displays a radical clonal shift (Supplementary Figure 8 (a) and (b)): the main clone in the early time points is replaced by a distinct new clone that appears only at the second time point and expands to become the predominant clone.\nUsing targeted sequencing data, Cloe{} obtained a very accurate reconstruction, identifying the correct number of clones, obtaining a single misassignment and average errors on clonal fractions of 1\\% (Figure~\\ref{fig:cll_error}, Supplementary Figure 9). On less data, our method opted for a solution with 4 clones that ignored the founding clone, only present in the first of five samples at a clonal fraction of 3\\%. Choosing the top solution with 5 clones recovered the correct clonal structure; on WGS data there were five incorrect mutation assignments (Supplementary Figure 10), whereas with the combined dataset only one (Supplementary Figure 11). Barring the rare founding clone, the 4-clone reconstructions are correct with one (combined data) and two (WGS data) misassignments.\n\nThe remaining cases showed more stable dynamics (Supplementary Figure 8 (c)--(f)). For CLL006, Cloe{} assigned the nine mutations of the TAR dataset to six clones without errors; three errors were observed with 18 mutations in the WGS dataset (Figure~\\ref{fig:cll_error}). Analysis of the combination of the two data types yielded an additional clone, though similar log-posterior probabilities and a higher log-likelihood were obtained by a six-clone solution. Removing clone C5 from the seven-clone solution yields a correct reconstruction (Supplementary Figure 12).\n\nFinally, for CLL077, Cloe's analysis resulted in a perfect reconstruction of the genotypes with targeted sequencing data. Two misassignments were obtained for the combined dataset, whereas four of the five clones were identified in the WGS data: the founding clone, with only four of the 20 mutations, was merged with one of its children. After the four-clone solutions, solutions with six-clones had high log-posterior probabilities. Indeed the first of these solutions is an accurate reconstruction with two misassignments and one clone repeated twice almost identically. In the middle, solutions with the expected number of clones, five, had six errors (Supplementary Figure 13).\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.75]{CLL_error_combined.pdf}\n \\caption{Performance metrics of Cloe{} on the CLL datasets. The correct number of clones for cases CLL003 and CLL077 is 5, whereas for CLL006 it is 6. TAR stands for targeted sequencing (average depth 101600\\( \\times \\)); WGS stands for whole-genome sequencing (average depth 39\\( \\times \\); BOTH is the WGS dataset with TAR data for shared mutations. The legend refers to the number of inferred clones. When the first solution inferred the wrong number of clones, the top solution for the correct number of clones is also shown.}\n \\label{fig:cll_error}\n\\end{figure}\n\nOverall, Cloe{} produced accurate reconstructions of the latent parameters. Higher errors were observed when an incorrect number of clones was inferred. However, even in these cases, our phylogenetic model allowed us to obtain close approximations of the ground truth. \n\nAssuming that our reconstruction of the ground truth is correct, Cloe's inference results in a better reconstruction than reported by CloneHD \\cite{Fischer2014}, both using high-depth and low-depth data: first, because of our phylogenetic modelling, we were able to identify the founding clone; second, we could confidently identify the rising clone's parent (the ambiguous green clone in \\cite{Fischer2014}). With low-depth data, Cloe{} did prefer a model with four clones, but could also provide a more accurate five-clone solution.\n\nOur results on targeted sequencing data largely agree with those obtained by PhyloSub~\\cite{Jiao2014}, with two small exceptions. For CLL003, Cloe{} predicts that clone 4 (clone \\( c \\) in \\cite{Jiao2014}, Figure 7, right) does not harbour the \\textit{IL11RA} mutation. This episode appears to be supported by the data (Supplementary Figure 14) as Cloe's reconstruction leads to closer fit with the data (sum of absolute errors on the allele fractions is 0.06 for Cloe, 0.13 for PhyloSub, for this mutation). Rather than a loss of mutation, this could be due to convergent evolution at the leaf nodes, leading to a sum of absolute errors of 0.07. For case CLL006, our reconstruction agrees with that of \\cite{Schuh2012}: five tumour clones are detected, and the \\textit{EGFR} mutation is predicted to stem from the founding clone. PhyloSub preferred to place the EGFR mutation in an additional clone after the founder, leading to a closer fit: the sum of absolute errors was 0.02 compared to Cloe's 0.07 for this mutation.\n\n\n\n\n\\subsection{Acute myeloid leukaemia}\n\nAML31 refers to a patient with acute myeloid leukaemia, whose case was studied in great depth with several sequencing experiments targeting bulk DNA at various scales, RNA and also single cells \\cite{Griffith2015}. As each layer of data refined the authors' understanding of the evolution of this tumour, seven clusters and driver mutations were identified. Integration of all sequencing data revealed over 1300 mutations curated in a ``platinum list''. The tumour genomes appeared to be devoid of copy-number aberrations. \n\nWe considered a subset of platinum-list mutations for three datasets: ALLDNA (median depth of 1841\\( \\times \\) for the primary tumour sample, 388\\( \\times \\) for the relapse), TORRENT (median depths 41\\( \\times \\) and 46.5\\( \\times \\)), and WGS (median depths 323\\( \\times \\) and 41\\( \\times \\)). For each dataset, we selected a random subset of 250 mutations, halving the number of mutant reads for hemizygous mutations, and adding reported driver mutations.\n\nCloe{} was run with \\( K \\in \\left\\{ 3, \\dots, 7 \\right\\} \\) on the datasets. Model selection on ALLDNA indicated \\( K = 5 \\) as the preferred solution, followed closely by \\( K = 6 \\) which provided a closer fit to the data (Supplementary figure 15). The inferred mutation dynamics for both models are shown in Figure~\\ref{fig:alldna_fit}. Whereas both model sizes could capture the trends in the data, the solution for \\( K = 6 \\) correctly identified two groups of mutations that rise in allele fraction in the relapse sample. \n\nOur reconstruction shows a decrease in tumour burden at relapse, a single origin for all clones, and branched evolution after the founding clone (Figure~\\ref{fig:alldna_params}). Clone 5 and its child, clone 6, become the main clones in the relapse sample, supplanting clones 3 and 4. The founding clone appears present only at very low clonal fractions.\n\nMatching our clones to the original clusters, we found a close correspondence (Table~\\ref{tab:alldna_match}), corroborating Cloe's inference. The only misassignment is \\textit{TP53} to clone 6, which in the original study required single-cell sequencing and additional time points to identify as belonging to a separate clone. Beyond the genotypes, there was also a close match between inferred and expected clonal fractions, with a maximum absolute difference of 4\\%.\n\nCluster 6 was not identified by our model. According to the original analysis, this cluster was present at less than 5.5\\% clonal fraction in the primary sample, to then disappear at relapse. Such a cluster would contribute half of its clonal fraction in allele fraction, due to the heterozygosity of the mutations. We do observe that nine of the 257 mutations were not assigned to any clone. Their average allele fraction was 2.2\\% in the primary sample and close to 0\\% in the relapse (Supplementary Figure 16). Because they did not fit the dynamics of the other clones, sequencing noise was used to fit them (eq.~\\ref{eq:seq_noise}). \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{ALLDNA_fit.pdf}\n \\caption{Observed and inferred mutation dynamics for 257 mutations from the ALLDNA dataset. Left: observed allele fractions; centre: allele fractions inferred by Cloe{} with 5 clones; right: allele fractions inferred by Cloe{} with 6 clones.}\n \\label{fig:alldna_fit}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{ALLDNA_params6.pdf}\n \\caption{Parameters inferred by Cloe{} running with 6 clones on 257 mutations from the ALLDNA dataset. Genotypes are shown on the left, where green denotes presence of a mutation; clonal fractions for each clone are shown on the right. C1 is fixed as the normal contamination.}\n \\label{fig:alldna_params}\n\\end{figure}\n\n\\begin{table}\n \\centering\n \\begin{tabular}{ccl}\n\\hline\nClone & Cluster & Drivers \\\\\n\\hline\nC2 & 1 & \\textit{DNMT3A} \\\\\nC4 & 4 & \\textit{FOXP1} \\\\\nC5 & 3 & \\textit{IDH2} \\\\\nC3 & 2 & \\textit{IDH1} \\\\\nC6 & 5 & \\textit{CXCL17}, \\textit{TP53} \\\\\n\\hline\n \\end{tabular}\n \\caption{Correspondence between Cloe's inferred clones, and the clusters in the original analysis by \\cite{Griffith2015}. While drivers are also present in the children of a clone, here we report the clone in which the mutations first appeared.}\n \\label{tab:alldna_match}\n\\end{table}\n\nOn the modest amount of data of the TORRENT dataset our model selection produced a more conservative estimate of the number of clones, preferring four clones. Using five or more clones improved the log-likelihood to the same extent. We compare here solutions for \\( K = 4 \\) and \\( K = 5 \\) (Figure~\\ref{fig:torrent_fit}).\n\nWith three tumour clones, our model matched the main trends: two large clones in the primary sample that disappear at relapse, and one growing clone. In addition, tumour content was accurately inferred: 89\\% for the primary sample and 44\\% for the relapse sample, compared to the expected values of 91\\% and 47\\%. The addition of a fourth tumour clone (\\( K = 5 \\)) allows a better disambiguation of the clones present in the primary, while the spread of allele fractions in the relapse sample makes it difficult to distinguish two rising clone.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{TORRENT_fit.pdf}\n \\caption{Observed and inferred mutation dynamics for 254 mutations from the TORRENT dataset. Left: observed allele fractions; centre: allele fractions inferred by Cloe{} with 4 clones; right: allele fractions inferred by Cloe{} with 5 clones.}\n \\label{fig:torrent_fit}\n\\end{figure}\n\nIdentifying seven clones including the normal in two samples with a median depth less than 45\\( \\times \\) is an arduous task. \\cite{Griffith2015} show that SciClone detects four clones up to around 100\\( \\times \\) depth using all mutations on the platinum list. While Cloe{} prefers four clones using a subset of mutations at a depth of 45\\( \\times \\), it is capable of splitting the observed dynamics further, obtaining closer approximations of the real clonal structure.\n\nFinally, for the WGS dataset, Cloe's solution with 5 clones obtained the highest posterior probability, while 6 and 7 clones obtained closer fits to the data (Supplementary Figure 17). With four tumour clones, Cloe{} identified three decreasing groups of mutations and one group that arose at relapse. This matches the observed dynamics, as the low depth at relapse accounts for a larger spread of the allele fractions that confounds the identification of two rising clones (Supplementary figure 18). Interestingly, the addition of another clone, rather than fitting this low-depth relapse data, matches a fourth group of mutations present only in the primary around 5\\% clonal fraction. These mutations overlap with the unassigned mutations in the ALLDNA dataset and the inferred clone does not harbour additional driver mutations other than \\textit{DNMT3A}, which derives from its parent.\n\nWith this case study we applied Cloe{} to a scenario with two samples, highlighting the difficulties of automatic model selection, especially when trying to identify a large number of clones with a moderate amount of data.\n\nThe running parameters for the two case studies differ from the ones listed in Table~\\ref{tab:params} in that we used five chains with \\( \\Delta T = 0.25 \\). In addition, for the AML datasets we ran 50000 iterations of our sampler with \\( \\mu = 0.2 \\), \\( \\rho = 0.04 \\), and \\( \\varepsilon = 0.001 \\).\n\n\n\n\n\\section{Discussion}\n\n\nAs tumour sequencing data grow in depth and breadth, the question of tumour heterogeneity will continue to be focal. In this study we presented Cloe, a novel latent feature model for direct clonal reconstruction. Our model discovers genotypes in the data by assigning observed mutations to latent features (clones) guided by a latent phylogeny. This phylogenetic deconvolution sets Cloe{} apart from other direct reconstruction methods \\cite{Fischer2014, Zare2014, Sengupta2015}. Compared to indirect reconstruction methods, our algorithm can handle multiple primary tumours, the loss of mutations and convergent evolution. In particular, to our knowledge this is the first method to allow and penalise convergent evolution.\n\nOur study on simulated data showed a good performance of our MCMCMC algorithm. However, tuning the MCMCMC parameters in order to correctly explore the spiked posterior landscape is not trivial. We empirically found parameters that would allow the chains to mix well. Regions of high posterior probability are quickly reached, yet finding the right peak is a slow process, complicated by each biological constraint on the model.\nMany parameters can be tuned in our model. We sought values that would work well for both simulated data and our validation data. Tuning the MCMCMC parameters to each dataset independently, thus optimising the exploration of the posterior space, might further improve results.\n\nIn our definition of the tree we assume that multiple primary tumours are less likely to occur than tumours with a single origin. If our understanding of clonal evolution were to suggest otherwise, the definition of the tree may be simplified to a discrete uniform distribution, giving equal weight to a single origin or multiple ones.\n\n\n\\paragraph{Limitations}\nThe main limitation of our method is the restriction to mutations from copy-number neutral regions. Whereas this may be amenable to certain types of cancer (e.g. mutation-driven rather than copy-number driven cancers), it may preclude the analysis of more genomically rearranged tumours.\n\nIn contrast to some models described in the literature, our method does not include the number of clones as a parameter. Instead, Cloe{} must be run for various choices of \\( K \\), and the best solution in terms of posterior probability will indicate the number of clones with good accuracy. On our simulation and validation datasets our model was indeed able to identify the correct number of clones in 58\/59 cases.\n\nAs shown in the case studies, model selection may not be trivial. We thus recommend manual review of the inferred parameters for various model sizes to ensure that the results of the inference are robust.\n\nAnalysing hundreds of mutations can result in a high computational burden. This limitation could be alleviated by preprocessing the input data, grouping mutations that exhibit similar dynamics throughout the samples. One way to do this is via a Chinese Restaurant Process with a product of binomials; mutant read counts and depths for all mutations in a cluster could then be summed and analysed as a single unit.\n\n\\paragraph{Extensions}\nWe see several avenues for future extensions.\nAt the theoretical level, future work should focus on optimising the inference and extending this framework to arbitrary copy-numbers. Also, to address the model selection problem, the phylogenetic latent feature model could be rephrased in a non-parametric perspective.\nIn terms of applications, our model could also be applied to epigenetics: by appropriately changing the likelihood function, Cloe{} could deconvolute methylation data into evolutionarily related epigenotypes.\n\nIn summary, Cloe{} is a rigorous and flexible framework for clonal deconvolution of cancer genomes that achieves high accuracy in benchmarking studies and leads to important insights into tumour evolution in clinical case studies. \n\n\n\n\n\n\\section*{Acknowledgements}\n\nWe would like to acknowledge the support of The University of Cambridge, Cancer Research UK and Hutchison Whampoa Limited. \nThis work was funded by CRUK core grant C14303\/A17197, in particular A19274 (Markowetz lab core grant).\nWe wish to thank Marta Grzelak and James Hadfield for their assistance with sequencing, and Malvina Josephidou for discussions on the CRP clustering of mutations.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\label{intro} Power-laws are intrinsic features of the economic\nand financial data. There are numerous studies of power-law\nprobability distributions in various economic systems\n\\cite{1,2,3,4,41,42,8,5}. The key result in recent findings is\nthat the cumulative distributions of returns and trading activity\ncan be well described by a power-law asymptotic behavior,\ncharacterized by an exponent $\\lambda \\approx 3$, well outside the\nLevy stable regime $0<\\lambda <2$ \\cite {5}.\n\nThe time-correlations in the financial time series are studied\nextensively as well \\cite{5,6,7}. Gopikrishnan \\emph{et al} \\cite\n{5,6} provided empirical evidence that the long-range correlations\nfor volatility were due to the trading activity, measured by a\nnumber of transactions $N$.\n\n\nRecently we adapted the model of $1\/f$ noise based on the Brownian\nmotion of time interval between subsequent pulses, proposed in\n\\cite{9,10,11,12}, to model the share volume traded in the\nfinancial markets \\cite{13}. The idea to transfer long time\ncorrelations into the stochastic process of the time interval\nbetween trades or time series of trading activity is in\nconsistence with the detailed studies of the empirical financial\ndata \\cite{5,6} and fruitfully reproduces the spectral properties\nof the financial time series \\cite{13,131}. Further, we\ngeneralized the model defining stochastic multiplicative point\nprocess to reproduce a variety of self-affine time series\nexhibiting the power spectral density $S(f)$ scaling as a power of\nfrequency, $S(f) \\propto f^{-\\beta}$ \\cite{132}.\n\nIn this contribution we analyze the applicability of the\nstochastic multiplicative point process as a model of trading\nactivity in the financial markets. We investigate the spectral density\nand counting statistics of model trading activity in comparison\nwith empirical data from the stock exchange. The model reproduces the\nspectral properties of trading activity and explains the mechanism\nof power law distribution in the real markets.\n\n\\section{The model}\n\nWe consider a point process $I(t)$ as a sequence of the $\\delta$-type\nrandom correlated pulses,\n\\begin{equation}\n\\label{eq:1}I(t)=\\sum\\limits_ka_k\\delta (t-t_k),\n\\end{equation}\nand define the number of trades $N_j$ in the time intervals\n$\\tau_d$ as an integral of the signal, $N_j=\\int\\limits_{t_j}^{t_j+\\tau\n_d}I(t)dt$. Here $a_k$ is a contribution of one transaction.\nWhen $a_k=1$, the signal (\\ref{eq:1}) counts the\ntransactions in the financial market. When $a_k$ describes asset price\nchange during one transaction, the signal counts the price changes.\nWhen $a_k=\\bar a$ is a constant, the point process is\ncompletely described by the set of times of the events $\\{t_k\\}$\nor equivalently by the set of interevent intervals $\\{\\tau\n_k=t_{k+1}-t_k\\}$. Various stochastic models for $\\tau _k$ can be\nintroduced to define a stochastic point process. In papers\n\\cite{9,10,11,12} it has been shown analytically that the\nrelatively slow Brownian fluctuations of the interevent time $\\tau\n_k$ yield $1\/f$ fluctuations of the signal (\\ref{eq:1}). In\n\\cite{132} we have generalized the model introducing stochastic\nmultiplicative process for the interevent time $\\tau _k$,\n\\begin{equation}\n\\label{eq:5}\\tau _{k+1}=\\tau _k+\\gamma \\tau _k^{2\\mu -1}+\\tau\n_k^\\mu \\sigma \\varepsilon _k.\n\\end{equation}\nHere the interevent time $\\tau _k$ fluctuates due to the external\nrandom perturbation by a sequence of uncorrelated normally\ndistributed random variable $\\{\\varepsilon _k\\}$ with zero\nexpectation and unit variance, $\\sigma $ denotes the standard\ndeviation of the white noise and $\\gamma \\ll 1$ is a damping\nconstant. From the big variety of possible stochastic processes we\nhave chosen the multiplicative one, which yields multifractal\nintermittency and power-law probability distribution functions.\nPure multiplicativity corresponds to $\\mu=1$. Other values\nof $\\mu$ may be considered, as well.\n\nThe iterative relation (\\ref {eq:5}) can be rewritten as Langevine\nstochastic differential equation in $k$-space\n\\begin{equation}\n\\label{eq:6}\\frac{d\\tau _k}{dk}=\\gamma \\tau _k^{2\\mu -1}+\\tau\n_k^\\mu \\sigma \\xi \\left( k\\right).\n\\end{equation}\nHere we interpret $k$ as continuous variable while $\\left\\langle\n\\xi \\left( k\\right) \\xi \\left( k^{\\prime }\\right) \\right\\rangle\n=\\delta (k-k^{\\prime })$.\n\nThe steady state solution of the stationary Fokker-Planck\nequation with zero flow, corresponding to (\\ref {eq:6}), gives the\nprobability density function for $\\tau _k$ in the $k$-space (see,\ne.g., \\cite{19})\n\\begin{equation}\n\\label{eq:7}P_k(\\tau\n_k)=C\\tau_k^\\alpha=\\frac{\\alpha+1}{\\tau_{max}^{(\\alpha+1)}-\\tau_{min}^{(\\alpha+1)}}\\tau\n_k^\\alpha ,\\quad \\alpha =2\\gamma \/\\sigma ^2-2\\mu.\n\\end{equation}\n\nThe steady state solution (\\ref{eq:7}) assumes Ito convention\ninvolved in the relation between expressions (\\ref{eq:5}),\n(\\ref{eq:6}) and (\\ref{eq:7}) and the restriction for the diffusion\n$0<\\tau_{\\min }<\\tau_k<\\tau _{\\max }$.\n\nWe have already derived the formula for the power spectral density of the\nmultiplicative stochastic point process model, defined by\nEqs.~(\\ref {eq:5}) and (\\ref{eq:6}) for the interevent time\n\\cite{132},\n\\begin{equation}\nS_{\\mu }(f)=\\frac{2C\\overline{a}^{2}}{\\sqrt{\\pi }\\overline{\\tau\n}(3-2\\mu )f}\\left(\\frac{\\gamma }{\\pi f}\\right)^{\\frac{\\alpha\n}{3-2\\mu }}{\\mathop{\\mathrm{Re}}}\\int\\limits_{x_{\\min }}^{x_{\\max\n}}\\exp \\left\\{-i\\bigg(x-\\frac \\pi\n4\\bigg)\\right\\}{\\mathop{\\mathrm{erfc}}} (\\sqrt{-ix})x^{\\frac\\alpha\n{3-2\\mu }-\\frac 12}dx \\label{eq:12}\n\\end{equation}\nwhere $\\bar \\tau =\\left\\langle \\tau _k\\right\\rangle =T\/(k_{\\max\n}-k_{\\min })$ is the expectation of $\\tau _k$. Here we introduce\nthe scaled variable $x=\\pi f \\tau ^{3-2\\mu }\/\\gamma$ and\n$x_{\\min }=\\pi f \\tau _{\\min }^{3-2\\mu }\/\\gamma,\\quad x_{\\max }=\n\\pi f \\tau _{\\max }^{3-2\\mu }\/\\gamma$.\n\nExpression (\\ref{eq:12}) is appropriate for the numerical calculations\nof the power spectral density of the generalized\nmultiplicative point process defined by Eqs. (\\ref{eq:1}) and\n(\\ref{eq:5}). In the limit $\\tau _{\\min }\\rightarrow 0$ and $\\tau\n_{\\max }\\rightarrow \\infty $ we obtain an explicit expression\n\\begin{equation}\n\\label{eq:14}S_\\mu (f)=\\frac{C\\overline{a}^2}{\\sqrt{\\pi\n}\\overline{\\tau } (3-2\\mu )f}\\left(\\frac \\gamma {\\pi\nf}\\right)^{\\frac \\alpha {3-2\\mu }}\\frac{\\Gamma (\\frac 12+\\frac\n\\alpha {3-2\\mu })}{\\cos (\\frac{\\pi \\alpha }{2(3-2\\mu )})}.\n\\end{equation}\n\nEquation (\\ref{eq:14}) reveals that the multiplicative point\nprocess (\\ref{eq:5}) results in the power spectral density\n$S(f)\\sim 1\/{f^\\beta }$ with the scaling exponent\n\\begin{equation}\n\\label{eq:15}\\beta =1+\\frac{2\\gamma \/\\sigma ^2-2\\mu }{3-2\\mu }.\n\\end{equation}\n\nLet us compare our analytical results (\\ref{eq:12}) and\n(\\ref{eq:14}) with the numerical calculations of the power\nspectral density according to equations (\\ref{eq:1}) and\n(\\ref{eq:5}). In Fig.~\\ref{fig:1} we present the numerically\ncalculated power spectral density $S(f)$ of the signal $I(t)$ for\n$\\mu =0.5$ and $\\alpha =2\\gamma\/\\sigma ^2-1=0$, -0.5 and +0.5.\nNumerical results confirm that the multiplicative point process\nexhibits the power spectral density scaled as $S(f)\\sim 1\/f^\\beta\n$. Equation (\\ref{eq:12}) describes the model power spectral\ndensity very well in a wide range of parameters. The explicit\nformula (\\ref{eq:14}) gives a good approximation of power spectral\ndensity for the parameters when $\\beta \\simeq 1$. These results\nconfirm the earlier finding \\cite {9,10,11,12} that the power\nspectral density is related to the probability distribution of the\ninterevent time $\\tau _k$ and $1\/f$ noise occurs when this\ndistribution is flat, i.e., when $\\alpha =0$.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Fig1.eps}\n\\end{center}\n\\caption{Power spectral density $S(f)$ vs frequency $f$ calculated\nnumerically according to Eqs. (\\ref{eq:1}) and (\\ref{eq:5}) with the parameters\n$\\mu=0.5$, $\\sigma=0.02$ and different relaxations of the signal\n$\\gamma$. We restrict the diffusion of the interevent time in the\ninterval $\\tau _{min}=10^{-6}$; $\\tau _{max}=1$ with the\nreflective boundary condition at $\\tau _{min}$ and transition to\nthe white noise, $\\tau _{k+1}=\\tau _{max}+\\sigma\\varepsilon_k$,\nfor $\\tau_k>\\tau _{max}$. The straight lines represent the results given\nby the explicit formula (\\ref{eq:14}).}\n\\label{fig:1}\n\\end{figure}\n\n\nIt is likely that such a stochastic model with parameters in the\nregion $0.5\\leq \\beta \\leq 1.5$ may be adaptable for a wide\nvariety of different systems. In this paper we will discuss\napplicability of the model for the financial market.\n\nWe derived pdf of $N$ for the pure multiplicative model with $\\mu\n=1$ in \\cite{132}\n\n\\begin{equation}\nP(N)=\\frac{C'\\tau_{d}^{2+\\alpha}(1+\\gamma N)}{N^{3+\\alpha}\n(1+\\frac{\\gamma}{2}N)^{3+\\alpha}}\\sim\\left\\{\n\\begin{array}{ll}\n\\frac{1}{N^{3+\\alpha}},& N\\ll \\gamma^{-1}, \\\\\n\\frac{1}{N^{5+2\\alpha}},& N\\gg \\gamma^{-1}.\n\\end{array}\n\\right.\n\\label{eq:18}\n\\end{equation}\n\n\nProbability distribution function for $N$ obtained from the\nnumerical simulation of the model is in a good agreement with the\nanalytical result (\\ref{eq:18}).\n\n\\section{Discussion and conclusions}\n\nWe have introduced a multiplicative stochastic model for the time\nintervals between events of point process. Such a model of time\nseries has only a few parameters defining the statistical\nproperties of the system, i.e., the power-law behavior of the\ndistribution function and the scaled power spectral density of the\nsignal. The ability of the model to simulate $1\/f$ noise as well\nas to reproduce signals with the values of power spectral density\nslope $\\beta$ between 0.5 and 1.5 promises a wide variety of\napplications of the model.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{Fig2.eps}\n\\end{center}\n\\caption{Power spectral density $S(f)$ vs frequency $f$ calculated\nnumerically from the empirical data of 3 most liquid stocks from the\nLithuanian Stock Exchange. Straight line fits $S(f)\\propto f^{-0.7}$.}\n\\label{fig:2}\n\\end{figure}\n\nLet us present shortly the possible interpretations of the\nempirical data of the trading activity in the financial markets.\nWith a very natural assumption of transactions in the financial\nmarkets as point events we can model the number of transactions\n$N_j$ in equal time intervals $\\tau _d$ as the outcome of the\ndescribed multiplicative point process. We already know from\navailable studies \\cite{5} that the empirical data exhibit power\nspectral density in the low frequency limit with the slope $\\beta\n\\simeq 0.7$. Empirical data from the Lithuanian Stock Exchange for\nthe most liquid assets confirm the same value $\\beta \\simeq 0.7$,\n(see Fig~\\ref{fig:2}). For the pure multiplicative model with\n$\\mu =1$ this results in $\\alpha =2\\gamma \/\\sigma ^2-2\\mu\n\\simeq -0.3$. The corresponding cumulative distribution of $N$ in\nthe tail of high values (see equation (\\ref{eq:18})) has the\nexponent $\\lambda=4+2\\alpha =3.4$. This is in an excellent\nagreement with the empirical cumulative distribution exponent 3.4\ndefined in \\cite{5} for 1000 stocks of the three major US stock\nmarkets.\n\nThe numerical results confirm that the multiplicative stochastic\nmodel of the time interval between trades in the financial market\nis able to reproduce the main statistical properties of trading\nactivity $N$ and its power spectral density. The power-law\nexponents of the pdf of the interevent time, $\\alpha $, and the\ncumulative distribution of the trading activity, $\\lambda $, as\nwell as the slope of power spectral density, $\\beta$, are defined\njust by one parameter of the model $2\\gamma \/\\sigma ^2$. The model\nsuggests a simple mechanism of the power-law statistics of trading\nactivity in the financial markets.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzaszh b/data_all_eng_slimpj/shuffled/split2/finalzzaszh new file mode 100644 index 0000000000000000000000000000000000000000..c4efd790b2afd1b547a713e4d430d78901c40a76 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzaszh @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nSince two-dimensional models with a continuous global symmetry\nwere recognized to be asymptotically free \\cite{free}\nthey became a famous laboratory for testing many ideas\nand methods before applying them to more complicated gauge theories.\nIn this paper we follow this common way and present a method,\ndifferent from the conventional perturbation theory (PT), \nwhich allows to investigate\nthese models in the weak coupling region.\nThe conventional PT is one of the main technical\ntools of the modern physics. In spite of the belief of the most\nof community that this method gives the correct asymptotic expansion\nof such theories as $4D$ QCD or $2D$ spin systems with continuous global\nsymmetry, the recent discussion of this problem\n\\cite{superinst}-\\cite{QA} has shown that it is rather far from\nunambiguous solution. \nIndeed, for the PT to be applicable it is necessary\nthat a system under consideration would possess a well ordered\nground state. In two dimensional models, like the $O(N)$-sigma models\nthe MW theorem guarantees the absence of such a state in \nthe thermodynamic limit (TL) however small coupling constant is \\cite{MW}.\nThen, the usual argument in support of the PT is that locally\nthe system is ordered and the PT is not supposed to be used for the calculation\nof long-distance observables. On the other hand it should reproduce the correct\nbehaviour of the fixed-distance correlations as well as\nall thermodynamical functions which can be expressed via\nshort-range observables. The example of $1D$ models shows that\neven this is not always true \\cite{rossi}, \nso why should one believe in correctness of the conventional PT in $2D$?\nIn fact, the only way to justify the PT is to\nprove that it gives the correct asymptotic expansion of \nnonperturbatively defined models in the TL.\nNow, it was shown in \\cite{superinst}\nthat PT results in $2D$ nonabelian models depend on the boundary\nconditions (BC) used to reach the TL.\nThis result could potentially imply that the low-temperature limit and\nthe TL do not commute in nonabelian models. Actually, the main \nargumentation of \\cite{superinst} regarding the failure\nof the PT expansion is based\non the fact that the conventional PT is an expansion around\na broken vacuum, i.e. the state which simply does not exist in\nthe TL of $2D$ models. According to \\cite{superinst},\nthe ground state of these systems can be described\nthrough special configurations -- the so-called gas of\nsuperinstantons (SI) and the correct expansion should take into account\nthese saddle points. \nAt the present stage it is rather unclear how one could\nconstruct an expansion in the SI background. Fortunately, there exists\nother, more eligible way to construct the low-temperature expansion\nwhich respects the MW theorem and is apriori not the expansion around \nthe broken vacuum. We develop this method on an example of\nthe $2D$ $SU(N)\\times SU(N)$ principal chiral model\nwhose partition function (PF) is given by\n\\begin{equation}\nZ = \\int\\prod_xDU_x\n\\exp \\left [\\beta \\sum_{x,n} {\\mbox {Re Tr}} U_xU_{x+n}^+\\right] ,\n\\label{pfsun}\n\\end{equation}\n\\noindent\nwhere $U_x\\in SU(N)$, $DU_x$ is the invariant measure and we\nimpose the periodic BC. The basic idea is the following.\nAs was rigorously proven, the conventional PT \ngives an asymptotic expansion which is uniform\nin the volume for the abelian $XY$ model \\cite{XYPT}. \nOne of the basic theorems which underlies the proof states \nthat the following inequality holds in the $3D$ $XY$ model\n\\begin{equation}\n< \\exp (\\sqrt{\\beta}A(\\phi_x)) > \\ \\leq \\ C \\ ,\n\\label{ineq3D}\n\\end{equation}\n\\noindent\nwhere $C$ is $\\beta$-independent and $A(\\phi +2\\pi)=\\phi$. $\\phi_x$\nis an angle parametrizing the action of the $XY$ model\n$S=\\sum_{x,n}\\cos (\\phi_x - \\phi_{x+n})$. It follows that at large\n$\\beta$ the Gibbs measure is concentrated around $\\phi_x\\approx 0$ \nproviding a possibility to construct an expansion around $\\phi_x=0$.\nThis inequality is not true in $2D$ in the thermodynamic limit\nbecause of the MW theorem, however the authors of \\cite{XYPT} prove \nthe same inequality for the link angle, i.e.\n\\begin{equation}\n< \\exp (\\sqrt{\\beta}A(\\phi_l)) > \\ \\leq \\ C \\ , \\ \n\\phi_l=\\phi_x-\\phi_{x+n} \\ ,\n\\label{ineq2D}\n\\end{equation}\n\\noindent\nwhere the expectation value refers to infinite volume limit.\nThus, in $2D$ the Gibbs measure at large $\\beta$ is concentrated\naround $\\phi_l\\approx 0$ and the asymptotic series can be\nconstructed expanding the action in powers of $\\phi_l$. In the abelian\ncase such an expansion is equivalent to the expansion around $\\phi_x=0$\nbecause i) the action depends only on the difference $\\phi_x - \\phi_{x+n}$\nand ii) the integration measure is flat, $DU_x=d\\phi_x$. \n\nIn $2D$ nonabelian models, again because of the MW theorem, one has to expect \nin the TL something alike to (\\ref{ineq2D}), namely\n\\begin{equation}\n< \\exp (\\sqrt{\\beta}{\\mbox {arg}}A({\\mbox {Tr}}V_l)) >\n\\ \\leq \\ C \\ , \\ V_l=U_xU_{x+n}^+ \\ .\n\\label{nabineq2D}\n\\end{equation}\n\\noindent\nDespite there is not a rigorous proof of (\\ref{nabineq2D}), \nthat such (or similar) inequality holds in $2D$ nonabelian models \nis intuitively clear and should follow from the chessboard \nestimates \\cite{chest} and from the Dobrushin-Shlosman proof of the MW theorem\n\\cite{MW} which shows that spin configurations are distributed uniformly\nin the group space in the TL. Namely, the probability $p(\\xi )$ \nthat ${\\mbox {Tr}}(V_l-I)\\leq -\\xi$ is bounded by \n\\begin{equation}\np(\\xi ) \\leq O(e^{-b\\beta\\xi}) \\ , \\ \\beta\\to\\infty \\ ,\n\\label{chestlink}\n\\end{equation}\n\\noindent\nif the volume is sufficiently large, $b$ is a constant. \nThus, until $\\xi\\leq O((\\sqrt{\\beta})^{-1})$ is not satisfied,\nall configurations are exponentially suppressed.\nThis is equivalent to the statement that the Gibbs \nmeasure at large $\\beta$ is concentrated around $V_l\\approx I$, therefore\n(\\ref{nabineq2D}) or its analog holds. \nIn what follows it is assumed that (\\ref{nabineq2D}) is correct, \nhence $V_l=I$ is the only saddle point for the invariant \nintegrals\\footnote{It follows already from (\\ref{chestlink}).\nWhat is important in (\\ref{nabineq2D}) is a factor $\\sqrt{\\beta}$,\notherwise the very possibility of the expansion in $1\/\\beta$\nbecomes problematic.}. \nThus, the correct asymptotic expansion, if exists,\nshould be given via an expansion around $V_l=I$, similarly to the\nabelian model. If the conventional PT gives the correct asymptotics, it\nmust reproduce the series obtained expanding around $V_l=I$.\nHowever, neither i) nor ii) holds in the nonabelian models, therefore\nit is far from obvious that two expansions indeed coincide.\nLet us parametrize $V_l=\\exp (i\\omega_l)$ and $U_x=\\exp (i\\omega_x)$.\nConsider the following expansion\n\\begin{equation}\nV_l = \\exp \\left [i\\omega_l \\right ] \\sim \nI + \\sum_{n=1}\\frac{1}{(\\beta)^{n\/2}}\\frac{(i\\omega_l)^n}{n!} \\ .\n\\label{ltexp}\n\\end{equation}\n\\noindent\nThe standard PT states that to calculate the asymptotic expansion\none has to re-expand this series as\n\\begin{equation}\n\\omega_l = \\omega_x-\\omega_{x+n}+ \\sum_{k=1}\n\\frac{1}{(\\beta)^{k\/2}}\\omega_l^{(k)} \\ ,\n\\label{ltosres}\n\\end{equation}\n\\noindent\nwhere $\\omega_l^{(k)}$ are to be calculated from the definition\n$U_xU_{x+n}^+=\\exp (i\\omega_l)$.\nThis is presumably true in a finite volume where one can fix\nappropriate BC (like the Dirichlet ones), or to break down \nthe global symmetry by fixing the global gauge \n(on the periodic lattice). Then, making $\\beta$ sufficiently large\none forces all the spin matrices to fluctuate around $U_x\\approx I$,\ntherefore the substitution (\\ref{ltosres}) is justified.\nWe do not see how this procedure could be justified when\nthe volume increases and fluctuations of $U_x$ spread up\nover the whole group space. In other words, it is not clear why\nin the series\n$$\n\\omega_l = \\omega_x-\\omega_{x+n}+ \\sum_{k=1}\\omega_l^{(k)} \\ \n$$\nthe term $\\omega_l^{(k+1)}$ is suppressed as $\\beta^{-1\/2}$\nrelatively to the term $\\omega_l^{(k)}$.\nIt is only (\\ref{ltexp}) which remains correct\nin the large volume limit and takes into account \nall the fluctuations contributing at a given order of \nthe low-temperature expansion.\n\nIt is a purpose of the present paper to develop\nan expansion around $V_l=I$ aiming to calculate the asymptotic\nseries for nonabelian models.\nFirst of all, one has to give a precise mathematical meaning to\nthe expansion (\\ref{ltexp}). It is done in the next section. \n\n\\section{Link representation for the partition and correlation functions}\n\nTo construct an expansion of the Gibbs measure \nand the correlation functions using (\\ref{ltexp}) \nwe use the so-called link representation for the partition and \ncorrelation functions. First, we make a change of variables\n$V_l=U_xU_{x+n}^+$ in (\\ref{pfsun}). PF becomes\n\\begin{equation}\nZ = \\int \\prod_l dV_l\n\\exp \\left[ \\beta \\sum_l {\\mbox {Re Tr}} V_l + \\ln J(V) \\right] \\ ,\n\\label{lPF}\n\\end{equation}\n\\noindent\nwhere the Jacobian $J(V)$ is given by \\cite{linkrepr}\n\\begin{equation}\nJ(V) = \\int \\prod_xdU_x\\prod_l\n\\left[ \\sum_r d_r \\chi_r \\left( V^+_lU_xU^+_{x+n} \\right) \\right]\n= \\prod_p \\left[ \\sum_r d_r \\chi_r \\left( \\prod_{l\\in p}V_l \\right) \\right].\n\\label{jacob}\n\\end{equation}\n\\noindent\n$\\prod_p$ is a product over all plaquettes of $2D$ lattice,\nthe sum over $r$ is sum over all representations of $SU(N)$, \n$d_r=\\chi_r(I)$ is the dimension of $r$-th representation. \nThe $SU(N)$ character $\\chi_r$ depends on a product of the link \nmatrices $V_l$ along a closed path (plaquette in our case):\n\\begin{equation}\n\\prod_{l\\in p}V_l = V_n(x)V_m(x+n)V_n^+(x+m)V_m^+(x) \\ .\n\\label{prod}\n\\end{equation}\n\\noindent\nThe expression $\\sum_r d_r \\chi _r(\\prod_{l\\in p}V_l)$ is the\n$SU(N)$ delta-function which reflects the fact that the product of\n$U_xU^+_{x+n}$ around plaquette equals $I$ (original model has\n$L^2$ degrees of freedom, $L^2$ is a number of sites; since a number\nof links on the $2D$ periodic lattice is $2L^2$, the Jacobian must\ngenerate $L^2$ constraints)\\footnote{Strictly speaking, on the periodic\nlattice one has to constraint two holonomy operators, i.e. closed\npaths winding around the whole lattice. We do not expect such global \nconstraints to influence the TL in $2D$ (see Discussion).}. \nThe solution of the constraint\n\\begin{equation}\n\\prod_{l\\in p}V_l = I\n\\label{constr}\n\\end{equation}\n\\noindent\nis a pure gauge $V_l=U_xU_{x+n}^+$, so that two forms of the PF\nare exactly equivalent.\n\nThe corresponding representation for the abelian $XY$ model reads\n\\begin{equation}\nZ_{XY} = \\int\\prod_ld\\phi_l\n\\exp \\left [\\beta \\sum_l\\cos\\phi_l \\right ]\\prod_pJ_p \\ ,\n\\label{PFLxy}\n\\end{equation}\n\\noindent\nwhere the Jacobian is given by the periodic delta-function\n\\begin{equation}\nJ_p = \\sum_{r=-\\infty}^{\\infty} e^{ir\\phi_p} \\ , \\\n\\phi_p=\\phi_n(x)+\\phi_m(x+n)-\\phi_n(x+m)-\\phi_m(x+n) \\ .\n\\label{PFLxyJ}\n\\end{equation}\n\\noindent\n\nConsider two-point correlation function\n\\begin{equation}\n\\Gamma (x,y) = < {\\mbox {Tr}} \\ U_xU_y^+ > \\ ,\n\\label{corf}\n\\end{equation}\n\\noindent\nwhere the expectation value refers to the ensemble defined in (\\ref{pfsun}). \nLet $C_{xy}$ be some path connecting points $x$ and $y$. Inserting \nthe unity $U_zU_z^+$ in every site $z\\in C_{xy}$ one gets\n\\begin{equation}\n\\Gamma (x,y) = < {\\mbox {Tr}} \\prod_{l\\in C_{xy}} (U_xU_{x+n}^+) > =\n< {\\mbox {Tr}} \\prod_{l\\in C_{xy}} W_l > \\ ,\n\\label{corf1}\n\\end{equation}\n\\noindent\nwhere $W_l = V_l$ if along the path $C_{xy}$ the link $l$ goes in\nthe positive direction and $W_l = V_l^+$, otherwise. \nThe expectation value in (\\ref{corf1}) refers now to the ensemble\ndefined in (\\ref{lPF}). Obviously, it does not depend on the path $C_{xy}$\nwhich can be deformed for example to the shortest path between sites $x$ \nand $y$.\n\nIn this representation the series (\\ref{ltexp}) acquires a well\ndefined meaning, therefore the expansion of the action,\nof the invariant measure, etc. can be done.\n\n\\section{$XY$ model: Weak coupling expansion of the free energy}\n\n\nIn this section we prove that for the abelian XY model \nthe large-$\\beta$ expansion in the link\nrepresentation gives the same results as the conventional PT \nin the thermodynamic limit. We consider only the free energy but \nthe generalization for the correlation functions is straightforward.\n\nThe first step is a standard one, i.e. we rescale \n$\\phi\\to\\frac{\\phi}{\\sqrt{\\beta}}$ and make an expansion \n\\begin{equation}\n\\exp \\left[ \\beta\\cos\\frac{\\phi}{\\sqrt{\\beta}} \\right] = \\exp \n\\left[ \\beta -\\frac{1}{2}(\\phi)^2 \\right] \n\\left[ 1 + \\sum_{k=1}^{\\infty} (\\beta)^{-k} \n\\sum_{l_1,..,l_k}\\frac{a_1^{l_1}...a_k^{l_k}}{l_1!...l_k!} \\right] \\ ,\n\\label{cos}\n\\end{equation}\n\\noindent \nwhere $l_1+2l_2+...+kl_k=k$ and\n\\begin{equation}\na_k = (-1)^{k+1}\\frac{\\phi^{2(k+1)}}{(2k+2)!} \\ .\n\\label{a_k}\n\\end{equation}\n\\noindent\nIn addition to this perturbation one has to extend the integration region\nto infinity. We do not treat this second perturbation, as usually supposing \nthat all the corrections from this perturbation go down exponentially with \n$\\beta$ (in the abelian case it can be proven rigorously \\cite{XYPT}). \nIt is more convenient now to go to a dual lattice identifying \nplaquettes of the original lattice with its center, i.e. $p\\to x$.\nLet $l=(x;n)$ be a link on the dual lattice.\nIntroducing sources $h_l$ for the link field, one then finds \n\\begin{equation}\nZ_{XY}(\\beta >> 1) = e^{\\beta DL^2 - L^2\\ln \\beta}\n\\prod_{l} \\left[ 1+\\sum_{k=1}^{\\infty}\\frac{1}{\\beta^k}\nA_k\\left( \\frac{\\partial^2}{\\partial h_l^2} \\right) \\right] M(h_l) \\ .\n\\label{asxyGll}\n\\end{equation}\n\\noindent\nCoefficients $A_k$ are defined in (\\ref{cos}) and (\\ref{a_k}).\nThe generating functional $M(h_l)$ is given by\n\\begin{eqnarray}\nM(h_l) = \\sum_{r_x=-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\n\\prod_ld\\phi_l \\exp \n\\left[-\\frac{1}{2}\\sum_l \\phi_l^2 \n+ i\\sum_l \\frac{\\phi_l}{\\sqrt{\\beta}} (r_x-r_{x+n}) \n+\\sum_l\\phi_l h_l \\right] .\n\\label{Gfunxy}\n\\end{eqnarray}\n\\noindent\nSums over representations are treated by the Poisson resummation formula.\nThe Gaussian ensemble appears in terms of the fluctuations of $r$-fields.\nThe integral over zero mode of $r$-field is not Gaussian and leads to \na delta-function in the Poisson formula. Thus, the zero mode \ndecouples from the expansion.\nAll the corrections to the integrals over representations in the\nPoisson formula fall down exponentially with $\\beta$ so that \nthe generating functional becomes\n\\begin{equation}\nM(h_l) = \\exp \n\\left[ \\frac{1}{4} \\sum_{l,l^{\\prime}}h_lG_{ll^{\\prime}}h_{l^{\\prime}} \\right] \\ ,\n\\label{Gfunxy1}\n\\end{equation}\n\\noindent\nwhere we have introduced the following function which we term \n``link'' Green function \n\\begin{equation}\nG_{ll^{\\prime}} = 2\\delta_{l,l^{\\prime}} - G_{x,x^{\\prime}} -\nG_{x+n,x^{\\prime}+n^{\\prime}} + G_{x,x^{\\prime}+n^{\\prime}} + G_{x+n,x^{\\prime}} \\ .\n\\label{Gll1}\n\\end{equation}\n\\noindent\n$G_{x,x^{\\prime}}$ is a ``standard'' Green function on the periodic lattice\n\\begin{equation}\nG_{x,x^{\\prime}} = \\frac{1}{L^2} \\sum_{k_n=0}^{L-1}\n\\frac{e^{\\frac{2\\pi i}{L}k_n(x-x^{\\prime})_n}}\n{D-\\sum_{n=1}^D\\cos \\frac{2\\pi}{L}k_n} \\ , \\ k_n^2\\ne 0 \\ .\n\\label{Gxx}\n\\end{equation}\n\\noindent\nNormalization is such that $G_{ll}=1$.\nAs far as we could check, the expression (\\ref{asxyGll}) reproduces \nthe well known asymptotics of the free energy of the $XY$ model \nin two dimensions. For example, the first order coefficient of the \nfree energy being expressed in $G_{ll^{\\prime}}$ reads\n\\begin{equation}\nC_1 = \\frac{1}{64 L^2} \\sum_{l} G^2_{ll} = \\frac{1}{32} \\ .\n\\label{C1Gll}\n\\end{equation}\n\\noindent\nLet us add some comments. In the standard expansion to avoid the zero mode problem\none has to fix appropriate BC, like Dirichlet ones or to fix a global gauge\nif one works on the periodic lattice \\cite{unitgauge}. In the present scheme\nthe zero mode decouples automatically due to using $U(1)$ delta-function\nwhich takes into account the periodicity of the integrand in link angles.\nMore important observation is that it is allowed\nto take the TL already in the formula (\\ref{asxyGll}):\nsince the generating functional\ndepends only on the link Green function which is infrared finite, \nthe uniformity of the expansion in the volume follows immediately. \nThis is a direct consequence of the fact that the Gibbs measure of \nthe $XY$ model is a function of the gradient $\\phi_l$ only. \n\n\n\\section{Weak coupling expansion in the $SU(2)$ model}\n\nWe turn now to the nonabelian models. As the simplest example we analyze\nthe $SU(2)$ principal chiral model. The method developed here \nhas straightforward generalization to arbitrary $SU(N)$ or $SO(N)$ model \nand we shall present it elsewhere. \n\n\\subsection{Representation for the partition function}\n\nWe take the standard form for the $SU(2)$ link matrix which is the most suitable\nfor the weak coupling expansion\n\\begin{equation}\nV_l = \\exp [i\\sigma^k\\omega_k(l)] \\ ,\n\\label{su2mtr}\n\\end{equation}\n\\noindent\nwhere $\\sigma^k, k=1,2,3$ are Pauli matrices.\nLet us define\n\\begin{equation}\nW_l = \\left[ \\sum_k\\omega^2_k(l) \\right]^{1\/2}\n\\label{Wl}\n\\end{equation}\n\\noindent\nand similarly\n\\begin{equation}\nW_p = \\left[ \\sum_k\\omega^2_k(p) \\right]^{1\/2} ,\n\\label{Wp}\n\\end{equation}\n\\noindent\nwhere $\\omega_k(p)$ is a plaquette angle defined as\n\\begin{equation}\nV_p = \\prod_{l\\in p}V_l = \\exp \\left [ i\\sigma^k\\omega_k(p) \\right ] \\ .\n\\label{plangle}\n\\end{equation}\n\\noindent\nThe exact relation between link angles $\\omega_k(l)$ and the plaquette\nangle $\\omega_k(p)$ is given in the Appendix B. Then, the partition function\n(\\ref{lPF}) can be exactly rewritten to the following form appropriate\nfor the weak coupling expansion\n\\begin{eqnarray}\nZ = \\int \\prod_l \\left[ \\frac{\\sin^2W_l}{W^2_l}\\prod_kd\\omega_k(l) \\right]\n\\exp \\left[ 2\\beta\\sum_l\\cos W_l \\right] \\prod_x \\frac{W_x}{\\sin W_x} \n\\nonumber \\\\\n\\prod_x \\sum_{m(x)=-\\infty}^{\\infty}\\int\\prod_kd\\alpha_k(x)\n\\exp \\left[ -i\\sum_k\\alpha_k(x)\\omega_k(x) + 2\\pi im(x)\\alpha (x) \\right] \\ ,\n\\label{PFwk}\n\\end{eqnarray}\n\\noindent\nwhere we have introduced auxiliary field $\\alpha_k(x)$ and\n\\begin{equation}\n\\alpha (x) = \\left[ \\sum_k\\alpha^2_k(x) \\right]^{1\/2} .\n\\label{Ax}\n\\end{equation}\n\\noindent\nThe representation for the Jacobian (\\ref{D14}) used here can be \ninterpreted as an analog of the $U(1)$ periodic delta-function \n(\\ref{PFLxyJ}): this is the $SU(2)$ delta-function which is periodic\nwith respect to a length of the vector $\\vec{\\omega}(x)$.\nWe give a derivation of the partition function (\\ref{PFwk})\nin the Appendix A. In rewriting the final formula of that derivation\n(\\ref{D19}), we went over to the dual lattice identifying \nthe links of the original lattice with dual links and the original\nplaquettes with dual sites located in the center of the original\nplaquettes. \n\n\n\\subsection{General expansion}\n\nTo perform the weak coupling expansion \nwe proceed in a standard way, i.e. first we make the substitution\n\\begin{equation}\n\\omega_k(l)\\to (2\\beta)^{-1\/2}\\omega_k(l) \\ , \\\n\\alpha_k(x)\\to (2\\beta)^{1\/2}\\alpha_k(x) \n\\label{subst}\n\\end{equation}\n\\noindent\nand then expand the integrand of (\\ref{PFwk}) \nin powers of fluctuations of the link fields.\nWe would like to give here some technical details of the expansion\nwhich could be useful for a future use. \nIt is straightforward to get the following power series:\n\\begin{enumerate}\n\n\\item Action\n\n\\begin{equation}\n\\exp \\left[ 2\\beta\\cos\\frac{W_l}{\\sqrt{2\\beta}} \\right] = \\exp \n\\left[ 2\\beta -\\frac{1}{2}(W_l)^2 \\right] \n\\left[ 1 + \\sum_{k=1}^{\\infty} (2\\beta )^{-k} \n\\sum_{l_1,..,l_k} \\frac{a_1^{l_1}...a_k^{l_k}}{l_1!...l_k!} \\right] \\ ,\n\\label{actexp}\n\\end{equation}\n\\noindent \nwhere $l_1+2l_2+...+kl_k=k$ and\n\\begin{equation}\na_k = (-1)^{k+1}\\frac{W_l^{2(k+1)}}{(2k+2)!} \\ .\n\\label{a_kW}\n\\end{equation}\n\\noindent\n\n\\item Invariant measure\n\n\\begin{equation}\n\\frac{\\sin^2W_l}{W^2_l} = 1 +\n\\sum_{k=1}^{\\infty}\\frac{(-1)^k}{(2\\beta )^k}C_kW_l^{2k} \\ , \\ \nC_k = \\sum_{n=0}^k \\frac{1}{(2n+1)!(2k-2n+1)!} \\ .\n\\label{invMexp}\n\\end{equation}\n\\noindent\n\n\\item Contribution from Jacobian I\n\n\\begin{equation}\n\\frac{W_x}{\\sin W_x} = \n1 + \\sum_{k=1}^{\\infty}\\frac{J_k}{(2\\beta )^k}W_x^{2k} \\ , \\\nJ_k = 2\\frac{2^{2k-1}}{(2k)!}\\mid B_{2k} \\mid \\ ,\n\\label{J1exp}\n\\end{equation}\n\\noindent\nwhere $B_{2k}$ are Bernoulli numbers.\n\n\\item Contribution from Jacobian II\n\n\\begin{eqnarray}\n\\alpha_k(x)\\omega_k(x) = \\alpha_k(x)\\left[ \\omega^{(0)}_k(x) +\n\\sum_{n=1}^{\\infty} \\frac{\\omega^{(n)}_k(x)}{(2\\beta )^{n\/2}} \\right] \n\\ , \\\\ \\nonumber\n\\exp \\left[ -i\\sum_k\\alpha_k(x)\\omega_k(x) \\right] = \n\\exp \\left[ -i\\sum_k\\alpha_k(x)\\omega^{(0)}_k(x) \\right] \\\\ \\nonumber\n\\left[ 1 + \\sum_{q=1}^{\\infty}\\frac{(-i)^q}{q!} \\left( \\sum_k\\alpha_k(x) \n\\sum_{n=1}^{\\infty} \\frac{\\omega^{(n)}_k(x)}{(2\\beta )^{n\/2}} \\right)^q \\right] \\ .\n\\label{J2exp}\n\\end{eqnarray}\n\\noindent\n\n\\end{enumerate}\nUsing the relations (\\ref{D9})-(\\ref{D12}) one can calculate \n$\\omega^{(n)}_k(x)$ up to an arbitrary order in $n$. In particular, \n$\\omega^{(0)}_k(x)$ is given by (see Fig.1 for our notations of\ndual links)\n\\begin{equation}\n\\omega^{(0)}_k(x) = \\omega_k(l_3) + \\omega_k(l_4) - \n\\omega_k(l_1) - \\omega_k(l_2) \\ .\n\\label{w0}\n\\end{equation}\n\\noindent\nWe shall use an obvious property\n\\begin{equation}\n\\sum_x\\alpha_k(x)\\omega^{(0)}_k(x) = \n\\sum_l\\omega_k(l) \\left [\\alpha_k(x+n) - \\alpha_k(x) \\right ] \\ , \\ l=(x;n) \\ .\n\\label{resum}\n\\end{equation}\n\\noindent\nIntroducing now the external sources $h_k(l)$ coupled to the link field \n$\\omega_k(l)$ and $s_k(x)$ coupled to the auxiliary field $\\alpha_k(x)$\nand adjusting the definitions\n\\begin{equation}\n\\omega_k(l) \\to \\frac{\\partial}{\\partial h_k(l)} \\ , \\ \n\\alpha_k(x) \\to \\frac{\\partial}{\\partial s_k(x)} \\ ,\n\\label{deriv}\n\\end{equation}\n\\noindent\nwe get finally the following formal weak coupling expansion \nfor the PF (\\ref{PFwk})\n\\begin{eqnarray}\nZ = C(2\\beta ) Z(0,0) \\prod_l\\left[ \\left( 1 + \\sum_{k=1}^{\\infty} (2\\beta )^{-k} \n\\sum_{l_1,..,l_k}^{\\prime}\\frac{a_1^{l_1}...a_k^{l_k}}{l_1!...l_k!} \\right) \n\\left( 1 + \\sum_{k=1}^{\\infty}\\frac{(-1)^k}{(2\\beta )^k}C_kW_l^{2k} \n\\right) \\right] \\\\ \\nonumber\n\\prod_x \\left[ \\left(1 + \\sum_{k=1}^{\\infty}\\frac{J_k}{(2\\beta )^k}W_x^{2k} \\right) \n\\left( 1 + \\sum_{q=1}^{\\infty}\\frac{(-i)^q}{q!} \\left( \\sum_k\\alpha_k(x) \n\\sum_{n=1}^{\\infty} \\frac{\\omega^{(n)}_k(x)}{(2\\beta )^{n\/2}} \\right)^q \\right) \\right]\n\\ M(h,s) \\ ,\n\\label{PFwkexp}\n\\end{eqnarray}\n\\noindent\nwhere\n\\begin{equation}\nC(\\beta ) = \\exp\\left[ 2\\beta L^2 - \\frac{3}{2}L^2\\ln\\beta \\right] \\ .\n\\label{preexp}\n\\end{equation}\n\\noindent\nAs usually, one has to put $h_k=s_k=0$ after taking all the derivatives. \n$M(h,s)$ is a generating functional which we study in the next subsection.\nIt is obvious that the ground state satisfies\n\\begin{equation}\n<(\\omega^{(0)}_k(x))^p> = 0 \\ , \\ p=1,2,... \\ ,\n\\label{mainst}\n\\end{equation}\n\\noindent\nprecisely like the abelian model.\n\n\n\\subsection{Generating functional and zero modes}\n\nHere we are going to study the generating functional $M(h,s)$ given by\n\\begin{equation}\nM(h,s) = \\frac{Z(h,s)}{Z(0,0)} \\ ,\n\\label{GF}\n\\end{equation}\n\\noindent\nand\n\\begin{eqnarray}\nZ(h,s) = \\int_{-\\infty}^{\\infty} \\prod_{x,k} d\\alpha_k(x)\n\\int_{-\\infty}^{\\infty} \\prod_{l,k} d\\omega_k(l) \n\\exp \\left[ -\\frac{1}{2}\\omega^2_k(l) \n-i\\omega_k(l)[\\alpha_k(x+n) - \\alpha_k(x) ] \\right] \\nonumber \\\\ \n\\sum_{m(x)=-\\infty}^{\\infty}\n\\exp \\left[ 2\\pi i\\sqrt{2\\beta}\\sum_xm(x)\\alpha (x) +\n\\sum_{l,k}\\omega_k(l)h_k(l) + \\sum_{x,k}\\alpha_k(x)s_k(x) \\right] .\n\\label{GF1}\n\\end{eqnarray}\n\\noindent\nAs in the abelian case we expect that integrals over zero modes\nof the auxiliary field are not Gaussian and should lead to some constraint\non the sums over $m_x$. To see this, we put $h_k=s_k=0$ and integrate out\nthe link fields. Partition function becomes\n\\begin{equation}\nZ(0,0) =\\sum_{m(x)=-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\prod_{x,k} d\\alpha_k(x)\n\\exp \\left[ -\\alpha_k(x)G_{x,x^{\\prime}}^{-1}\\alpha_k(x^{\\prime})\n+ 2\\pi i\\sqrt{2\\beta}m(x)\\alpha (x) \\right] \\ , \n\\label{GF00}\n\\end{equation}\n\\noindent\nwith $G_{x,x^{\\prime}}$ given in (\\ref{Gxx}) and sum over repeating indices is\nunderstood here and in what follows. \nIt is clear that in this case the zero modes should be controlled via integration \nover the radial component of the vector $\\vec{\\alpha}(x)$. To see this, we change\nto the spherical coordinates and treat only a constant mode in the angle variables\n(since the zero mode problem could only arise from this configuration). One has\n\\begin{eqnarray}\nZ(0,0) \\sim \\sum_{m_x=-\\infty}^{\\infty} \\int_0^{\\infty} \n\\prod_x \\alpha^2_x d\\alpha_x\n\\exp \\left[ -\\alpha_xG_{x,x^{\\prime}}^{-1}\\alpha_{x^{\\prime}}\n+ 2\\pi i\\sqrt{2\\beta}m_x\\alpha_x \\right] \\nonumber \\\\\n\\sim \\sum_{m_x=-\\infty}^{\\infty} \\delta \\left( \\sum_xm_x \\right) \n\\exp \\left[ -2\\pi^2\\beta \\sum_{x,x^{\\prime}}m_x\nG_{x,x^{\\prime}} m_{x^{\\prime}}\\right] + O(m_x^2) \\ ,\n\\label{GF0mod}\n\\end{eqnarray}\n\\noindent\nand we used the notation $\\alpha_x$ for the radial component of the vector \n$\\vec{\\alpha}(x)$. Since the zero mode of the radial component in the $x$\nspace is\n$$\n\\alpha (p=0) = \\left( \\frac{1}{L^D}\\sum_k(\\sum_x\\alpha_k(x))^2 \\right)^{1\/2}\n$$ \none has to omit the zero mode from the Green function in each term \nof the sum over $k$ in the integrand of (\\ref{GF00}).\nAs in the abelian case the integration over zero modes produces \ndelta-function in (\\ref{GF0mod}). Since however only $m_x=0$ for all $x$\ncontribute to the asymptotics of the free energy and fixed distance correlation\nfunction (other values of $m_x$ being exponentially suppressed) \nwe have to put $m_x=0$ omitting at the same time all zero modes. \nCalculating resulting Gaussian integrals we come to\n\\begin{equation}\nM(h,s) = \\exp \n\\left[ \\frac{1}{4}s_k(x)G_{x,x^{\\prime}}s_k(x^{\\prime}) +\n\\frac{i}{2}s_k(x)D_l(x)h_k(l) +\n\\frac{1}{4}h_k(l)G_{ll^{\\prime}}h_k(l^{\\prime}) \\right] \\ ,\n\\label{GFfin}\n\\end{equation}\n\\noindent\nwhere $G_{ll^{\\prime}}$ was introduced in (\\ref{Gll1}) and\n\\begin{equation}\nD_l(x^{\\prime}) = G_{x,x^{\\prime}} - G_{x+n,x^{\\prime}} \\ , \\ l=(x,n) \\ .\n\\label{Dxl}\n\\end{equation}\n\\noindent\nFrom (\\ref{GFfin}) one can deduce the following simple rules\n\\begin{eqnarray}\n<\\omega_k(l)\\omega_n(l^{\\prime})> =\n\\frac{\\delta_{kn}}{2}G_{ll^{\\prime}} \\ , \\ \n<\\alpha_k(x)\\alpha_n(x^{\\prime})> =\n\\frac{\\delta_{kn}}{2}G_{xx^{\\prime}} \\ , \\nonumber \\\\ \n- i<\\omega_k(l)\\alpha_n(x^{\\prime})> =\n\\frac{\\delta_{kn}}{2}D_l(x^{\\prime}) \\ .\n\\label{Frules}\n\\end{eqnarray}\n\\noindent\nWe describe some simple properties of the functions $G_{ll^{\\prime}}$\nand $D_l(x^{\\prime})$ in the Appendix C.\nThe expansion (\\ref{PFwkexp}), representation (\\ref{GFfin}) for the \ngenerating functional and rules (\\ref{Frules}) \nare main formulas of this section which allow \nto calculate the weak coupling expansion of both the free energy and any\nshort-distance observable. Let us now comment on the infrared\nfinitness of the expansion.\nIt follows from the representation for the generating functional\nthat all expectation values of the link fields are expressed only via\nthe link Green functions $G_{ll^{\\prime}}$ and $D_l(x)$ which are infrared finite\nby construction. All combinations of auxiliary fields which contain\nodd overall powers of the fields are expressed only via $D_l(x)$\nand, therefore are also infrared finite. However, even powers\ninclude $G_{x,x^{\\prime}}$ and the infrared finitness is not provided\nautomatically. In particular, it means that unlike $XY$ \nmodel we are not allowed to take the TL at this stage.\n\nOur last comment concerns the partition function (\\ref{GF00}). We believe it \ncan be regarded as an analog of the corresponding expression in the $XY$ model,\ni.e. this is a nonabelian analog of the so-called ``spin-wave--vortex'' representation\nfor the partition function. One can see that the nonabelian model\nis not factorized into three abelian components and is periodic in\nthe length of the vector $\\vec{\\alpha}(x)$ rather than in \nits components $\\alpha_k(x)$. \n\n\\subsection{First order coefficient of the correlation function}\n\nAs the simplest example we would like to calculate the first\norder coefficient of the correlation function (\\ref{corf1}).\nExpanding (\\ref{corf1}) in $1\/\\beta$ one has\n\\begin{equation}\n\\Gamma (x,y) = 1 - \\frac{1}{4\\beta}\n< \\sum_{k=1}^3\\left (\\sum_l\\omega_k(l) \\right )^2 > + O(\\beta^{-2}) = \n1 - \\frac{3}{8\\beta}\n\\sum_{l,l^{\\prime}\\in C^d_{xy}} G_{ll^{\\prime}} + O(\\beta^{-2}) \\ ,\n\\label{GLXY1}\n\\end{equation}\n\\noindent\nwhere $C^d_{xy}$ is a path dual to the path $C_{xy}$, i.e consisting\nof the dual links which are orthogonal to the original\nlinks $l,l^{\\prime}\\in C_{xy}$.\nThe form of $G_{ll^{\\prime}}$ ensures independence of $\\Gamma (x,y)$\nof a choice of the path $C_{xy}$. After some algebra it is easy to get\nthe result\n\\begin{equation}\n\\Gamma (x,y) = 1 - \\frac{3}{4\\beta} D(x-y) \\ , \\\nD(x) = \\frac{1}{L^2} \\sum_{k_n=0}^{L-1}\n\\frac{1 - e^{\\frac{2\\pi i}{L}k_n x_n}}\n{D-\\sum_{n=1}^D\\cos \\frac{2\\pi}{L}k_n} \\ , \\ k_n^2\\ne 0 ,\n\\label{Dx}\n\\end{equation}\n\\noindent\nwhich coincides with the result of the conventional PT.\n\n\n\\section{First order coefficient of the free energy}\n\nThe main result of our study is the first order\ncoefficient of the $SU(2)$ free energy \n\\begin{equation}\nF=\\frac{1}{2L^2}\\ln Z = 2\\beta - \\frac{3}{4}\\ln\\beta +\n\\frac{1}{2\\beta L^2}C^1 + O(\\beta^{-2}) \\ .\n\\label{fren}\n\\end{equation}\n\\noindent\nThere are four contributions at this order to $C^1$\n\\begin{equation}\nC^1 = C^1_{ac} + C^1_{meas} + C^1_{J1} + C^1_{J2} \\ .\n\\label{frensum}\n\\end{equation}\n\\noindent\nContribution from the action (\\ref{actexp}) is given by\n\\begin{equation}\n\\frac{1}{2L^2}C^1_{ac}=\\frac{5}{128L^2}\\sum_lG^2_{ll}=\n\\frac{5}{64} \\ .\n\\label{c1ac}\n\\end{equation}\n\\noindent\nContribution from the measure (\\ref{invMexp}) is given by\n\\begin{equation}\n\\frac{1}{2L^2}C^1_{meas}= - \\frac{1}{8L^2}\\sum_lG_{ll}\n= - \\frac{1}{4} \\ .\n\\label{c1meas}\n\\end{equation}\n\\noindent\nContribution from third brackets in (\\ref{PFwkexp})\nis proportional to $[\\omega_k^{(0)}]^2$ and equals 0,\nbecause of (\\ref{mainst}). There are two contributions\nfrom the expansion of the Jacobian (\\ref{J2exp}). The first one\nis given by the expectation value of the operator\n$-i\\sum_x\\sum_{k=1}^3\\alpha_k(x)\\omega_k^{(2)}(x)$.\n$\\omega_k^{(2)}(x)$ is given in the Appendix B (\\ref{wkx2}). \nFrom the form of the generating functional (\\ref{GFfin}) one can see that\nthe expectation value of this operator depends only on\nlink Green functions $G_{ll^{\\prime}}$ and $D_l(x^{\\prime})$.\nOne gets after long but straightforward algebra\\footnote{Our previous\nversion suffered from incorrect sign in this term which led to \nwrong final result.}\n\\begin{eqnarray}\n\\frac{1}{2L^2}C^1_{J1}=\\frac{1}{4L^2}\\sum_l\\sum_xD_l(x) \\nonumber \\\\\n(\\ \\frac{1}{2}\\sum_{i=1}^4(\\delta_{ll_i}(G_{l_3l_i}+G_{l_4l_i}-\nG_{l_1l_i}-G_{l_2l_i})+G_{l_il_i}(\\delta_{ll_1}+\\delta_{ll_2}-\n\\delta_{ll_3}-\\delta_{ll_4}))+ \\nonumber \\\\\n\\delta_{ll_1}(G_{l_3l_4}+2G_{l_3l_2}+2G_{l_4l_2})+ \n\\delta_{ll_2}(G_{l_3l_4}-G_{l_3l_1}-G_{l_4l_1})+ \\nonumber \\\\\n\\delta_{ll_3}(G_{l_4l_1}+G_{l_4l_2}-G_{l_1l_2})-\n\\delta_{ll_4}(2G_{l_3l_1}+2G_{l_3l_2}+G_{l_1l_2}) \\ ) \\ ,\n\\label{c1J1Gr}\n\\end{eqnarray}\n\\noindent\nwhere links $l_i$ are defined in Appendix C (see Fig.1), $l=(x,n)$.\nIn terms of standard $D$-functions defined in Appendix C the result reads\n\\begin{equation}\n\\frac{1}{2L^2}C^1_{J1}=\\frac{1}{4}[6-2D(2,0)-D(1,1)]=\n\\frac{1}{2}+\\frac{3}{2\\pi} \\ .\n\\label{c1J1res}\n\\end{equation}\n\\noindent\nThe second term is given by the operator\n$< \\frac{1}{2}\\left\n(\\sum_x\\sum_{k=1}^3\\alpha_k(x)\\omega_k^{(1)}(x)\\right )^2 >$.\n$\\omega_k^{(1)}(x)$ is given in the Appendix B (\\ref{wkx1}).\nIn terms of Green functions it reads\n\\begin{equation}\n\\frac{1}{2L^2}C^1_{J2} = - \\ \\frac{3}{16} (Q^{(1)} + Q^{(2)}) \\ ,\n\\label{c1J2res}\n\\end{equation}\n\\noindent\n\\begin{equation}\nQ^{(1)}= \\frac{1}{2L^2}\\sum_{x,x^{\\prime}}\n\\sum_{is) \\sim s^{-\\zeta}~,\n\\end{equation}\nwhere the exponent $\\zeta=3.03\\pm0.41$ ranging from 2.4 to 3.9\n\\cite{Farmer-Gillemot-Lillo-Mike-Sen-2004-QF,Mike-Farmer-2007-JEDC},\nwhich is well consistent with the inverse cubic law\n\\cite{Gopikrishnan-Meyer-Amaral-Stanley-1998-EPJB,Gabaix-Gopikrishnan-Plerou-Stanley-2003-PA,Gabaix-Gopikrishnan-Plerou-Stanley-2003-Nature}.\nIn addition, Mike and Farmer found that the spread possesses long\nmemory with the Hurst index being $0.751$ has been\ntranslated vertically for clarity.} \\label{Fig:s3:cdf}\n\\end{center}\n\\end{figure}\n\nThere are also significant discrepancies. Comparing the cumulative\ndistributions in Fig.~\\ref{Fig:s3:cdf} and that on the NYSE\n\\cite{Plerou-Gopikrishnan-Stanley-2005-PRE}, significant differences\nare observed. The distribution of the spreads on the SSE decays much\nfaster than that on the NYSE for small spreads. In other words, the\nproportion of small spreads is much larger on China's SSE. Possible\ncauses include the absence of market orders, no short positions, the\nmaximum percentage of fluctuation (10\\%) in each day, and the $t+1$\ntrading mechanism in the Chinese stock markets on the one hand and\nthe hybrid trading system containing both specialists and\nlimit-order traders in the NYSE on the other hand. The exact cause\nis not clear for the time being, which can however be tested when\nnew data are available after the introduction of market orders in\nJuly 1, 2006. Moreover, the PDF's in SSE drop abruptly after the\npower-law parts for the largest spreads, which is not observed in\nthe NYSE case \\cite{Plerou-Gopikrishnan-Stanley-2005-PRE}.\n\n\n\n\\section{Long memory}\n\\label{s1:memory}\n\nAnother important issue about financial time series is the presence\nof long memory, which can be characterized by its Hurst index $H$.\nIf $H$ is significantly larger than $0.5$ the time series is viewed\nto possess long memory. Long memory can be defined equivalently\nthrough autocorrelation function $C(\\ell) \\sim \\ell^{-{\\gamma}}$ and\nthe power spectrum $p(\\omega)\\sim\\omega^{-\\eta}$, where the\nautocorrelation exponent ${\\gamma}$ is related to the Hurst index\n$H$ by $\\gamma=2-2H$\n\\cite{Kantelhardt-Bunde-Rego-Havlin-Bunde-2001-PA,Maraun-Rust-Timmer-2004-NPG},\nand the power spectrum exponent $\\eta$ is given by $\\eta=2H-1$\n\\cite{Talkner-Weber-2000-PRE,Heneghan-McDarby-2000-PRE}.\n\nThere are many methods proposed for estimating the Hurst index such\nas the rescaled range analysis (RSA)\n\\cite{Hurst-1951-TASCE,Mandelbrot-Ness-1968-SIAMR,Mandelbrot-Wallis-1969a-WRR,Mandelbrot-Wallis-1969b-WRR,Mandelbrot-Wallis-1969c-WRR,Mandelbrot-Wallis-1969d-WRR},\nfluctuation analysis (FA)\n\\cite{Peng-Buldyrev-Goldberger-Havlin-Sciortino-Simons-Stanley-1992-Nature},\ndetrended fluctuation analysis (DFA)\n\\cite{Peng-Buldyrev-Havlin-Simons-Stanley-Goldberger-1994-PRE,Hu-Ivanov-Chen-Carpena-Stanley-2001-PRE,Kantelhardt-Bunde-Rego-Havlin-Bunde-2001-PA},\nwavelet transform module maxima (WTMM) method\n\\cite{Holschneider-1988-JSP,Muzy-Bacry-Arneodo-1991-PRL,Muzy-Bacry-Arneodo-1993-JSP,Muzy-Bacry-Arneodo-1993-PRE,Muzy-Bacry-Arneodo-1994-IJBC},\ndetrended moving average (DMA)\n\\cite{Alessio-Carbone-Castelli-Frappietro-2002-EPJB,Carbone-Castelli-Stanley-2004-PA,Carbone-Castelli-Stanley-2004-PRE,Alvarez-Ramirez-Rodriguez-Echeverria-2005-PA,Xu-Ivanov-Hu-Chen-Carbone-Stanley-2005-PRE},\nto list a few. We adopt the detrended fluctuation analysis.\n\nThe method of detrended fluctuation analysis is widely used for its\neasy implementation and robust estimation even for a short time\nseries\n\\cite{Taqqu-Teverovsky-Willinger-1995-Fractals,Montanari-Taqqu-Teverovsky-1999-MCM,Heneghan-McDarby-2000-PRE,Audit-Bacry-Muzy-Arneodo-2002-IEEEtit}.\nThe idea of DFA was invented originally to investigate the\nlong-range dependence in coding and noncoding DNA nucleotides\nsequence\\cite{Peng-Buldyrev-Havlin-Simons-Stanley-Goldberger-1994-PRE}\nand then applied to various fields including finance. The method of\nDFA consists of the following steps.\n\nStep 1: Consider a time series $x(t)$, $t=1,2,\\cdots,N$. We first\nconstruct the cumulative sum\n\\begin{equation}\nu(t) = \\sum_{i = 1}^{t}{x(i)}, ~~t=1,2,\\cdots,N~.\n \\label{Eq:DFA_u}\n\\end{equation}\n\nStep 2: Divide the series $u(t)$ into $N_\\ell$ disjoint segments\nwith the same length $\\ell$, where $N_\\ell = [N\/\\ell]$. Each segment\ncan be denoted as $u_v$ such that $u_v(i) = u(l + i)$ for\n$1\\leqslant{i}\\leqslant{\\ell}$, and $l = (v - 1)\\ell$. The trend of\n$u_v$ in each segment can be determined by fitting it with a linear\npolynomial function $\\widetilde{u}_v$. Quadratic, cubic or higher\norder polynomials can also be used in the fitting procedure while\nthe simplest function could be linear. In this work, we adopted the\nlinear polynomial function to represent the trend in each segment\nwith the form:\n\\begin{equation}\n\\widetilde{u}_v(i) = ai+b~,\n \\label{Eq:DFA_wu}\n\\end{equation}\nwhere $a$ and $b$ are free parameters to be determined by the least\nsquares fitting method and $1\\leqslant{i}\\leqslant\\ell$.\n\nStep 3: We can then obtain the residual matrix $\\epsilon_{v}$ in\neach segment through:\n\\begin{equation}\n\\epsilon_{v}(i)=u_{v}(i)-\\widetilde{u}_{v}(i)~,\n\\end{equation}\nwhere $1\\leqslant{i}\\leqslant{\\ell}$. The detrended fluctuation\nfunction $F(v,\\ell)$ of the each segment is defined via the sample\nvariance of the residual matrix $\\epsilon_{v}$ as follows:\n\\begin{equation}\nF^2(v,\\ell) = \\frac{1}{\\ell}\\sum_{i = 1}^{\\ell}[\\epsilon_{v}(i)]^2~.\n \\label{Eq:DFA_F1}\n\\end{equation}\nNote that the mean of the residual is zero due to the detrending\nprocedure.\n\nStep 4: Calculate the overall detrended fluctuation function\n$F(\\ell)$, that is,\n\\begin{equation}\nF^2(\\ell) = \\frac{1}{N_\\ell}\\sum_{v = 1}^{N_\\ell}F^2(v,\\ell)~.\n \\label{Eq:DFA_F2}\n\\end{equation}\n\nStep 5: Varying the value of $\\ell$, we can determine the scaling\nrelation between the detrended fluctuation function $F(\\ell)$ and\nthe size scale $\\ell$, which reads\n\\begin{equation}\nF(\\ell) \\sim \\ell^{H}~,\n \\label{Eq:DFA_H}\n\\end{equation}\nwhere $H$ is the Hurst index of the time series\n\\cite{Taqqu-Teverovsky-Willinger-1995-Fractals,Kantelhardt-Bunde-Rego-Havlin-Bunde-2001-PA}.\n\nFigure~\\ref{Fig:dfa} plots the detrended fluctuation function\n$F(\\ell)$ of the bid-ask spreads from different definitions using\nlinear prices. The bottom $F(\\ell)$ curve is for the average spread\nafter removing the intraday pattern. All the curves show evident\npower-law scaling with the Hurst indexes $H_{\\rm{I}} = 0.91 \\pm\n0.01$ for definition I, $H_{\\rm{II}} = 0.92 \\pm 0.01$ for definition\nII, $H_{\\rm{III}} = 0.75 \\pm 0.01$ for definition III, and\n$H_{\\rm{III}} = 0.77 \\pm 0.01$ for definition without intraday\npattern, respectively. Quite similar results are obtain for\nlogarithmic prices where $H_{\\rm{I}} = 0.89 \\pm 0.01$ for definition\nI, $H_{\\rm{II}} = 0.91 \\pm 0.01$ for definition II, $H_{\\rm{III}} =\n0.77 \\pm 0.01$ for definition III, and $H_{\\rm{III}} = 0.76 \\pm\n0.01$ for definition III without intraday pattern. The two Hurst\nindexes for definitions I and II are higher than their counterparts\non the London Stock Exchange where ``even time'' is adopted\n\\cite{Mike-Farmer-2007-JEDC}. It is interesting to note that the\npresence of intraday pattern does not introduce distinguishable\ndifference in the Hurst index and the two indexes for definition III\nare also very close to those of average spreads in the Brazilian\nstock market and on the New York Stock Exchange where real time is\nused\n\\cite{Plerou-Gopikrishnan-Stanley-2005-PRE,Cajueiro-Tabak-2007-PA}.\nDue to the large number of data used in the analysis, we argue that\nthe bid-ask spreads investigated exhibit significant long memory.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig-DFA.eps}\n\\caption{Detrended fluctuation function $F(\\ell)$ for the spreads\nobtained from three definition with linear prices. The curves have\nbeen shifted vertically for clarity.} \\label{Fig:dfa}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\section{Multifractal analysis}\n\\label{s1:MFA}\n\nIn this section, we investigate whether the time series of bid-ask\nspread obtained from definition III possesses multifractal nature.\nThe classical box-counting algorithm for multifractal analysis is\nutilized and described below\n\\cite{Halsey-Jensen-Kadanoff-Procaccia-Shraiman-1986-PRA}.\n\nConsider the spread time series $S(t)$, $t=1,2,\\cdots,N$. First, we\ndivide the series $S(t)$ into $N_\\ell$ disjoint segments with the\nsame length $\\ell$, where $N_\\ell = [N\/\\ell]$. Each segment can be\ndenoted as $S_v$ such that $S_v(i) = S(l + i)$ for\n$1\\leqslant{i}\\leqslant{\\ell}$, and $l = (v - 1)\\ell$. The sum of\n$S_v$ in each segment is calculated as follows,\n\\begin{equation}\n\\Gamma(v,\\ell) = \\sum_{i = 1}^{\\ell}{S_v(i)},\n~~v=1,2,\\cdots,N_\\ell~.\n \\label{Eq:MF_F}\n\\end{equation}\nWe can then calculate the $q$th order partition function $\\Gamma(q;\n\\ell)$ as follows,\n\\begin{equation}\n\\Gamma(q; \\ell) = \\sum_{v = 1}^{N_\\ell}[\\Gamma(v,\\ell)]^q~.\n \\label{Eq:MF_Fq}\n\\end{equation}\nVarying the value of $\\ell$, we can determine the scaling relation\nbetween the partition function $\\Gamma(q; \\ell)$ and the time scale\n$\\ell$, which reads\n\\begin{equation}\n\\Gamma(q; \\ell) \\sim \\ell^{\\tau(q)}~.\n \\label{Eq:DFA_M_h}\n\\end{equation}\n\nFigure~\\ref{Fig:mffq} illustrates the power-law scaling dependence\nof the partition function $\\Gamma(q; \\ell)$ of the bid-ask spreads\nafter removing the intraday pattern in definition III for different\nvalues of $q$, where both linear prices and logarithmic prices are\ninvestigated. The continuous lines are the best linear fits to the\ndata sets. The collapse of the data points on the linear lines\nindicates evident power-law scaling between $\\Gamma(q; \\ell)$ and\n$\\ell$. The slopes $\\tau(q)$ of the fitted lines are $\\tau(-4) =\n-5.02 \\pm 0.01$, $\\tau(-2) = -3.01 \\pm 0.01$, $\\tau(0) = -1.01 \\pm\n0.01$, $\\tau(2) = 0.99 \\pm 0.01$, and $\\tau(4) = 2.98 \\pm 0.01$ for\nlogarithmic prices and $\\tau(-4) = -5.02 \\pm 0.01$, $\\tau(-2) =\n-3.01 \\pm 0.01$, $\\tau(0) = -1.01 \\pm 0.01$, $\\tau(2) = 0.99 \\pm\n0.01$, and $\\tau(4) = 2.98 \\pm 0.01$ for linear prices. We notice a\nnice relation $\\tau(q)=q-1$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=7cm]{Fig-MFDFA_Fs.eps}\n\\caption{Log-log plots of the partition function $\\Gamma(q; \\ell)$\nof the bid-ask spreads calculated from definition III with the\nintraday pattern removed for five different values of $q$. Both\nlinear and logarithmic prices are investigated (shown in the\nlegend). The markers stand for the results calculated from the real\ndata and the continuous lines are the best fits. The plots for $q =\n-2, 0, 2$, and $4$ are shifted upwards for clarity.}\n \\label{Fig:mffq}\n\\end{center}\n\\end{figure}\n\nQuantitatively similar results are obtained when the intraday\npattern is not removed. The scaling exponents are $\\tau(-4) = -5.03\n\\pm 0.01$, $\\tau(-2) = -3.01 \\pm 0.01$, $\\tau(0) = -1.01 \\pm 0.01$,\n$\\tau(2) = 0.99 \\pm 0.01$, and $\\tau(4) = 2.96 \\pm 0.01$ for\nlogarithmic prices and $\\tau(-4) = -5.03 \\pm 0.01$, $\\tau(-2) =\n-3.02 \\pm 0.01$, $\\tau(0) = -1.01 \\pm 0.01$, $\\tau(2) = 0.99 \\pm\n0.01$, and $\\tau(4) = 2.97 \\pm 0.01$ for linear prices. Again, we\nobserve that $\\tau(q)=q-1$.\n\nIn the standard multifractal formalism based on partition function,\nthe multifractal nature is characterized by the scaling exponents\n$\\tau(q)$. It is easy to obtain the generalized dimensions $D_q=\n{\\tau}(q)\/(q - 1)$\n\\cite{Grassberger-1983-PLA,Hentschel-Procaccia-1983-PD,Grassberger-Procaccia-1983-PD}\nand the singularity strength function $\\alpha(q)$, the multifractal\nspectrum $f(\\alpha)$ via the Legendre transform\n\\cite{Halsey-Jensen-Kadanoff-Procaccia-Shraiman-1986-PRA}:\n$\\alpha(q) = {\\rm d}{\\tau}(q)\/{\\rm d}q$ and $f(q) = q{\\alpha} -\n{\\tau}(q)$.\n\nFigure~\\ref{Fig:falpha:tau} shows the multifractal spectrum\n$f(\\alpha)$ and the scaling function $\\tau(q)$ in the inset for\nlinear and logarithmic prices. One finds that the two $\\tau(q)$\ncurves are linear and $\\tau(q)=q-1$, which is the hallmark of the\npresence of monofractality, not multifractality. The strength of the\nmultifractality can be characterized by the span of singularity\n$\\Delta\\alpha=\\alpha_{\\max}-\\alpha_{\\min}$. If $\\Delta\\alpha$ is\nclose to zero, the measure is almost monofractal. The maximum and\nminimum of $\\alpha$ can be reached when $q\\to\\pm\\infty$, which can\nnot be achieved in real applications. However, $\\Delta\\alpha$ can be\napproximated with great precision with mediate values of $q$. The\nsmall value of $\\Delta\\alpha$ shown in Fig.~\\ref{Fig:falpha:tau}\nindicates a very narrow spectrum of singularity. Indeed, one sees\nthat $f(\\alpha)\\approx1$ and $\\alpha\\approx1$ for all values of $q$.\nWe thus conclude that there is no multifractal nature in the bid-ask\nspread investigated.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig-MFDFA_fAlpha_tau.eps}\n\\caption{Multifractal function $f({\\alpha})$ of the spreads in\ndefinition III with the intraday pattern removed. Inset: Scaling\nexponents ${\\tau}(q)$ of partition functions as a function of $q$.\nFor clarity, the $\\tau(q)$ curve for logarithmic price is shifted\nupwards by 1.}\n \\label{Fig:falpha:tau}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{s7:conclusion}\n\nThe bid-ask spread defined by the difference of the best ask price\nand the best bid price is considered as the benchmark of the\ntransaction cost and a measure of the market liquidity. In this\npaper, we have carried out empirical investigations on the\nstatistical properties of the bid-ask spread using the limit-order\nbook data of a stock SZ000001 (Shenzhen Development Bank Co., LTD)\ntraded on the Shenzhen Stock Exchange within the whole year of 2003.\nThree different definitions of spread are considered based on event\ntime at transaction level and on fixed interval of real time.\n\nThe distributions of spreads at transaction level decay as power\nlaws with tail exponents well below 3. In contrast the average\nspread in real time fulfils the inverse cubic law for different time\nintervals $\\Delta{t}= 1$, $2$, $3$, $4$, and $5$ min. We have\nperformed the detrended fluctuation analysis on the spread and found\nthat the spread time series exhibits evident long-memory, which is\nin agreement with other stock markets. However, an analysis using\nthe classic textbook box-counting algorithm does not provide\nevidence for the presence of multifractality in the spread time\nseries. To the best of our knowledge, this is the first time to\ncheck the presence of multifractality in the spread.\n\nOur analysis raises an intriguing open question that is not fully\naddressed. We have found that the spread possesses a\nwell-established intraday pattern composed by a large L-shape and a\nsmall L-shape separated by the noon closing of the Chinese stock\nmarket. This feature will help to understand the cause of the wide\nspread at the opening of the market, which deserves further\ninvestigation.\n\n\\begin{acknowledgement}\nWe are grateful to Dr. Tao Wu for fruitful suggestions. This work\nwas partially supported by the National Natural Science Foundation\nof China (Grant No. 70501011) and the Fok Ying Tong Education\nFoundation (Grant No. 101086).\n\\end{acknowledgement}\n\n\\bibliographystyle{h-elsevier3}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nStudies in linguistics are often characterized by small and unbalanced data sets.\nThis is particularly true for quantitative sociolinguistic studies that investigate the use of non-standard speech forms since such forms often occur at much lower token frequencies than standard speech forms (e.g.\\ Buschfeld 2020).\nThe present paper outlines ways to statistically meet the problems of low token frequencies and unbalanced data sets.\nDrawing on child data from Singapore and England, collected by one of the authors, we discuss a new resampling method which builds on the traditional method of repeated random undersampling but enhances it in two important respects.\nWe evaluate the outcome of undersampling by means of two criteria, i.e.\\ interpretability of the results and predictive power.\nWe therefore assess potential models from both their linguistic and statistical perspectives, which, to our best knowledge, is so far an unprecedented approach.\n\nIn this paper, we first introduce the linguistic study, i.e.\\ its data and research questions, in Section~\\ref{sec:2}. Section~\\ref{sec:3} outlines the general statistical approach we take, i.e.\\ classification rules and, in particular, conditional inference trees, and discusses their power and value against the background of notions such as prediction and interpretation, unbalanced classes, and re- and undersampling. In Section~\\ref{sec:int}, we conflate the statistical and linguistic perspectives. We show how and why interpretability and predictive power are crucial criteria for linguistic studies and how these criteria can be jointly utilized. To this end, we introduce our newly developed statistical approach {\\bf PrInDT}. The results of our study are presented and discussed in Section~\\ref{sec:5} on the basis of the three best trees generated by {\\bf PrInDT}. Last but not least, we look into ensembles of our {\\bf PrInDT}-generated trees and show how they can add important information to the findings of individual trees. We finish with some general statements that point the way ahead for both statistical and linguistic research.\n\n\\section{Linguistic Hypotheses and Data}\\label{sec:2}\n\nThe data were collected for a large-scale research project investigating the acquisition of English as a first language (L1) in Singapore (Buschfeld 2020).\nSingapore English (SingE) is one of the most extensively researched second language (L2) varieties of English.\nHowever, for an ever-increasing number of children, it is (one of) their home language(s). According to the Straits Times, one of the leading newspapers in Singapore, the number of primary one students who speak mostly English at home has risen to around 70\\% in all three major ethnic groups in Singapore, i.e.\\ the Chinese, Indian, and Malay (Chan, 2020).\nTo place L1 SingE on the map of L1 varieties of English and investigate the linguistic characteristics of SingE as well as the acquisitional route Singaporean children take in their linguistic development, we compare the data to data collected from monolingual and bi-\/multilingual children in England.\nThe relevant hypotheses for the study on subject pronoun realization read as follows (Buschfeld 2020, 85):\n\\begin{itemize}\n\\itemsep0pt\n\\item[]{\\bf Hypothesis 1}: All children (from Singapore and from England) drop subject pronouns in their early acquisitional stages. The amount of zero subjects is higher in the Singapore group due to the differences in input (SingE behaves similar to null-subject languages, at least at a surface level).\n\\item[]{\\bf Hypothesis 2}: In later acquisitional stages, the Singaporean children drop subject pronouns at a much higher rate than children acquiring English in a traditional native English setting. The children from England ultimately realize subjects as obligatory sentence constituents.\n\\end{itemize}\nTo sum up our hypotheses, we expect to find both age-related and ethnicity-related differences for the realization of subject pronouns.\n\nThe data were elicited systematically in video-recorded task-directed dialogue between the researcher and the children, consisting of several parts: a grammar elicitation task, a story retelling task, elicited narratives, and free interaction. The recorded material was orthographically transcribed and manually coded for the realization of subject pronouns (realized vs. zero):\n\nThe following {\\bf examples} illustrate the respective SingE variants (Buschfeld 2020, 144):\n\\begin{itemize}\n\\itemsep0pt\n\\item[1.]\tResearcher: [\\ldots ] what do you do with your friends? Do you play with them?\\\\\nChild: [ {\\bf \\o} I] Play with them. Sometimes drawing. [\\ldots]\\\\\nChild: Sometimes [ {\\bf \\o} WE] play some fun things.\n\\item[2.]\tChild: I think in MH370, I think they can find because [ {\\bf \\o} IT] is easy to go there [\\ldots].\n\\end{itemize}\n\n\\noindent The forms in Examples 1 and 2 are variably used by children acquiring English as an L1 in Singapore. They are also typical for adult SingE and thus part of the input the children receive.\n\nThe aim of our analysis is to find prediction rules for the use of subject pronouns (realized vs. zero) by means of extra- and intralinguistic variables.\nThe {\\bf extralinguistic variables} considered as independent variables in the statistical analysis are \\textsc{ethnicity} (ETH), \\textsc{age} (AGE), \\textsc{sex} (SEX), \\textsc{linguistic background} (LiBa), and \\textsc{mean length of utterance} (MLU). \\textsc{pronoun} (PRN) is taken into account as the {\\bf intralinguistic variable}.\nWe aim to determine whether any of these variables has a statistically significant influence on the results.\nAll in all, we extracted 6146 tokens of the values of the subject pronoun variable, each with the full set of extra- and intra-linguistic variables. 528 of these are realized as zero pronouns.\n\n\\section{Statistical Modeling}\\label{sec:3}\n\nStatistical modeling has long found its way into quantitative linguistics, but still, we would like to first address one of the central questions raised by such endeavors:\nWhat can we expect from statistical modeling?\n\nFrom a statistical perspective, the answer is quite straightforward. Statistical analyses can yield two kinds of insights. Description elicits information from the data sample. Inference generalizes from a sample to a population.\nBoth description and inference aim at the {\\bf interpretation} of properties of either the observed data (description) or of a more general population (inference).\nInference often comprises the {\\bf prediction} of such properties for the more general population.\n\nIn order to approximate the relationships between a class variable (e.g.\\ pronoun type with possible values `zero' and `realized') and influential variables like extra- and intra-linguistic variables, {\\bf classification rules} can be employed. Such rules can be used either for description or inference.\nNote that models never depict reality, but are only a hopefully adequate approximation thereof.\nTo construct such models, we need \\, n \\, observations (tokens) comprising a value of the class variable and of all influential variables ({\\bf learning sample}).\n\n\\subsection{Conditional Inference Trees}\\label{page:lowfit}\n\nA variety of different methods exists for constructing classification rules.\nIn the present paper, we concentrate on so-called {\\bf decision trees}, which\ncombine different decisions on individual variables into a set of rules.\nThe type of tree most often used in linguistics is called {\\bf conditional inference tree} ({\\bf c-tree}) (cf., e.g., Tagliamonte\/Baayen, 2012; Gries, 2020). Such trees include only those decisions that significantly improve the correct prediction about the realization of the dependent variable and employ statistical tests for inference. Therefore, such trees are geared towards generalization \/ prediction.\n\nAn example of a c-tree for subject realization can be found in Figure~\\ref{fig:simple}.\nThe tree has to be interpreted as follows: decision variables and p-values of the individual decisions are presented in the `node' corresponding to the decision. e.g.\\ PRN and p < 0.001 in node 1. The terminal nodes indicate the relative frequency of each class in the node, e.g.\\ 20\\% of zeros and 80\\% of realized subject pronouns in node 4. The class with the highest frequency is predicted for each token assigned to the node. This is of particular interest for tokens not used for rule construction, i.e.\\ {\\bf prediction}. Note that the tree in Figure~\\ref{fig:simple} predicts only one of the two classes, namely `realized', i.e.\\ the zero class is never predicted. The statistical repercussions and approaches to meet this problem are discussed in the following section.\n\n\\begin{figure}[h]\n\\centering\n\\vspace{-0.6cm}\n\\includegraphics[scale=0.4]{zeroTreeSimple.pdf}\\\\\n\\raggedright prediction: realized \\hspace{0.9cm} realized \\hspace{1.7cm} realized \\hspace{1.65cm} realized\\\\\n\\caption{Simple c-tree for pronoun realization.}\n\\label{fig:simple}\n\\end{figure}\n\n\\subsection{Evaluating trees with unbalanced classes}\\label{subsec:eval}\n\nEvaluation of classification rules is indispensable to assess their suitability.\nThe standard evaluation measure for classification rules is\\\\\n\n\\hspace{0.6cm}{\\bf overall accuracy} = (no. of correct predictions) \/ (no. of tokens) .\\\\\n\n\\noindent For the tree in Figure~\\ref{fig:simple}, the overall accuracy is 0.914, i.e.\\ very high. This is caused by the strong imbalance between the two classes. The zero class is extremely small (528 tokens) whereas the larger class contains more than ten times as many tokens (5618). Although the smaller class is never predicted, the accuracy is very high but clearly misleading. Two things can be done to improve the predictive power of the model: the model quality has to be assessed in a different way and the high imbalance between the classes has to be resolved (cf. Section~\\ref{subsec:under}).\nFor unbalanced classes model quality can be assessed more adequately:\n\\begin{equation*}\n\\begin{split}\n\\text{\\bf balanced accuracy} & = \\text{mean of accuracies in the two classes} \\\\\n& = {(\\text{accuracy(zero)} + \\text{accuracy(realized)})\/2} .\n\\end{split}\n\\end{equation*}\n\\noindent For our example, the value of the {balanced accuracy} = ({\\bf 0} + 1.0)\/2 = {\\bf 0.5} confirms that the high overall accuracy is misleading, since the average of the accuracies of the two classes amounts to only 50\\%. The low balanced accuracy and the fact that the smaller class is totally ignored in the predictions render the tree unsuitable for linguistic application.\n\n\\subsection{Resampling}\n\nThe suitability of a tree-like classification rule strongly depends on whether it is to be used for description or prediction. Do we only want to describe the data set at hand or do we want to predict the behavior of an unknown subject?\n\nIf classification rules are used for prediction, their {\\bf predictive power} has to be assessed. In order to simulate prediction, we divide our learning sample (i.e.\\ the full data set) into training and test sets. This is called {\\bf resampling}. Different resampling methods exist. In this paper, we utilize two resampling methods: hold-out and subsampling.\n\nThe simplest way of resampling is called {\\bf hold-out}, where one part of the observations (say 1\/3) is randomly withheld from rule construction (training). Subsequently, a rule is constructed on the training set. This rule is then tested on the hold-out and the test accuracy is used for evaluation. The disadvantage of this method is that the hold-out is generated only once and may not be fully representative of the data set.\n\n{\\bf Subsampling} solves this problem by repeating this procedure B times (e.g., B = 200). Then, for each observation (token) the class most often predicted by the B different rules is compared to the observed value of the class variable in the actual data set. The resulting accuracy is used as the evaluation measure for the predictive power of all the B trees generated in the subsampling process.\nIn the following, we introduce a subsampling method which meets the problem of unbalanced classes.\n\n\\subsection{Undersampling}\\label{subsec:under}\n\nIn order to construct a classification rule with acceptable balanced accuracy, we stochastically reduce the larger class, i.e.\\ we apply the method of undersampling.\n{\\bf Undersampling} takes, in the simplest case, the full sample of the smaller class together with a small sample of the larger class for constructing a rule (training). This procedure is repeated B times (cf. Weiss 2004).\n\nFor the {\\bf evaluation of undersampling}, we combine goodness of fit and predictive power. The idea is to compute the overall balanced accuracy on all observations of both classes.\nFor the smaller class, the accuracy is computed on the training sample (fit),\ni.e.\\ the full data sample of the small class.\nFor the larger class, it is computed not\nonly on the training sample (fit), but also on the hold-out from rule construction (prediction).\nThis way, rules from different training samples can be adequately compared and the best\nrule can be identified by looking for the highest balanced accuracy.\n\nIn our example, the smaller class (zero) comprises 528 tokens. For undersampling, we decided to subsample the larger class for training. To create a data set with roughly equally-sized classes, we randomly selected 9\\% of the larger class, i.e.\\ we kept 506 tokens for training. We repeated the process of random subsampling B = 1001 times.\n\n\\section{Statistics meets Linguistics}\\label{sec:int}\n\n\\subsection{Limits on interpretability}\\label{page:uninter}\n\nUnfortunately, undersampling might produce trees with high accuracies that are linguistically uninterpretable.\nFigure~\\ref{fig:uninter} shows such a tree with a balanced accuracy of 0.6898. We later see that this balanced accuracy is only marginally (< 0.02) lower than the best interpretable tree we found (Figure~\\ref{fig:best}). The problem in the uninterpretable tree in Figure~\\ref{fig:uninter} is the split in node 4. Here, the Singapore Chinese (\\textit{S\/C}) cluster with the ancestral English (\\textit{E\/a}) children. This contradicts the linguistic, typologically motivated expectation\nto find differences in pronoun realization between these two\ngroups. First of all, the Chinese languages the Singapore Chinese\nchildren speak as additional languages all allow for zero subjects and it is widely\naccepted that languages acquired bilingually influence each other\nstructurally. Therefore, the Chinese children have a stronger inclination towards zero\nsubjects than monolingual children growing up in England.\nFor the Indian children, the linguistic situation is not as clear since the Indian languages spoken by the children do not as unambiguously prefer the zero subject option as the Chinese languages. If any significant differences are to be found between the\ngroups, these clearly have to be found between the Chinese children and the\nmonolingual ancestral English ones. Therefore, a combination of the two ethnicities in the same group of values has to be excluded.\n\n\\begin{figure}[thp]\n\\centering\n\\hspace*{-0.5cm}\\includegraphics[angle=90,scale=0.5]{treeT7.pdf}\\\\\n\\caption{Uninterpretable tree from undersampling.}\n\\label{fig:uninter}\n\\end{figure}\n\n\nTo make sure that the resulting tree is linguistically interpretable, we have identified the following combinations of values of our variables as not permissible in splits of interpretable trees:\\\\\n\n\\noindent {\\bf Excluded groups of values in our linguistic example}: \\\\\nETH == \\{E\/a, S\/C\\},\\\\ETH == \\{E\/a, E\/m, S\/C\\},\\\\ ETH == \\{E\/a, E\/migr, S\/C\\},\\\\\nETH == \\{E\/a, E\/m, E\/migr, S\/C\\},\\\\\nETH == \\{E\/a, E\/m, E\/migr, S\/C, S\/I\\},\\\\ ETH == \\{E\/a, E\/m, E\/migr, S\/C, S\/m\\},\\\\\nETH == \\{E\/a, E\/m, S\/C, S\/I\\},\\\\ ETH == \\{E\/a, E\/migr, S\/C, S\/I\\},\\\\\nETH == \\{E\/a, S\/C, S\/I\\},\\\\ MLU == \\{1, 3\\}\\\\\n\n\\noindent The above restrictions mainly concern \\textsc{ethnicity} excluding all combinations entailing \\{E\/a, S\/C\\} for the reasons stated above. Moreover, splits with MLU are restricted by excluding the combination of MLU groups 1 and 3. The decision to exclude this combination is again linguistically motivated. Group 1 contains children of 45 months and younger. Language acquisition research has shown that in this age group, children often produce subjectless sentences, not only in Singapore but also in England (e.g. Roeper \\& Rohrbacher 2000). The children in group 3 are all 7 years and older. The children from England in this group have thus clearly moved beyond the zero-subject stage. The children from Singapore still use zero subjects, due to the reasons outlined earlier, but to a lesser extent. This is also due to the decreasing acquisition effect but also since they enter formal schooling and thus more standard language exposure from age 7 onwards. Therefore, the children in MLS groups 1 and 3 show completely different inclinations towards zero subjects.\n\n\\subsection{Interpretability meets predictive power}\n\nOn the basis of the above deliberations, we are able to generate trees with both predictive power and interpretability if the following criteria are met:\n\\begin{itemize}\n\\itemsep0pt\n\\item[] {\\bf Interpretability}: no excluded grouping is included in the tree.\n\\item[] {\\bf Predictive power}: the balanced accuracy is `high enough', e.g.\\ greater than a threshold $c$.\n\\end{itemize}\n\n\\noindent We propose two kinds of tree-like approaches that meet these criteria. As a first step, we use that tree from undersampling which is interpretable and has the highest balanced accuracy. As a second step, we employ {\\bf ensembles of rules} that meet the criteria. Such ensembles are defined in the following way:\n\nThe B rules from undersampling are assessed by the two criteria defined above.\nTo assess predictive power, in our example, the threshold $c$ is set to the value of the median of balanced accuracies in undersampling.\nAll rules from undersampling meeting the two criteria are combined to a so-called {\\bf ensemble} of rules.\nFor each observation, the ensemble of rules predicts that class which is most often `predicted' by the individual trees in the ensemble. These predictions are the basis for calculating balanced accuracies.\n\n\\section{Results}\\label{sec:5}\n\nAll results were created by means of the software R (R Core Team 2019). The decision trees were generated by the R-package `party'. Note that, since the significance limit of 0.01 is used, the trees are smaller than for the standard limit of 0.05. This facilitates straightforward interpretability in order to illustrate our line of argumentation. The following results are generated by means of the R-function PrInDT, that implements our newly developed procedure.\\footnote{The source code of this function can be requested from the first author by e-mail.}\n\nIn Sections 5.1 and 5.2, we show and discuss the three best trees meeting the two criteria discussed in the previous sections. In Section 5.3, we present the results from the ensembles.\n\n\\subsection{The three best individual trees}\\label{subsec:5.1}\n\nThe three best trees presented in Figures~\\ref{fig:best}--\\ref{fig:best3} have balanced accuracies of 0.702, 0.701, and 0.700, respectively. We argue that, for a linguistic study, this is at least an acceptable if not high accuracy (cf. Winter 2020, 77, for a similar line of argumentation). Furthermore, all three trees fulfill the interpretability criterion and reveal a similar structure.\n\nIn all three trees, the most prominent split separates the pronoun \\textit{it} from the rest (\\textit{he, I, she, they, we, you} singular\\footnote{Plural \\textit{you} does not exist in the data.}). If further significant splits are modeled for \\textit{it} (Figures~\\ref{fig:best} and \\ref{fig:best2}), the variables ETH and AGE are involved. The ETH split mainly separates the Singaporean children from the children growing up in England. The second-best tree (Figure~\\ref{fig:best2}) further shows an AGE-related split (node 10) for the ancestral English and mixed-English children. Those children approximately four and a half years and younger use considerably more zero pronouns than the older ones in these groups (Figure~\\ref{fig:best2}; nodes 11 and 12).\n\nFor the realization of the remaining pronouns, MLU, LiBa, and AGE play a role in all three trees. MLU is the most prominent variable for the rest of the pronouns and splits the children into the outliers (OL)\\footnote{Four of the children in the Singapore cohort were identified as outliers on the basis of the MLU-categorization since the syntactic complexity of their utterances does not match the expected age-related performance.} and group 1, i.e. those children younger than 45 months, and the two groups of older children (MLU groups 2 and 3). In all three trees, no further splits occur for groups 2 and 3. The group 1 and OL children are split into further subgroups by LiBa. ETH determines significant splits of the data in the best and second-best tree (Figures~\\ref{fig:best} and \\ref{fig:best2}) and SEX has a significant impact in the best and third-best tree (Figure~\\ref{fig:best} and \\ref{fig:best3}).\n\nLooking into the extreme frequencies in the manifestations of our dependent variable, the trees reveal the following: In the best tree (Figure~\\ref{fig:best}), the highest share of zero pronouns can be found for multilingual male children (node 10). In the second-best tree (Figure~\\ref{fig:best2}), all multilingual children show a particularly high share of zero pronouns (node 8). The third-best tree (Figure~\\ref{fig:best3}) again shows that male gender and multilingual linguistic background are predictors for high rates of zero pronouns, but only for the pronouns \\textit{he, I, she}, and \\textit{they} (node 8).\n\nExtreme frequencies of realized subject pronouns can be found for the following sub-rules. In the best tree (Figure~\\ref{fig:best}), monolingual children older than 28 months exclusively use the realized variant (node 8). Furthermore, the realized variant clearly dominates for pronoun \\textit{you} singular (node 4).\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[angle=90,scale=0.5]{tree1.pdf}\\\\\n\\caption{Best interpretable tree from undersampling.}\n\\label{fig:best}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[angle=90,scale=0.48]{tree2.pdf}\\\\\n\\caption{Second-best interpretable tree from undersampling.}\n\\label{fig:best2}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[angle=90,scale=0.48]{tree3.pdf}\\\\\n\\caption{Third-best interpretable tree from undersampling.}\n\\label{fig:best3}\n\\end{figure}\n\\noindent In the second-best tree (Figure~\\ref{fig:best2}), again, monolingual children older than 28 months exclusively use the realized variant (node 7). MLU groups 2 and 3 also clearly favor the realized variant (node 3). In the third-best tree (Figure~\\ref{fig:best3}), those male children that are multilingual use exceptionally high proportions of realized \\textit{we} and \\textit{you} singular (node 7) and again, MLU groups 2 and 3 make exceptionally high use of the realized variant (node 3).\n\n\\subsection{Discussion of trees}\\label{subsec:5.2}\n\nTo summarize, the three best trees from our {\\bf PrInDT} analysis show that all considered variables (ETH, AGE, SEX, LiBa, MLU, PRN) have an effect on the data to different degrees and in different combinations of sub-rules. The intralinguistic variable PRN has the strongest effect on the results in all three trees. Zero pronouns most prominently occur for the pronoun \\textit{it}, this means either for the semantically empty dummy \\textit{it} (\"It is raining\") or the referential \\textit{it} (\"My cat, it is black\").\n\nMLU, LiBa, and AGE play a role in all three trees in very similar ways: MLU always determines a split into group 1 and the outliers on the one hand and the groups 2 and 3 on the other, with the former being more prone to using zero forms than the latter.\nAGE plays a different role for the pronoun \\textit{it} and the group of all other pronouns. For \\textit{it}, AGE splits the English ancestral and mixed children at 55 months. For the rest of the pronouns and the OL and young children (group 1), AGE splits the children at 28 months. Therefore, hypothesis 1 is confirmed: AGE and MLU are strong predictors for the realization of subject pronouns.\n\nHypothesis 2 is also confirmed: ETH clearly guides the realization of subject pronouns in that Singaporean children make significantly stronger use of zero subjects throughout the age groups (at least in two of the three best trees).\n\nFurther findings that go beyond our initial hypotheses have revealed that SEX has a significant impact on the realization of subject pronouns in the best and the third-best trees, with the male children showing a stronger inclination towards using the zero variant than the female children. This is not surprising either, as it is a common finding in sociolinguistic research that women often use more standard speech forms than men.\n\nAs we have seen, the trees not only have satisfactory if not high balanced accuracies, they are also fully interpretable and consistent in their linguistic findings. This nicely illustrates the methodological power of our {\\bf PrInDT} approach. Note that a clear-cut threshold for linguistic studies does not exist and depends on the individual data set (cf. Section~\\ref{subsec:5.3}).\n\n\\subsection{Ensembles of decision trees}\\label{subsec:5.3}\n\nLet us now consider four possible ensembles of interpretable trees with acceptable predictive power.\\footnote{Note that random forests, as often used in linguistics, are another type of ensembles.} Figure~\\ref{fig:ba} illustrates the distribution of the balanced accuracies of all 1001 trees from undersampling. Note that the range of balanced accuracies is quite small (from 0.6755 to 0.7021). In particular, this means that none of the trees has a totally unacceptable predictive power.\n\n\\begin{figure}[H]\n\\centering\n\\vspace{-0.9cm}\n\\includegraphics[width=0.75\\textwidth]{histba.pdf}\\\\\n\\vspace{-0.3cm}\n\\caption{Histogram of balanced accuracies from undersampling; the median is represented by the red vertical line.}\n\\label{fig:ba}\n\\end{figure}\n\\vspace{-0.2cm}\n\\noindent In this paper, we consider\n\\begin{itemize}\n\\itemsep0pt\n\\item[a)] the ensemble comprising only the three interpretable trees with the highest balanced accuracies (discussed in Section~\\ref{subsec:5.1}),\n\\item[b)] the ensemble consisting of {\\bf all} interpretable trees from undersampling,\n\\item[c)] the ensemble including all interpretable trees with a predictive power greater than the median of the balanced accuracies of all trees from undersampling (c = 0.6904; cf. Figure~\\ref{fig:ba}).\n\\end{itemize}\n\n\\begin{table}[H]\n\\centering\n\\caption{No. of trees and balanced accuracies for the different ensembles}\n\\label{tab:acc}\n\\vspace{0.3cm}\n \\begin{tabular}{l|rr}\n ensemble & trees & accuracy \\\\\n \\hline\n a) & 3 & 0.699\\\\\n b) & 941 & 0.690\\\\\n c) & 137 & 0.696\n \\end{tabular}\n\\end{table}\n\n\\noindent Table~\\ref{tab:acc} summarizes the ensembles and their respective accuracies. Note that only 60 of the 1001 trees turned out to be uninterpretable (cf. b)). Overall, ensemble a), which comprises the three interpretable trees with the best balanced accuracies, comes with the {\\bf best balanced accuracy} (0.699) of the ensembles.\nThe second-best ensemble with an accuracy of 0.696 is ensemble c), which consists of 137 trees, which have greater balanced accuracies than the median of all balanced accuracies.\nEnsemble b), which consists of all interpretable trees from undersampling, includes \n941 trees and has the lowest accuracy of the ensembles (0.690). Still, its accuracy is close to the median. \n\n\\subsection{Discussion of ensembles}\n\nFinally, we would like to argue that ensembles can add important information to the findings of individual trees. Small samples in resampling represent only small parts of the data set. Therefore, trees based on these samples also only depict parts of the reality. Ensembles ensure that as many of these parts are considered as determined by the number of repetitions (in our example 3, 941, and 137). It is further interesting to investigate how strongly the accuracies vary when the underlying samples are changing as this gives some indication of the robustness of the findings.\n\nFrom a linguistic perspective, working with ensembles makes sense for three reasons. Similar to the first statistical reason outlined above, working with trees trained on resampled data sets runs the risk of neglecting important parts of the data. Ensembles are more representative of the data set and thus of the overall population. Furthermore, randomly selected ensembles mirror the randomness of the data collected for a linguistic study. In both cases, only parts of the overall population can be captured with basically everybody or every token having an equal chance of being selected. Moreover, ensembles account for the nature of sociolinguistic variation, since it is neither fully random nor deterministic. As the last 60 years of sociolinguistic research have clearly shown, linguistic variation is guided by intra- and extralinguistic factors but at the same time shows inter- and even intraspeaker variation (e.g. Meyerhoff 2019, 12). Our study once more confirms these findings.\nEnsembles, therefore, aptly represent the strong systematicity of linguistic data on the one hand while at the same time leaving room for randomness and variation.\nThe one shortcoming of ensembles for linguistic studies, however, is that the interpretability of the model is blurred since it is difficult to synthesize the interpretation of a high number of trees.\n\n\\section{Conclusion}\n\nIn this paper, we have seen that statistical models and their evaluation should be integral parts of linguistic analyses, but only if the following crucial points are taken into consideration:\n\\begin{itemize}\n\\itemsep0pt\n\\item[1.]\t Perfectly interpretable trees with allegedly clear and strong linguistic findings can be misleading due to extremely low predictive power (see Figure~\\ref{fig:simple}).\n\\item[2.]\tTrees with high accuracies can be linguistically uninterpretable (see Figure~\\ref{fig:uninter}).\n\\end{itemize}\nTo study and illustrate this finding, we have developed and presented the statistical method {\\bf PrInDT} ({\\bf Pr}ediction and {\\bf In}terpretation with {\\bf D}ecision {\\bf T}rees),\nwhich prescribes interpretability and high predictive power as properties for trees generated by resampling. Either the individual interpretable trees with highest predictive power or a whole ensemble of interpretable trees with `high enough' predictive power are then used for the prediction of relationships between linguistic variables. We successfully applied this method to find prediction rules for the use of subject pronouns (realized vs. zero) depending on extra- and intralinguistic variables.\n\nWe would like to conclude with some general remarks on statistical modeling, not only but in particular in linguistics:\n\\begin{description}\n\\itemsep0pt\n\\item[\\bf Predictive power:] {\\bf Bad models can never be interpreted.} Therefore, evaluation is a crucial step of model building. Significant splits alone do not qualify a model for usage. Interpretation of models with a bad fit or predictive power intrinsically carries the risk of drawing the wrong conclusions.\n\\item[\\bf Interpretability of trees and ensembles:] {\\bf Use interpretable models only.} Restricting models by means of prior knowledge, i.e.\\ findings from earlier studies, or on the basis of theoretical assumptions enhances the interpretability of trees and the homogeneity of ensembles. This way, all trees are in accordance with prior knowledge or theory. If, despite of this, an ensemble turns out to have low predictive power, then our data do not match the underlying theoretical assumptions and the model cannot be interpreted (see Predictive power above).\n\\end{description}\n\n\\noindent Last but not least, we would like to stress that our method {\\bf PrInDT} is not restricted to undersampling but can be applied to any ensemble, e.g. to random forests and other more general resampling methods. Therefore, as a next step we would also like to withhold (small) parts of the smaller class from the training of the trees. This way, the smaller class is also predicted for some tokens. Moreover, in order to facilitate the interpretation of ensembles, we will implement a ranking of the importance of predictors in ensembles, similar to the ranking of variables in random forests.\n\n\\section{References}\n\\begin{hangparas}{.25in}{1}\nBuschfeld, S. (2020), `Children's English in Singapore: Acquisition, Properties, and Use', Routledge.\n\nChan, M. (2020), `English, mother tongue and the Singapore identity', The Straits Times, url: https:\/\/www.straitstimes.com\/opinion\/english-mother-tongue-and-the-spore-identity.\n\nGries, S.Th. (2020), `On classification trees and random forests in corpus linguistics: Some words of caution and suggestions for improvement', Corpus Linguistics and Ling. Theory 16(3): 617--647.\n\nMeyerhoff, M. (2019), `Introducing Sociolinguistics', 3rd edn; Routledge.\n\nR Core Team (2019), `R: A language and environment for statistical\n computing'. R Foundation for Statistical Computing, https:\/\/www.R-project.org\/.\n\nRoeper, T., Rohrbacher, B. (2000). `Null subjects in early child language and the economy of projection'. In S. Powers, C. Hamann (Eds.), Acquisition of Scrambling and Cliticization, 345--397. Berlin: Springer.\n\nTagliamonte, S.A., Baayen, R.H. (2012), `Models, forests, and trees of York English: Was\/were variation as a case study for statistical practice', Language Variation and Change 24, 135--178.\n\nWeiss, G.M. (2004), `Mining with rarity: A unifying framework', ACM SIGKDD Explorations 6: 7--19.\n\nWinter, B. (2020), `Statistics for Linguists. An Introduction using R', Routledge.\n\\end{hangparas}\n\n\\end{document} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n \n\nThe understanding to non-perturbative dynamics of supersymmetric gauge theory\nhas made rapid progress in recent years following the \nseminal contribution by Seiberg and Witten$\\cite{sw}$, combining the ideas \nof holomorphicity$\\cite{sei}$ and duality$\\cite{mo}$.\nThe web of arguments leading to the explicit results consists of a skillful\ncombination of perturbative and nonperturbative arguments, formal\nconsiderations and physical reasoning. It should be checked by explicit\ncomputations, whenever possible, that no unexpected failure of these \narguments occurs. \nIn a recent work we have made an investigation in this\ndirection$\\cite{ref4}$ and this paper is intended as a review.\n\nThe starting point in Seiberg and Witten's work is the low-energy\neffective action of an $N=2$ supersymmetric Yang-Mills theory \nwith the gauge group $SU(2)$ of the following form,\n\\begin{eqnarray}\n\\Gamma =\\frac{1}{16\\pi}\\mbox{Im}{\\int}d^4xd^2{\\theta}d^2\\widetilde{\\theta}\n\\left[ \\frac{1}{2}{\\tau}{\\Psi}^2\n+\\frac{i}{2\\pi}{\\Psi}^2\\,\\ln\\frac{{\\Psi}^2}{{\\Lambda}^2}\n+\\sum_{n=1}^{\\infty}A_n\n\\left(\\frac{{\\Lambda}^2}{{\\Psi}^2}\\right)^{2n}{\\Psi}^2\\right] \\, ,\n\\label{eq1}\n\\end{eqnarray}\nwhere $\\displaystyle \\tau=\\frac{\\theta}{2\\pi}+\\frac{4\\pi i}{g^2}$\nis the modular parameter and ${\\Psi}$ the $N=2$ chiral superfield describing\nthe light degrees of freedom. The logarithmic term represents \nthe one-loop perturbative result and was first obtained by Di Vecchia \net al.${\\cite{dmnp}}$ in a calculation where they coupled the gauge \nsuperfield to an $N=2$ matter supermultiplet and integrated out the \nlatter. Subsequently, Seiberg$\\cite{sei}$ used the anomalous transformation \nbehaviour under $U(1)_R$ and holomorphicity to argue that the full \nlow-energy effective action should take the form ({\\ref{eq1}), where \nthe infinite series arises from nonperturbative instanton contributions.\nThe Seiberg-Witten solution $\\cite{sw}$ gives the explicit form of this \npart of $\\Gamma$.\n\nThe form ({\\ref{eq1}) has been confirmed by calculations in $N=1$ superspace\nand in harmonic superspace, extending the result to nonleading terms in the\nnumber of derivatives $\\cite{dgr,pw,ov,ma,ke}$. Independent confirmation has\nbeen obtained from $M$-theory $\\cite{oo}$. \n\nOur intention is\nto check the perturbative part of the effective \naction in Wess-Zumino gauge by a very down-to-earth calculation. In the Higgs \nphase of the theory, the $SU(2)$ gauge symmetry breaks down to $U(1)$, \nand the super-Higgs mechanism splits the supermultiplet into a massive \none and a massless one. The effective action of the massless fields \nshould be obtained by integrating out the heavy fields. In comparison\nwith other approaches, our method is quite conventional and is along\nthe lines of the standard definition of the low-energy effective theory. \n\nIt should be emphasized that this conventional calculation is very\ncomplicated. Even this modest programme we cannot carry out fully. \nWhat we have actually accomplished is the \ncomputation of the heavy fermion determinant. Reassuringly, we find that the form\n(\\ref{eq1}) is reproduced. Although no unexpected surprises were unearthed\nby our calculation, we still hope that it has some pedagogical value in\nshowing explicitly how the effective action arises. \n\nThe outline of this review is as follows. \nIn section 2 we describe the model and exhibit the Higgs mechanism. Section 3\ncontains the computation of the heavy fermion determinant using the constant\nfield approximation. The detailed calculations of the fermion eigenvalues and\ntheir degeneracies, which contain certain subtle points, are given \nin Appendix B. In section 4 we present a discussion of the results. \nIn the pedagogical vein of this paper, we give in Appendix A the \ncomponent form of the low-energy effective action (\\ref{eq1}). \n\n\n\\section{Splitting of $N=2$ Supermultiplet }\n\n\nThe classical action of \n$N=2$ supersymmetric $SU(2)$ Yang-Mills theory is$\\cite{dhd}$, \n\\begin{eqnarray}\nS&=& {\\int}d^4 x\\left[-\\frac{1}{4}G_{\\mu\\nu}^aG^{\\mu\\nu a}\n+D_{\\mu}{\\varphi}^{\\dagger a}D^{\\mu}{\\varphi}^a\n+i\\overline{\\psi}^a{\\gamma}^{\\mu}D_{\\mu}^{ab}{\\psi}^b\\right.\\nonumber\\\\[2mm]\n&&+\\left. \\frac{ig}{\\sqrt{2}}{\\epsilon}^{abc}\\overline{\\psi}^c\n[(1-{\\gamma}_5){\\varphi}^a\n+(1+{\\gamma}_5){\\varphi}^{\\dagger a}]{\\psi}^b\n+\\frac{g^2}{2}{\\epsilon}^{abc}{\\epsilon}^{ade}\n{\\varphi}^b{\\varphi}^{\\dagger c}{\\varphi}^{d}{\\varphi}^{\\dagger e}\\right]\\,,\n\\label{eq2} \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nG_{\\mu\\nu}^a&=&{\\partial}_{\\mu}K_{\\nu}^a-{\\partial}_{\\nu}K_{\\mu}^a\n-g{\\epsilon}^{abc}K_{\\mu}^bK_{\\nu}^c,~~\n D_{\\mu}{\\varphi}^a={\\partial}_{\\mu}{\\varphi}^a\n-g {\\epsilon}^{abc}K_{\\mu}^b{\\varphi}^c,\n\\nonumber\\\\[2mm]\n{\\varphi}^a&=&\\frac{1}{\\sqrt{2}}(S^a+iP^a),~~ \n{\\varphi}^{\\dagger a}=\\frac{1}{\\sqrt{2}}(S^a-iP^a), ~~a=1,2,3\\,.\\nonumber\n\\end{eqnarray}\n The bosonic part of the action (\\ref{eq2}) \nis just the Georgi-Glashow model\nin the Bogomol'nyi-Prasad-Sommerfield (BPS) limit. \nIn addition to the fermionic term and Yukawa interaction term, this action\n has the scalar potential\n\\begin{eqnarray}\nV(\\varphi)=-\\frac{g^2}{2}{\\epsilon}^{abc}{\\epsilon}^{ade}\n{\\varphi}^b{\\varphi}^{\\dagger c}\n{\\varphi}^d{\\varphi}^{\\dagger e}\n{\\equiv}g^2\\mbox{Tr}\\left([\\varphi,{\\varphi}^{\\dagger}]\\right)^2\\,.\n\\end{eqnarray}\nThe unbroken supersymmetry requires that in the ground state the\nscalar potential must vanish, which leads to\n\\begin{eqnarray}\n[\\varphi,{\\varphi}^{\\dagger}]&=&0\\,.\n\\label{eq:h2}\n\\end{eqnarray}\n(\\ref{eq:h2}) means that ${\\varphi}^{\\dagger}$ and ${\\varphi}$ \ncommute. Since the theory is gauge invariant, we can always choose$\\cite{dhd}$ \n\\begin{eqnarray}\n\\langle S^a \\rangle =v{\\delta}^{a3},~~\\langle P^a \\rangle =0 \\, ,\n\\label{eq:c1}\n\\end{eqnarray}\nwhere $v$ is a real constant. \nFor $v{\\neq}0$ the theory is in the Higgs phase\nand exhibits a spontaneous breaking of the gauge symmetry.\nIn a unitary gauge \n\\begin{eqnarray}\nS^{T}=\\left(0,0, S+v \\right) \\, .\n\\label{ug} \n\\end{eqnarray}\nThe corresponding classical Lagrangian can be written as follows,\n\\begin{eqnarray}\n{\\cal L}={\\cal L}_V+{\\cal L}_S+{\\cal L}_P+{\\cal L}_F+{\\cal L}_Y\\, ,\n\\end{eqnarray}\nwhere ${\\cal L}_V$, ${\\cal L}_S$, ${\\cal L}_P$, ${\\cal L}_F$ and ${\\cal L}_Y$\ndenote respectively the vector field, the scalar field,\nthe scalar potential, \nthe fermionic and the Yukawa interaction parts, \n\\begin{eqnarray}\n{\\cal L}_V&=& -\\frac{1}{4}({\\partial}^{\\mu}A^{\\nu}\n-{\\partial}^{\\nu}A^{\\mu}) ({\\partial}_{\\mu}A_{\\nu}-{\\partial}_{\\nu}A_{\\mu}) \n-\\frac{1}{2}({\\partial}_{\\mu}W_{\\nu}^+-{\\partial}_{\\nu}W_{\\mu}^+) \n({\\partial}^{\\mu}W^{-\\nu}-{\\partial}^{\\nu}W^{-\\mu})\\nonumber \\\\[2mm]\n&&-i g[({\\partial}^{\\mu}W_{\\nu}^+W_{\\mu}^-\n-{\\partial}^{\\mu}W_{\\nu}^-W_{\\mu}^+)A^{\\nu}+\n({\\partial}_{\\nu}W_{\\mu}^-W^{+\\mu}\n-{\\partial}_{\\nu}W^+_{\\mu}W^{\\mu -})A^{\\nu}\\nonumber\\\\[2mm]\n&&+({\\partial}^{\\mu}A^{\\nu}\n-{\\partial}^{\\nu}A^{\\mu})W_{\\mu}^+W_{\\nu}^-]\n+g^2(-W_{\\mu}^+W^{-\\mu}A_{\\nu}A^{\\nu}+W_{\\mu}^+W_{\\nu}^-A^{\\mu}A^{\\nu})\n\\nonumber\\\\\n&&+ \\frac{g^2}{2}W^{+\\mu}W^{-\\nu}(W^+_{\\mu}W^-_{\\nu}-W^-_{\\mu} W^+_{\\nu});\n\\label{eq8}\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\cal L}_S&=&\\frac{1}{2}\\partial_{\\mu}P\\partial^{\\mu}P\n+\\frac{1}{2}\\partial_{\\mu}S\\partial^{\\mu}S+\\partial_{\\mu}P^+\\partial^{\\mu}P^-\n+igA^{\\mu}(\\partial_{\\mu}P^-P^+-\\partial_{\\mu}P^+P^-)\\nonumber\\\\\n&&+igP(\\partial^{\\mu}P^+W_{\\mu}^--\\partial^{\\mu}P^-W_{\\mu}^+)+\nig\\partial^{\\mu}P(W_{\\mu}^+P^--W_{\\mu}^-P^+)+g^2P^2W^{+\\mu}W^-_{\\mu}\\nonumber\\\\\n&&+g^2(S+v)^2W^{+\\mu}W^-_{\\mu}+g^2A^{\\nu}A_{\\nu}P^{+}P^--g^2(W_{\\mu}^+P^--\nW_{\\mu}^-P^+)A^{\\mu}P\\nonumber\\\\\n&&-\\frac{g^2}{2}(W^{\\mu +}P^--W^{\\mu -}P^+)^2.\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\cal L}_P=g^2(S+v)^2P^+P^-,\n\\end{eqnarray}\nwhere the various quantities are defined as follows:\n\\begin{eqnarray}\n&& W_{\\mu}^+{\\equiv}\\frac{1}{\\sqrt{2}}(K_{\\mu}^{1}\n-iK_{\\mu}^{2})~,~\nW_{\\mu}^-{\\equiv}\\frac{1}{\\sqrt{2}}(K_{\\mu}^{1}\n+iK_{\\mu}^{2})~,~K_{\\mu}^3 {\\equiv}A_{\\mu} \\,;\\nonumber\\\\\n&&P^+ {\\equiv}\\frac{1}{\\sqrt{2}}(P^1-iP^2)~,~P^-{\\equiv}\n \\frac{1}{\\sqrt{2}}(P^1+iP^2)~,~P^3{\\equiv}P.\n\\end{eqnarray}\nThe above Lagrangians clearly show that $W_{\\mu}^{\\pm}$ and $P^{\\pm}$\nbecome massive with mass $m{\\equiv}|gv|$ while $A_{\\mu}$, $S$ and $P$\nremain massless.\n\nUp to some total derivative terms, the bosonic part of the Lagrangian \ncan be rewritten in the following form:\n\\begin{eqnarray} \n{\\cal L}_B&=&{\\cal L}_V+{\\cal L}_S+{\\cal L}_P\\nonumber\\\\\n&=&-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}+\\frac{1}{2}\\partial_{\\mu}P\\partial^{\\mu}P\n+\\frac{1}{2}\\partial_{\\mu}S\\partial^{\\mu}S+\\frac{1}{2}\nW^{+\\mu}\\left[\n{\\eta}_{\\mu\\nu}D^{\\dagger\\alpha}D_{\\alpha}-D_{\\nu}^{\\dagger}D_{\\mu}\n-igF_{\\mu\\nu}\\right]W^{-\\nu}\\nonumber\\\\[2mm]\n&&+\\frac{1}{2}W^{-\\mu}\\left[\n{\\eta}_{\\mu\\nu}D^{\\alpha}D^{\\dagger}_{\\alpha}\n-D_{\\nu}D_{\\mu}^{\\dagger}+igF_{\\mu\\nu}\\right]W^{+\\nu}\n+g^2[P^2+(S+v)^2]W_{\\mu}^+W^{\\mu -}\\nonumber\\\\\n&&+\\frac{1}{2}P^+(-\\partial^{\\mu}\\partial_{\\mu}\n+2igA_{\\mu}\\partial^{\\mu}+g^2A_{\\mu}A^{\\mu})P^-\n+\\frac{1}{2}P^-(-\\partial^{\\mu}\\partial_{\\mu}\n-2igA_{\\mu}\\partial^{\\mu}+g^2A_{\\mu}A^{\\mu})P^+\n\\nonumber\\\\\n&&+\\frac{1}{2}W_{\\mu}^+(-igP\\partial^{\\mu}+ig\\partial^{\\mu}P-g^2A_{\\mu}P)P^-\n+\\frac{1}{2}P^-(igP\\partial^{\\mu}+2ig\\partial^{\\mu}P-g^2A_{\\mu}P)W_{\\mu}^+\n\\nonumber\\\\\n&&+\\frac{1}{2}P^+(-2ig\\partial^{\\mu}P-igP\\partial^{\\mu}-g^2A^{\\mu}P)W_{\\mu}^-\n+\\frac{1}{2}W_{\\mu}^-(-ig\\partial^{\\mu}P+igP\\partial^{\\mu}-g^2A^{\\mu}P)P^+\n\\nonumber\\\\\n&&+ \\frac{g^2}{2}W^{+\\mu}W^{-\\nu}(W^+_{\\mu}W^-_{\\nu}-W^-_{\\mu} W^+_{\\nu})\n-\\frac{g^2}{2}(W^+_{\\mu}P^--W^-_{\\mu}P^+)^2\\nonumber\\\\ \n&=&-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}\n+\\partial_{\\mu}{\\phi}^{*}\\partial^{\\mu}{\\phi}\n+\\frac{1}{2}W^{+\\mu}{\\Delta}_{\\mu\\nu}W^{-\\nu}\n+\\frac{1}{2}W^{-\\mu}{\\Delta}^{\\dagger}_{\\mu\\nu}W^{+\\nu}\n+\\frac{1}{2}P^+\\Delta P^-\\nonumber\\\\\n&&+\\frac{1}{2}P^-\\Delta^{\\dagger}P^+\n+\\frac{1}{2}W^{+\\mu}\\Delta_{\\mu}P^-\n+\\frac{1}{2}P^-\\widetilde{\\Delta}_{\\mu}W^{+\\mu}\n+\\frac{1}{2}P^+\\widetilde{\\Delta}_{\\mu}^{\\dagger}W^{-\\mu}\n+\\frac{1}{2}W^{-\\mu}\\Delta_{\\mu}^{\\dagger}P^+\\nonumber\\\\\n&&+ \\frac{g^2}{2}W^{+\\mu}W^{-\\nu}(W^+_{\\mu}W^-_{\\nu}-W^-_{\\mu} W^+_{\\nu})\n-\\frac{g^2}{2}(W^+_{\\mu}P^--W^-_{\\mu}P^+)^2 \\, ,\n\\end{eqnarray} \nwhere \n\\begin{eqnarray} \n{\\Delta}_{\\mu\\nu}\n&{\\equiv}&{\\eta}_{\\mu\\nu}D^{\\dagger\\alpha}D_{\\alpha}-D_{\\nu}^{\\dagger}D_{\\mu}\n-igF_{\\mu\\nu}+g^2|\\sqrt{2}\\phi+v|^2{\\eta}_{\\mu\\nu},\\nonumber\\\\[2mm]\n{\\Delta}_{\\mu\\nu}^{\\dagger}\n&=&{\\eta}_{\\mu\\nu}D^{\\alpha}D_{\\alpha}^{\\dagger}\n-D_{\\nu}D_{\\mu}^{\\dagger}\n+igF_{\\mu\\nu}+g^2|\\sqrt{2}\\phi+v|^2{\\eta}_{\\mu\\nu};\\nonumber\\\\[2mm]\n\\Delta_{\\mu}&{\\equiv}&-igP\\partial^{\\mu}+ig\\partial^{\\mu}P-g^2A_{\\mu}P,~\n\\widetilde{\\Delta}_{\\mu}{\\equiv}igP\\partial^{\\mu}\n+2ig\\partial^{\\mu}P-g^2A_{\\mu}P,\n\\nonumber\\\\\n\\Delta_{\\mu}^{\\dagger}&=&-ig\\partial^{\\mu}P+igP\\partial^{\\mu}-g^2A^{\\mu}P,\n~\\widetilde{\\Delta}_{\\mu}^{\\dagger}\n=-2ig\\partial^{\\mu}P-igP\\partial^{\\mu}-g^2A^{\\mu}P;\\nonumber\\\\\n\\Delta &=& -\\partial^{\\mu}\\partial_{\\mu}\n+2igA_{\\mu}\\partial^{\\mu}+g^2A_{\\mu}A^{\\mu},~\n\\Delta^{\\dagger}=-\\partial^{\\mu}\\partial_{\\mu}\n-2igA_{\\mu}\\partial^{\\mu}+g^2A_{\\mu}A^{\\mu}; \\nonumber\\\\\nD_{\\mu}&=&\\partial_{\\mu}-igA_{\\mu}, ~D_{\\mu}^{\\dagger}\n=\\partial_{\\mu}+igA_{\\mu}; ~\\phi{\\equiv}\\frac{1}{\\sqrt{2}}(S+iP).\t\n\\end{eqnarray}\n\nTo explicitly show that the spinor fields split into massive and massless ones, \nwe need some operations on ${\\cal L}_F$ and ${\\cal L}_Y$. The fermionic \npart is\n\\begin{eqnarray}\n{\\cal L}_F&=&i\\overline{\\psi}^1{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\psi}^1+\ni\\overline{\\psi}^2{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\psi}^2\n+i\\overline{\\psi}^3{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\psi}^3\\nonumber\\\\[2mm]\n&&+ \\frac{g}{\\sqrt{2}}\\overline{\\psi}^1(W_{\\mu}^+-W_{\\mu}^-)\n{\\gamma}^{\\mu}{\\psi}^3+ig \\overline{\\psi}^1A_{\\mu}{\\gamma}^{\\mu}{\\psi}^2\n\\nonumber\\\\[2mm]\n&&+ig \\overline{\\psi}^1A_{\\mu}{\\gamma}^{\\mu}{\\psi}^2+\n \\frac{ig}{\\sqrt{2}}\\overline{\\psi}^2\n{\\gamma}^{\\mu}(W_{\\mu}^++W_{\\mu}^-){\\psi}^3\n\\nonumber\\\\[2mm]\n&&-\\frac{ig}{\\sqrt{2}}\\overline{\\psi}^3{\\gamma}^{\\mu}\n(W_{\\mu}^++W_{\\mu}^-){\\psi}^2-\n\\frac{g}{\\sqrt{2}}\\overline{\\psi}^3{\\gamma}^{\\mu}\n(W_{\\mu}^+-W_{\\mu}^-){\\psi}^1 \\, .\n\\label{fp}\n\\end{eqnarray}\nAs for the Yukawa part, we first write it in terms of chiral spinors,\n\\begin{eqnarray}\n{\\cal L}_Y=i\\sqrt{2}gf^{abc}\\overline{\\psi}_L^c{\\varphi}^a{\\psi}_R^b+i\n\\sqrt{2}gf^{abc}\\overline{\\psi}_R^c{\\varphi}^{\\dagger a}{\\psi}_L^b \\, ,\n\\label{yu}\n\\end{eqnarray}\nwhere ${\\psi}_L=\\displaystyle \\frac{1}{2}(1-{\\gamma}_5){\\psi}$ and \n${\\psi}_R=\\displaystyle \\frac{1}{2}(1+{\\gamma}_5){\\psi}$.\nIn the unitary gauge, Eq.(\\ref{yu}) becomes\n\\begin{eqnarray}\n{\\cal L}_Y&=&ig(\\sqrt{2}{\\phi}+v)\n(\\overline{\\psi}_L^2{\\psi}_R^1-\\overline{\\psi}_L^1{\\psi}_R^2)\n+ig (\\sqrt{2}{\\phi}^*+v) (\\overline{\\psi}_R^2{\\psi}_L^1\n-\\overline{\\psi}_R^1{\\psi}_L^2)\\nonumber\\\\\n&&+\\frac{g}{\\sqrt{2}}(P^++P^-)\\left[(\\overline{\\psi}_R^3{\\psi}_L^2\n-\\overline{\\psi}_R^2{\\psi}_L^3)-(\\overline{\\psi}_L^3{\\psi}_R^2\n-\\overline{\\psi}_L^2{\\psi}_R^3)\\right]\\nonumber\\\\\n&&+\\frac{ig}{\\sqrt{2}}(P^+-P^-)\\left[(\\overline{\\psi}_R^1{\\psi}_L^3\n-\\overline{\\psi}_R^3{\\psi}_L^1)-(\\overline{\\psi}_L^1{\\psi}_R^3\n-\\overline{\\psi}_L^3{\\psi}_R^1)\\right] \\, .\n\\end{eqnarray}\nWith the combination\n\\begin{eqnarray}\n{\\Psi}_1{\\equiv} \\frac{1}{\\sqrt{2}}({\\psi}^1+i{\\psi}^2)\\,,\\,\n {\\Psi}_2{\\equiv} \\frac{1}{\\sqrt{2}}({\\psi}^1-i{\\psi}^2)\\, ,\\,\n{\\Psi}{\\equiv}{\\psi}^3 \\, , \n\\end{eqnarray}\n${\\cal L}_F$ and ${\\cal L}_Y$ can be formulated in these new fields, \n\\begin{eqnarray}\n{\\cal L}_Y&=&\n-g\\overline{\\Psi}_1\\left[\\frac{1}{\\sqrt{2}}(1-\\gamma_5)\\phi\n+\\frac{1}{\\sqrt{2}}(1+\\gamma_5)\\phi^*+v\\right]{\\Psi}_1\\nonumber\\\\\n&&+g\\overline{\\Psi}_2 \\left[\\frac{1}{\\sqrt{2}}(1-\\gamma_5)\\phi\n+\\frac{1}{\\sqrt{2}}(1+\\gamma_5)\\phi^*+v\\right]{\\Psi}_2\n\\nonumber\\\\\n&&- igP^+\\overline{\\Psi}\\gamma_5\\Psi_1+igP^-\\overline{\\Psi}\\gamma_5\\Psi_2\n-ig\\overline{\\Psi}_1 \\gamma_5\\Psi P^-+ig\\overline{\\Psi}_2\\gamma_5\\Psi P^+ \\, ,\n\\\\[2mm]\n{\\cal L}_F&=&i\\overline{\\Psi}_1{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\Psi}_1+\ni\\overline{\\Psi}_2{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\Psi}_2\n+i\\overline{\\Psi}{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\Psi}\\nonumber\\\\[2mm]\n&&+g\\overline{\\Psi}_1{\\gamma}^{\\mu}A_{\\mu}{\\Psi}_1-\ng\\overline{\\Psi}_2{\\gamma}^{\\mu}A_{\\mu}{\\Psi}_2 \\nonumber\\\\[2mm]\n&&+g\\overline{\\Psi}_2{\\gamma}^{\\mu}W_{\\mu}^+{\\Psi}-\ng\\overline{\\Psi}_1{\\gamma}^{\\mu}W_{\\mu}^-{\\Psi}\\nonumber\\\\[2mm]\n&&-g\\overline{\\Psi}{\\gamma}^{\\mu}W_{\\mu}^+{\\Psi}_1+\ng\\overline{\\Psi}{\\gamma}^{\\mu}W_{\\mu}^-{\\Psi}_2\\, .\n\\end{eqnarray}\nSo now the whole classical action is given by the Lagrangian\n\\begin{eqnarray}\n{\\cal L}&=&-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}+\\partial_{\\mu}\\phi^*\n\\partial^{\\mu}\\phi +i\\overline{\\Psi}{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\Psi}+\n\\frac{1}{2}W^{+\\mu}{\\Delta}_{\\mu\\nu}W^{-\\nu}\n+\\frac{1}{2}W^{-\\mu}{\\Delta}^{\\dagger}_{\\mu\\nu}W^{+\\nu}\\nonumber\\\\\n&&+\\frac{1}{2}P^+\\Delta P^-+\\frac{1}{2}P^-{\\Delta}^{\\dagger}P^+\n+\\frac{1}{2}W^{+\\mu}\\Delta_{\\mu}P^-+\\frac{1}{2}P^-\n\\widetilde{\\Delta}_{\\mu}W^{+\\mu}\n\\nonumber\\\\\n&&+\\frac{1}{2}P^+\\widetilde{\\Delta}_{\\mu}^{\\dagger}W^{-\\mu}\n+\\frac{1}{2}W^{-\\mu}\\Delta_{\\mu}^{\\dagger}P^+ \n+\\overline{\\Psi}_1\\Delta_F{\\Psi}_1\n+\\overline{\\Psi}_2\\widetilde{\\Delta}_F{\\Psi}_2\n\\nonumber \\\\[2mm]\n&&- igP^+\\overline{\\Psi}\\gamma_5\\Psi_1+igP^-\\overline{\\Psi}\\gamma_5\\Psi_2\n-ig\\overline{\\Psi}_1 \\gamma_5\\Psi P^-+ig\\overline{\\Psi}_2\\gamma_5\\Psi P^+ \n\\nonumber\\\\[2mm]\n&&+ g\\overline{\\Psi}_2{\\gamma}^{\\mu}W_{\\mu}^+{\\Psi}-\ng\\overline{\\Psi}_1{\\gamma}^{\\mu}W_{\\mu}^-{\\Psi}\n-g\\overline{\\Psi}{\\gamma}^{\\mu}W_{\\mu}^+{\\Psi}_1+\ng\\overline{\\Psi}{\\gamma}^{\\mu}W_{\\mu}^-{\\Psi}_2\\nonumber\\\\\n&&+ \\frac{g^2}{2}W^{+\\mu}W^{-\\nu}(W^+_{\\mu}W^-_{\\nu}-W^-_{\\mu} W^+_{\\nu})\n-\\frac{g^2}{2}(W^+_{\\mu}P^--W^-_{\\mu}P^+)^2 \\, \n\\label{eq20}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\Delta_F&{\\equiv}&i{\\gamma}^{\\mu}D_{\\mu}\n-\\frac{g}{\\sqrt{2}}(1-\\gamma_5)\\phi-\\frac{g}{\\sqrt{2}}(1+\\gamma_5)\\phi^*-gv,\n\\nonumber\\\\\n\\widetilde{\\Delta}_F&{\\equiv}&i{\\gamma}^{\\mu}D_{\\mu}^{\\dagger}\n+\\frac{g}{\\sqrt{2}}(1-\\gamma_5)\\phi +\\frac{g}{\\sqrt{2}}(1+\\gamma_5)\\phi^*+gv.\n\\end{eqnarray}\n\n\\section{Low-energy Effective Action: Calculation of the Fermionic \nDeterminant in Constant Field Approximation}\n\nThe standard definition of the low-energy effective action is given by \n\\begin{eqnarray}\n\\mbox{exp}\\left\\{i\\,{\\Gamma}_{\\rm eff}\n[A_{\\mu},{\\phi},{\\Psi},\\overline{\\Psi}]\\right\\}\n{\\equiv}{\\int} {\\cal D} W_{\\mu}^+ {\\cal D} W_{\\mu}^-\n{\\cal D}\\overline{\\Psi}_1{\\cal D}\\overline{\\Psi}_2 {\\cal D}{\\Psi}_1\n{\\cal D}{\\Psi}_2 {\\cal D}P^+{\\cal D}P^- \\mbox{exp}\n\\left[i\\, \\int \\, d^4 x\\, {\\cal L}\\right]\\, .\n\\end{eqnarray}\nAt tree level\n\\begin{eqnarray}\n{\\Gamma}_{\\rm eff}^{(0)}=S_{\\rm tree}={\\int}\\,d^4x\\left[ -\\frac{1}{4}\nF_{\\mu\\nu}F^{\\mu\\nu}\n+{\\partial}^{\\mu}{\\phi}^*{\\partial}_{\\mu}{\\phi}+\ni\\overline{\\Psi}{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\Psi}\\right]\\,.\n\\end{eqnarray}\nAt one-loop level, the integration over the heavy modes \nwill lead to the determinants of the dynamical operators. \nIn practical calculation we cannot evaluate\nthe determinant exactly. Here we shall\nemploy a technique called constant field \napproximation to compute the determinant, which\nwas invented by Schwinger${\\cite{sch}}$ \nand later was used in\nin \\cite{dmnp} and \\cite{ddds} to extract \nthe anomaly term in $N=2$ supersymmetric Yang-Mills\ntheory and the one-loop effective action of the supersymmetric $CP^{N-1}$ model.\nTo apply this method\n we first rewrite the the quadratic part of the\n classical action (\\ref{eq20}) as \n\\begin{eqnarray}\nS_{\\rm quad}=S_{\\rm tree}+{\\int}d^4x\\left(\\Phi^{\\dagger} M_{bb} \\Phi+\n\\overline{\\widetilde{\\Psi}}M_{fb}\\Phi +\\Phi^{\\dagger}M_{bf}\\widetilde{\\Psi}\n+\\overline{\\widetilde{\\Psi}}M_{ff}\\widetilde{\\Psi}\\right),\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\Phi&{\\equiv}&\\left(\\begin{array}{c} W^{-\\mu}\\\\W^{+\\mu}\\\\P^-\\\\P^+\\end{array}\\right),\n~~\\widetilde{\\Psi}{\\equiv}\\left(\\begin{array}{c}\\Psi_1\\\\ \\Psi_2\\\\\n\\Psi_1\\\\ \\Psi_2\\\\ \\end{array}\\right);\\nonumber\\\\\nM_{bb}&{\\equiv}&\\frac{1}{2}\n\\left(\\begin{array}{cccc}\n{\\Delta}_{\\mu\\nu} & 0 &\\Delta_{\\mu} & 0\\\\\n0 & {\\Delta}^{\\dagger}_{\\mu\\nu} &0 &\\Delta_{\\mu}^{\\dagger} \\\\\n\\widetilde{\\Delta}_{\\nu}^{\\dagger} & 0 &\\Delta &0 \\\\\n0 & \\widetilde{\\Delta}_{\\nu} & 0 & \\Delta^{\\dagger}\n\\end{array}\\right);\\nonumber\\\\\nM_{fb}&{\\equiv}&\\frac{1}{2} \\left(\\begin{array}{cccc}\n-g\\gamma_{\\mu}\\Psi & 0 & -ig\\gamma_5 \\Psi & 0 \\\\\n0 & g\\gamma_{\\mu}\\Psi & 0 & ig\\gamma_5 \\Psi \\\\\n-g\\gamma_{\\mu}\\Psi & 0 & -ig\\gamma_5 \\Psi & 0\\\\\n0 & g\\gamma_{\\mu}\\Psi & 0 & ig\\gamma_5 \\Psi\\\\\n \\end{array}\\right); \\nonumber\\\\\nM_{bf}&{\\equiv}&\\frac{1}{2} \\left(\\begin{array}{cccc}\n-g\\overline{\\Psi}\\gamma_{\\mu} & 0 & -g\\overline{\\Psi}\\gamma_{\\mu} & 0 \\\\\n0 &g\\overline{\\Psi}\\gamma_{\\mu} & 0 & g\\overline{\\Psi}\\gamma_{\\mu} \\\\\n-ig\\overline{\\Psi} \\gamma_5 & 0 & -ig\\overline{\\Psi} \\gamma_5 & 0\\\\\n0 & ig\\overline{\\Psi} \\gamma_5 & 0 & ig\\overline{\\Psi} \\gamma_5\n\\end{array}\\right);\\nonumber\\\\\nM_{ff}&{\\equiv}&\\frac{1}{2} \\left(\\begin{array}{cccc}\n\\Delta_F & 0 & 0 & 0\\\\\n0 & \\widetilde{\\Delta}_F & 0 & 0 \\\\\n0 & 0 & \\Delta_F & 0 \\\\\n0 & 0 & 0 &\\widetilde{\\Delta}_F\n \\end{array}\\right).\n\\label{eq27x}\n\\end{eqnarray}\nUsing the standard formulas\n\\begin{eqnarray} \nI&=&\\int {\\cal D}b^{\\dagger}{\\cal D}b{\\cal D}\\overline{f}{\\cal D}f\\,\n\\mbox{exp}\\left[\\int (dx)\n\\left(b^{\\dagger}M_{bb}b+\\overline{f}M_{fb}b+b^{\\dagger}M_{bf}f\n+\\overline{f}M_{ff}f\\right)\\right] \\nonumber\\\\[2mm]\n&=&\\int{\\cal D}b^{\\dagger}{\\cal D}b{\\cal D}\\overline{f}{\\cal D}f\\,\n \\mbox{exp}\\left\\{\\int (dx) \\left[b^{\\dagger}(M_{bb}-M_{bf}M_{ff}^{-1}M_{fb})b\n\\right.\\right.\\nonumber\\\\\n&&+\\left.\\left.(\\overline{f}\n+b^{\\dagger}M_{bf}M_{ff}^{-1})M_{ff}(M_{ff}^{-1}M_{fb}b+f)\n\\right]\\right\\}\n\\nonumber\\\\[2mm]\n&=&\\det M_{ff}\\det{}^{-1}(M_{bb}-M_{bf}M_{ff}^{-1}M_{fb}); \\nonumber\\\\\n\\det M&=&\\mbox{exp}\\,\\mbox{Tr}\\,\\ln M,\n\\end{eqnarray}\n$b$ and $f$ representing the general \nbosonic and fermionic fields, respectively,\nwe get\n\\begin{eqnarray}\nZ[A, \\phi,\\Psi,\\overline{\\Psi}]\n&=&\\mbox{exp}\\left\\{i\\,{\\Gamma}_{\\rm eff}\n[A_{\\mu},{\\phi},{\\Psi},\\overline{\\Psi}]\\right\\}\n{\\equiv}{\\int} {\\cal D} W_{\\mu}^+ {\\cal D} W_{\\mu}^-\n{\\cal D}\\overline{\\Psi}_1{\\cal D}\\overline{\\Psi}_2 {\\cal D}{\\Psi}_1\n{\\cal D}{\\Psi}_2 \\mbox{exp}\\left[ iS \\right] \\nonumber\\\\\n&=& \\mbox{exp}\\left[iS_{\\rm tree}\\right]\\det M_{ff}\\det{}^{-1}(M_{bb}\n-M_{bf}M_{ff}^{-1}M_{fb})\\nonumber\\\\\n&=&\\mbox{exp}\\left[iS_{\\rm tree}+\\mbox{Tr}\\ln M_{ff}-\n\\mbox{Tr}\\ln(M_{bb}-M_{bf}M_{ff}^{-1}M_{fb})\\right];\\nonumber\\\\\n{\\Gamma}_{\\rm eff}&=&S_{\\rm tree}-i\\left[\\mbox{Tr}\\ln M_{ff}-\n\\mbox{Tr}\\ln(M_{bb}-M_{bf}M_{ff}^{-1}M_{fb})\\right].\n\\label{eq27} \n\\end{eqnarray}\n\nThe following task is to evaluate the above determinants. \nLet us first consider the fermionic \npart. Since $M_{ff}$ has the form of a reducible matrix,\n\\begin{eqnarray} \n\\det M_{ff}=\\frac{1}{16}(\\det\\Delta_F)^2(\\det\\widetilde{\\Delta}_F)^2\n=\\frac{1}{16}\\mbox{exp}[2(\\mbox{Tr}\\ln\\Delta_F\n+\\mbox{Tr}\\ln\\det\\widetilde{\\Delta}_F)].\n\\end{eqnarray}\nNow we switch on the constant field approximation to work out the eigenvalues\nand eigenvectors of the above operators and further evaluate the determinant.\nWe choose only the third components of the electric\nand magnetic fields to be the constants different from zero,\n\\begin{eqnarray} \n-E_3=F^{03}{\\neq}0, ~~B_3=F^{12}{\\neq}0,\n\\end{eqnarray}\nand $\\phi$ the non-vanishing constant field.\nConsequently, the potential becomes\n\\begin{eqnarray} \nA^1=-F^{12}x_2, ~~A^3=-F^{30}x_0, ~~A^0=A^2=0.\n\\end{eqnarray}\nTo get the eigenvalues of the operators, it is necessary\nto rotate into Euclidean space,\n\\begin{eqnarray}\n&& x^4=x_4=-ix^0, ~~\\partial_0=\\frac{\\partial}{\\partial x^0}\n=i\\frac{\\partial}{\\partial x^4},\\nonumber\\\\\n&& f^{34}=f_{34}=iF^{30}, ~~f^{12}=f_{12}=F^{12}.\n\\end{eqnarray}\nLet us first consider $\\det \\Delta_F$.\nThe eigenvalue equation for $\\Delta_F$ is\n\\begin{eqnarray}\n\\Delta_F\\psi(x)=\\left[i\\gamma^{\\mu}D_{\\mu}-\\frac{g}{\\sqrt{2}}(1-\\gamma_5)\\phi\n-\\frac{g}{\\sqrt{2}}(1+\\gamma_5)\\phi^* -gv\\right]\\psi(x) \n=\\omega \\psi_1,\n\\label{eq32}\n\\end{eqnarray}\nwhere $\\psi$ is a four-component spinor wave function.\nIn order to get normalizable eigenstates, we consider the system in a\nbox of finite size $L$ in the $x_1$ and $x_3$ directions with periodic\nboundary conditions, so the eigenvector should be of the following form,\n\\begin{eqnarray}\n\\psi(x)=\\frac{1}{L}e^{ip_1{\\cdot}x_1}e^{ip_3{\\cdot}x_3}{\\chi}(x_2,x_4), \n\\nonumber\\\\\np_1=\\frac{2\\pi l}{L}, ~~p_3=\\frac{2\\pi k}{L},~~k,l=\\mbox{integers}.\n\\end{eqnarray}\nTo find the eigenvalues and eigenvectors, \nwe write the operators and the wave function in two-component forms,\n\\begin{eqnarray}\n\\Delta_F=\\left(\\begin{array}{cc} -g(\\sqrt{2}\\phi^*+v){\\bf 1} & \\Delta^-\\\\\n\\Delta^+ & -g(\\sqrt{2}\\phi +v){\\bf 1} \\end{array}\\right),~~~\n\\chi =\\left(\\begin{array}{c}\\chi_{1}\\\\ \\chi_{2}\\end{array}\\right),\n\\end{eqnarray}\nwhere ${\\bf 1}$ is the $2{\\times}2$ identity matrix and\n\\begin{eqnarray}\n\\Delta^{\\pm}=\\partial_4{\\pm}i\\left[\\sigma_1(\\partial_1+igf_{12}x_2)\n+\\sigma_2\\partial_2+\\sigma_3(\\partial_3+igf_{34}x_4)\\right].\n\\end{eqnarray}\nThe eigenvalue equation (\\ref{eq32}) \nis thus reduced to the following set of equations,\n\\begin{eqnarray}\n-g(\\sqrt{2}\\phi^*+v){\\chi}_{1}+ \\Delta^-{\\chi}_{2}\n&=&\\omega {\\chi}_{1},\\nonumber\\\\\n\\Delta^+{\\chi}_{1}-g(\\sqrt{2}\\phi +v){\\chi}_{2}&=&\\omega {\\chi}_{2},\n\\label{eq54}\n\\end{eqnarray}\nand now\n\\begin{eqnarray}\n\\Delta^{\\pm}=\\partial_4{\\mp}[\\sigma_1(p_1+gf_{12}x_2)\n-i\\sigma_2\\partial_2+\\sigma_3(p_3+gf_{34}x_4)].\n\\end{eqnarray}\nA detailed calculation and discussion of the eigenvalues \nare collected in Appendix B.\nWe obtain two series of eigenvalues,\n\\begin{eqnarray}\n\\omega_{\\pm}(m,n)\n=-g\\left[\\frac{(\\phi +\\phi^*)}{\\sqrt{2}} +v\\right]\n{\\pm}\\sqrt{\\frac{1}{2}g^2(\\phi -\\phi^*)^2-2mgf_{12}-2ngf_{34}},\n\\end{eqnarray} \nwhere for $m{\\geq}1$, $n{\\geq}1$ both eigenvalues are doubly degenerate,\nwhile $\\omega_{\\pm}(m.0)$ and $\\omega_{\\pm}(0,n)$ are nondegenerate, \nand for $m=n=0$, there exists only a nondegenerate eigenvalue \n$\\omega_-(0,0)$.\n\nIn a similar way we can solve the eigenvalue equation \n\\begin{eqnarray}\n\\widetilde{\\Delta}_F\\widetilde{\\psi}=\\left[i\\gamma^{\\mu}D_{\\mu}^{\\dagger}\n+\\frac{g}{\\sqrt{2}}(1-\\gamma_5)\\phi \n+\\frac{g}{\\sqrt{2}}(1+\\gamma_5)\\phi^*+gv \\right]\\widetilde{\\psi}\n=\\widetilde{\\omega}\\widetilde{\\psi},\n\\end{eqnarray}\n and obtain the eigenvalues,\n\\begin{eqnarray}\n\\widetilde{\\omega}_{\\pm}(m,n)\n=g\\left[\\frac{(\\phi +\\phi^*)}{\\sqrt{2}} +v\\right]\n{\\pm}\\sqrt{\\frac{g^2}{2}(\\phi -\\phi^* )^2-2m gf_{12}-2n gf_{34}}.\n\\end{eqnarray}\nThe degeneracies of $\\widetilde{\\omega}_{\\pm}(m,n)$, \n$\\widetilde{\\omega}_{\\pm}(m,0)$ and $\\widetilde{\\omega}_{\\pm}(0,n)$ with\n$m{\\geq}1$, $n{\\geq}1$ are the same as those of the $\\omega$s. There\n still only exists a nondegenerate eigenvalue $\\omega_-(0,0)$. \n\nWith the above eigenvalues\n$\\mbox{Tr}\\ln\\Delta_F$ and $\\mbox{Tr}\\ln\\widetilde{\\Delta}_F$ can be computed\nstraightforwardly, \n\\begin{eqnarray}\n\\mbox{Tr}\\ln\\Delta_F=\\ln\\det\\Delta_F\n=\\ln\\left[\\Pi\\omega_{\\pm (lk)}(m,n)\\right]^r\n=\\sum_{l,k=-\\infty}^{+\\infty}\\sum_{m,n=0}^{\\infty} r\n\\ln\\omega_{\\pm (lk)}(m,n),\n\\end{eqnarray}\nwhere $r$ is the degeneracy of $\\omega_{\\pm}(m,n)$.\nDue to the relation $x_2=2\\pi l\/(gf_{12} L)$ and $x_4=2\\pi k\/(gf_{34} L)$,\nthe summation over the momenta $k$ and $l$ is actually equivalent to\nan integration over $x_2$ and $x_4$. Since the fields are constants,\nthis integration will yield only a Euclidean \nspace volume factor, which tends to infinity\nin the continuous limit ($L{\\rightarrow}\\infty$), \n\\begin{eqnarray}\n\\sum_{l,k}=\\frac{L^2}{4\\pi^2}g^2f_{12}f_{34}{\\int}dx_2dx_4\n=\\frac{V}{4\\pi^2}g^2f_{12}f_{34},\n\\end{eqnarray}\nwhile the Lagrangian will be well defined.\nConsider the degeneracy of each eigenvalue, we have\n\\begin{eqnarray}\n\\mbox{Tr}\\ln\\Delta_F&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left\\{\\left[\\ln\\omega_-(0,0)+\\sum_{m=1}^{\\infty}\\ln\\omega_+(m,0)\n+\\sum_{n=1}^{\\infty}\\ln\\omega_+(0,n)\\right.\\right.\\nonumber\\\\\n&+&\\left.\\left.2\\sum_{m,n=1}^{\\infty}\\ln\\omega_+(m,n)\n\\right] +\\left[\\sum_{m=1}^{\\infty}\\ln\\omega_-(m,0)+\n\\sum_{n=1}^{\\infty}\\ln\\omega_-(0,n)\n+2\\sum_{m,n=1}^{\\infty}\\ln\\omega_-(m,n)\n\\right]\\right\\}\\nonumber\\\\\n&=&\\frac{V}{4\\pi^2}ge^2f_{12}f_{34}\\left\\{\\ln\\omega_-(0,0)\n+\\sum_{m=1}^{\\infty}\\ln[\\omega_+(m,0)\\omega_-(m,0)]\n\\right.\\nonumber\\\\\n&&+\\left.\\sum_{n=1}^{\\infty}\\ln[\\omega_+(0,n)\\omega_-(0,n)]\n+2\\sum_{m,n=1}^{\\infty}\\ln\\left[\\omega_+(m,n)\\omega_-(m,n)\\right]\n\\right\\} \\nonumber\\\\ \n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left\\{\\ln[-g(\\sqrt{2}\\phi +v)]+\n\\sum_{m=1}^{\\infty}\n\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)+2mgf_{12}]\\right.\\nonumber\\\\\n&&+\\sum_{n=1}^{\\infty}\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)\n+2ngf_{34}]\\nonumber\\\\\n&&+\\left.2\\sum_{m,n=1}^{\\infty}\n\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)+2mgf_{12}+2ngf_{34}]\\right\\}.\n\\end{eqnarray}\nSimilarly, we get\n\\begin{eqnarray}\n\\mbox{Tr}\\ln\\widetilde{\\Delta}_F\n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\\left\\{\\ln\\left[g(\\sqrt{2}\\phi^*+v) \\right]\n+\\sum_{m=1}^{\\infty}\n\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)+2mgf_{12}]\\right.\\nonumber\\\\\n&&+\\sum_{n=1}^{\\infty}\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)+2ngf_{34}]\n\\nonumber\\\\\n&&+\\left.2\\sum_{m,n=1}^{\\infty}\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)\n+2mgf_{12}+2ngf_{34}]\\right\\}.\n\\end{eqnarray}\nThus we finally obtain\n\\begin{eqnarray}\n\\mbox{Tr}\\ln\\Delta_F +\\mbox{Tr}\\ln\\widetilde{\\Delta}_F\n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left\\{\\ln\\left[\\frac{\\sqrt{2}\\phi^*+v }{\\sqrt{2}\\phi +v }\n\\right]\\right.\\nonumber\\\\\n&&+2\\sum_{m=1}^{\\infty} \\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)\n+2mgf_{12}]\\nonumber\\\\\n&&+2\\sum_{n=1}^{\\infty}\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)\n+2ngf_{34}]\\nonumber\\\\\n&&+ \\left.4\\sum_{m,n=1}^{\\infty}\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)\n+2mgf_{12}+2ngf_{34}] \\right\\}.\n\\end{eqnarray}\nMaking using of the proper-time regularization,\n\\begin{eqnarray}\n\\ln \\alpha=-\\int^{\\infty}_{{1}\/{\\Lambda^2}}\\frac{ds}{s}e^{-\\alpha s}\n\\end{eqnarray}\nwith $\\Lambda^2$ being the cut-off to regularize the infinite sum,\nwe have\n\\begin{eqnarray}\n&&\\mbox{Tr}\\ln\\Delta_F +\\mbox{Tr}\\ln\\widetilde{\\Delta}_F\n=\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\\left\\{\n\\ln\\left[\\frac{\\sqrt{2}\\phi^*+v}{\\sqrt{2}\\phi+v) }\\right]\n-2\\int^{\\infty}_{{1}\/{\\Lambda^2}}\n\\frac{ds}{s}e^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\\right.\\nonumber\\\\\n&&{\\times}\\left.\\left[\\sum_{m=1}^{\\infty}e^{-2mgf_{12}s}\n+\\sum_{n=1}^{\\infty}e^{-2ngf_{34}s}\n+2\\sum_{m,n=1}^{\\infty}e^{-(2mgf_{12}+2ngf_{34})s}\\right]\\right\\}\n\\nonumber\\\\\n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left\\{\\ln\\left[\\frac{\\sqrt{2}\\phi^*+v }{\\sqrt{2}\\phi+v }\\right]\n-2\\int^{\\infty}_{{1}\/{\\Lambda^2}}\n\\frac{ds}{s}e^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\n\\left[\\frac{e^{-gf_{34}s}}{\\sinh (gf_{34}s)}\\right.\\right.\\nonumber\\\\\n&&+\\frac{e^{-gf_{12}s}}{\\sinh (gf_{12}s)}\n+\\left.\\left.\\frac{e^{-(gf_{12}+gf_{34})s}}\n{\\sinh (gf_{12}s)\\,\\sinh (gf_{34}s)}\\right]\\right\\}\n\\nonumber\\\\\n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left[\\ln\\left(\\frac{\\sqrt{2}\\phi^*+v}{\\sqrt{2}\\phi+v }\\right)\n-\\int^{\\infty}_{{1}\/{\\Lambda^2}}\\frac{ds}{s}\ne^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)s}\\coth(gf_{12}s)\n\\coth(gf_{34}s)\\right]\\nonumber\\\\\n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left[\\ln\\left(\\frac{\\sqrt{2}\\phi^*+v}{\\sqrt{2}\\phi+v}\\right)\\right.\\nonumber\\\\\n&&-\\left.\\int^{\\infty}_{{1}\/{\\Lambda^2}}\\frac{ds}{s}\ne^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\n\\frac{\\cosh[g(f_{12}+f_{34})s]+\\cosh[g(f_{12}-f_{34})s]}\n{\\cosh[g(f_{12}+f_{34})s]-\\cosh[g(f_{12}-f_{34})s]}\\right]\\, ,\n\\label{eq50n}\n\\end{eqnarray}\nwhere we have used\n\\begin{eqnarray}\n\\sum_{m=1}^{\\infty}e^{-2mt}=\\frac{e^{-t}}{2\\sinh t},~~~\n\\cosh (x{\\pm}y)=\\cosh x\\cosh y{\\pm}\\sinh x\\sinh y.\n\\end{eqnarray}\nRotating back to Minkowski space and denoting ${\\bf X}{\\equiv}{\\bf H}+i{\\bf E}$,\nwe write (\\ref{eq50n}) as\n\\begin{eqnarray}\n&&\\mbox{Tr}\\ln\\Delta_F +\\mbox{Tr}\\ln\\widetilde{\\Delta}_F\n=\\frac{V}{4\\pi^2}g^2iE_z H_z\n\\left[\\ln\\left(\\frac{\\sqrt{2}\\phi^*+v }{\\sqrt{2}\\phi+v}\\right)\\right.\n\\nonumber\\\\\n&-&\\left.\\int^{\\infty}_{{1}\/{\\Lambda^2}}\n\\frac{ds}{s}e^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\n\\frac{ \\cosh[g(H_z+iE_z)s]+\\cosh[g(H_z-iE_z)s]}\n{\\cosh[g(H_z+iE_z)s]-\\cosh[g(H_z-iE_z)s]}\\right]\\nonumber\\\\\n&{\\equiv}&\\frac{V}{4\\pi^2}g^2i{\\bf E}{\\cdot}{\\bf H}\n\\left[\\ln\\left(\\frac{\\sqrt{2}\\phi+v }{\\sqrt{2}\\phi^*+v }\\right)\n\\nonumber\\right.\\\\\n&&-\\left.\\int^{\\infty}_{{1}\/{\\Lambda^2}}\\frac{ds}{s}\ne^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\n\\frac{ \\cosh[g{\\bf X}s]+\\cosh (g{\\bf X}^*s)}\n{\\cosh (g{\\bf X} s)-\\cosh (g{\\bf X}^*s)}\\right],\n\\label{eq84}\n\\end{eqnarray}\nTo extract the divergence, we must analyze the small-$s$ behaviour of\nthe integrand of (\\ref{eq84}) by\nusing the identities\n\\begin{eqnarray} \ni{\\bf E}{\\cdot}{\\bf H}=\\frac{1}{4}({\\bf X}^2-{\\bf X}^{*2})\n=\\frac{1}{4}F_{\\mu\\nu}\\widetilde{F}^{\\mu\\nu}, ~~\n{\\bf H}^2-{\\bf E}^2=\\frac{1}{2}({\\bf X}^2+{\\bf X}^{*2})\n=\\frac{1}{2}F_{\\mu\\nu}F^{\\mu\\nu},\n\\end{eqnarray}\nand the series expansion near $s\\sim 0$\n\\begin{eqnarray} \n\\frac{ \\cosh[g{\\bf X}s]+\\cosh[g {\\bf X}^*s]}\n{\\cosh[g{\\bf X} s]-\\cosh[g{\\bf X}^*s]}\n&=&\\frac{1}{({\\bf X}^2-{\\bf X}^{*2})}\\left[\\frac{4}{g^2s^2}+\n\\frac{2}{3}({\\bf X}^2+{\\bf X}^{*2})+{\\cal O}(s^2)\\right].\n\\label{eq84x}\n\\end{eqnarray}\nIt can be easily seen from (\\ref{eq84x}) that the integral in \n(\\ref{eq84}) has a quadratic divergence and a logarithmic one. Thus\nthe divergence term can be extracted by writing (\\ref{eq84}) \nas the form,\n\\begin{eqnarray}\n&&\\mbox{Tr}\\ln\\Delta_F +\\mbox{Tr}\\ln\\widetilde{\\Delta}_F\n=\\frac{V}{4\\pi^2}\\left\\{\\frac{1}{4}g^2F_{\\mu\\nu}\\widetilde{F}^{\\mu\\nu}\n \\ln\\left[\\frac{\\sqrt{2}\\phi^*+v}{\\sqrt{2}\\phi +v}\\right]\\right.\\nonumber\\\\\n&-&\\int^{\\infty}_{{1}\/{\\Lambda^2}}ds\n\\left(\\frac{1}{s^3}+\\frac{1}{6}\\frac{1}{s}g^2F_{\\mu\\nu}F^{\\mu\\nu}\\right)\ne^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)s}\\nonumber\\\\\n&-&\\int^{\\infty}_0\\frac{ds}{s^3}\ne^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\\left[\\frac{1}{4}g^2s^2\nF_{\\mu\\nu}\\widetilde{F}^{\\mu\\nu}\\frac{ \\cosh (g{\\bf X}s)+\\cosh (g{\\bf X}^*s)}\n{\\cosh (g{\\bf X} s)-\\cosh (g{\\bf X}^*s)}\\right.\\nonumber\\\\\n&-&\\left.\\left.\\frac{1}{s^3}\n-\\frac{1}{6}\\frac{1}{s}g^2F_{\\mu\\nu}F^{\\mu\\nu}\\right]\\right\\}.\n\\label{eq86}\n\\end{eqnarray}\nThe second term (\\ref{eq86}) is the UV divergent term, \nso the cut-off $1\/\\Lambda^2$ is preserved to regularize the \nintegral, while the last term is a finite term\nand hence the cut-off has been removed.\n\nNow we turn to the bosonic determinant. From (\\ref{eq27}) we have \n\\begin{footnotesize}\n\\begin{eqnarray}\n&&M_{ff}^{-1}=2\\left(\\begin{array}{cccc} 1\/{\\Delta}_F & 0 & 0 & 0\\\\\n0 & 1\/\\widetilde{\\Delta}_F & 0 & 0 \\\\\n0 & 0 & 1\/{\\Delta}_F & 0 \\\\\n0 & 0 & 0 & 1\/\\widetilde{\\Delta}_F \\end{array}\\right), ~~\nM_{bb}-M_{bf}M_{ff}^{-1}M_{fb}=\\nonumber\\\\ && \\frac{1}{2}\n\\left(\\begin{array}{cccc} \\Delta_{\\mu\\nu}\n-2g^2\\overline{\\psi}{\\gamma}_{\\mu}\\frac{1}{\\widetilde{\\Delta}_F}{\\gamma}_{\\nu}\\psi & 0 &\n\\Delta_{\\mu}-2ig^2\\overline{\\psi}{\\gamma}_{\\mu}\\frac{1}{\\Delta_F}{\\gamma}_5\\psi & 0\\\\\n0 & {\\Delta}^{\\dagger}_{\\mu\\nu}\n-2g^2\\overline{\\psi}{\\gamma}_{\\mu}\\frac{1}{\\widetilde{\\Delta}_F}{\\gamma}_{\\nu}\\psi & 0\n& {\\Delta}^{\\dagger}_{\\mu}\n-2ig^2\\overline{\\psi}{\\gamma}_{\\mu}\\frac{1}{\\widetilde{\\Delta}_F}{\\gamma}_5\\psi\\\\\n\\widetilde{\\Delta}_{\\nu}^{\\dagger}-2ig^2\\overline{\\psi}{\\gamma}_5\n\\frac{1}{\\Delta_F}{\\gamma}_{\\nu}\\psi & 0 & \\Delta+2g^2\\overline{\\psi}{\\gamma}_5\n\\frac{1}{\\Delta_F}{\\gamma}_5\\psi & 0\\\\\n0 & \\widetilde{\\Delta}_{\\nu}-2ig^2\\overline{\\psi}{\\gamma}_5\n\\frac{1}{\\Delta_F}{\\gamma}_{\\nu}\\psi & 0 & \\Delta^{\\dagger}\n+2g^2\\overline{\\psi}{\\gamma}_5\n\\frac{1}{\\Delta_F}{\\gamma}_5\\psi\n\\end{array}\\right).\\nonumber\\\\\n\\end{eqnarray}\n\\end{footnotesize}\nIn constant field approximation, $\\overline{\\psi}$ and $\\psi$ can be regarded\nGrassman numbers, so we can expand the bosonic determinant only \nto the quartic terms in $\\overline{\\psi}$ and $\\psi$.\nNow the key problem is how to find the eigenvalues and eigenstates of\nthe operator matrix $M_{bb}-M_{bf}M_{ff}^{-1}M_{fb}$. If they could be worked\nout, then with the eigenvalues and eigenvectors of fermionic \noperator, we can use the technique \ndeveloped in \\cite{dmnp} to evaluate this determinant. Unfortunately,\nit seems to us that in the constant field approximation\nit is very to find the eigenvalues and eigenstates of such \na horrible operator matrix. This difficulty is waiting to be overcome.\n\nDespite the fact that the bosonic part cannot be evaluated, we can see from \n(\\ref{eq27}) and (\\ref{eq86}) that the effective\nLagrangian associated with the fermionic part has already shown\nthe features of the perturbative part of the low-energy effective action.\nFirst, we believe that the quadratic divergence of Eq.(\\ref{eq86})\nwill be canceled owing to the nonrenormalization theorem. \nSecond, for the logarithmic divergence of Eq.(\\ref{eq86}), with\n\\begin{eqnarray}\n\\int^{\\infty}_{{1}\/{\\Lambda^2}}\\frac{ds}{s}\ne^{-g^2(\\sqrt{2}\\phi +v)(\\sqrt{2}\\phi^*+v) s}{\\sim}\n-\\ln\\left[\\frac{g^2(\\sqrt{2}\\phi +v)(\\sqrt{2}\\phi^*+v) }{\\Lambda^2}\\right],\n\\end{eqnarray}\nEq.(\\ref{eq86}) shows that the Wilson effective action has one term\nproportional to\n\\begin{eqnarray}\nF_{\\mu\\nu}F^{\\mu\\nu}\\ln\\left[\n\\frac{g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)}{\\Lambda^2}\\right].\n\\end{eqnarray}\nComparing with the component field form given by (\\ref{eq:last}), \nwe can conclude that the complete calculation should give the \nform (\\ref{eq1}) of the low-energy effective action. One can even guess \nthis from the requirement of\nsupersymmetry since the constant field approximation and the \nproper-time regularization preserve the supersymmetry explicitly. \nFurther, there is a finite term proportional \nto $F\\widetilde{F}\\ln\\left[(\\sqrt{2}\\phi +v)\/(\\sqrt{2}\\phi^*+v)\\right]$ in\n(\\ref{eq86}). As pointed out in \\cite{dmnp}, this is the \nreflection of the axial $U(1)_R$ anomaly in the effective action. \nThis anomaly term had played a crucial role\nin the nonperturbative analysis\\cite{sei}.\n \n\\section{Summary}\n\nIn summary, we have tried to calculate the perturbative part of\nthe low-energy effective action of \n$N=2$ supersymmetric Yang-Mills\ntheory based on a standard effective field theory technique.\nIt is well known that the Seiberg-Witten effective action is the\ncornerstone for all those new developments in $N=2$ supersymmetric\ngauge theory, and that this effective action has been obtained\nin an indirect way. Therefore, it is worthwhile to\ntry to compute this effective action using a straightforward\nintegration of the heavy degrees of freedom.\n Unfortunately, we have encountered an insurmountable difficulty \nin evaluating the bosonic operator adopting the constant field\napproximation. This prevents us from getting the complete result and\ngiving a thorough comparison with the form of (\\ref{eq1}). However, \nthe calculation of the fermionic determinant has indeed shown the \nbasic features of the low-energy effective action. This gives \na partial verification of the abstract symmetry analysis in extracting \nthe low-energy effective action. The complete calculation presents \nan interesting problem for further investigation.\n\n\\vspace{8mm}\n\n\\noindent{\\bf Acknowledgments:}\n We acknowledge the financial support by the Academy of Finland \nunder the project No. 37599 and 44129. W.F.C is partially supported \nby the Natural Sciences and Engineering Research Council of Canada. \n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzcypm b/data_all_eng_slimpj/shuffled/split2/finalzzcypm new file mode 100644 index 0000000000000000000000000000000000000000..2f5bad626164403fc831cf82cf4b438fc19a80be --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzcypm @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \\label{sec1}\nIt is known that every connected orientable 3-manifold $M$ has a fibered link $L$. \nThis is equivalent to that $M$ admits a surface $F$ bounding a link $L$ such that $M \\setminus (N(L) \\cup N(F))$ is homeomorphic to $F \\times [0,1]$, where $N(\\cdot)$ stands for a (sufficiently small) open tubular neighborhood. \nHowever, finding fibered links in a given 3-manifold is difficult in general. \nSince existence of a fiber surface of fixed type in a manifold concerns with the structure of the manifold, there are many works about these. \nAbout fibered links with planar fiber surface in lens spaces, \nlens space of type $(p,q)$ has such a fibered link whose number of components is as the same number as one plus the length of a continued fraction expansion of $\\frac{p}{q}$. \nHowever it is unknown that there exists a universal upper bound for the number of boundary components of planar fibered surface for every lens space. \\\\\nThere is an analogue of fibered links, $\\it{homologically \\ fibered \\ links}$ \\cite{homfib}; \nA link $L$ in a 3-manifold $M$ is called a $\\it{homologically \\ fibered \\ link}$ \nif there exists a Seifert surface $F$ of $L$ such that $M \\setminus (N(L) \\cup N(F))$ is homeomorphic as sutured manifold to a homology cobordism over the surface. \nThis is equivalent to that $M$ is obtained by pasting the upper boundary and the lower boundary of a homology cobordism over the surface. \nWe may get refinements of notions concerning with surface cross intervals for homology cobordisms. (See \\cite{hc} for example.) \nRecently, Nozaki \\cite{nozaki} showed that every lens space contains a genus one homologically fibered knot by solving some algebraic equations. \nIn this paper, like \\cite{nozaki}, we give the relation between the existence of homologically fibered link of a given number of components whose Seifert surface is a planar surface in a given lens space and some algebraic equation as follows. \nWe denote by $L(p,q)$ a lens space of type $(p,q)$, and $\\Sigma_{g,n}$ a surface of genus $g$ with $n$ boundary components.\\\\\n\n \\begin{prop} \\label{prop}\n $L(p,q)$ has a realization as a closure of a homology cobordism over $\\Sigma_{0,n+1}$ ($n\\geq 1$)\nif and only if there exist integers $a_{h}$, $l_{i,j}$ and $t_k$($h,i,j,k=1,2,\\dots,n$, $i\\lneq j$) satisfying following equation: \\\\\n\n\\begin{eqnarray}\n\\left|\n\\begin{array}{ccccc}\n p & -qa_{1} & -qa_{2} & \\cdots & -qa_{n}\\\\\n a_{1} & t_{1} & l_{1,2} & \\cdots & l_{1,n} \\\\\n a_{2} & l_{1,2} & t_{2} & \\cdots & l_{2,n}\\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n a_{n} & l_{1,n} & l_{2,n} & \\cdots & t_{n}\\\\\n\\end{array}\n\\right| = \\pm 1 \n\\end{eqnarray}\n\n\\end{prop}\n\nNote that since any manifold which has a realization as a closure of a homology cobordism over $\\Sigma_{0,1}$ is a homology 3-sphere, no lens spaces have such a realization. \nNote also that if we have a solution of $(1)$ for some $n$, by setting $a_{n+1}=l_{i,n+1}=0$ for all $i$ and $t_{n+1}=1$ we get that for $n+1$. \nHence a lens space which has a realization as a closure of homology cobordism over $\\Sigma_{0,n+1}$ has that of $\\Sigma_{0,n+2}$, too. \nBecause of this reason, it seems meaningful to find the minimal $n$ of $(1)$ for a given lens space. \nHowever, this question is completely answered as follows:\n\n\\vspace{1.0cm}\n \\begin{thm} \\label{thm1}\n $L(p,q)$ has a realization as a closure of a homology cobordism over $\\Sigma_{0,2}$ \nif and only if $q$ or $-q$ is a quadratic residue modulo $p$.\n \\end{thm}\n\n\n \\begin{thm} \\label{thm2}\n Every lens space has a realization as a closure of a homology cobordism over $\\Sigma_{0,3}$.\n \\end{thm}\n\nThe remainder of this paper is organized as follows. \nIn Section \\ref{sec2}, we give preliminary terminologies. \nIn Section \\ref{sec3}, we prove Proposition \\ref{prop}. \nSection \\ref{sec4} is devoted to the proofs for theorems.\nFinally, we give some applications for the invariant hc($\\cdot$) defined in \\cite{hc} in Section \\ref{sec5}.\n\n\n\\subsection*{Acknowledgements}\nThe author would like to give his thanks to Prof. Takuya Sakasai for giving many meaningful\nadvices, and Prof. Yuta Nozaki for telling algebraic operations to the author. \n\n\n\n\n\n\n\\section{Preliminary} \\label{sec2}\n \\subsection{Homology cobordisms (Section $2.4$ of \\cite{homcob})}\nA $\\it {homology \\ cobordism \\ over \\ \\Sigma_{g,n}}$ ($n\\geq 1$) is a triad ($X,\\partial_{+}X,\\partial_{-}X$), \nwhere $X$ is an oriented compact 3-manifold and $\\partial_{+}X \\cup \\partial_{-}X$ is a partition of $\\partial X$, and $\\partial_{\\pm} X$ are homeomorphic to $\\Sigma_{g,n}$ satisfying:\\\\\n\n\\begin{itemize}\n \\item $\\partial_{+}X \\cup \\partial_{-}X = \\partial X$.\n \\item $\\partial_{+}X \\cap \\partial_{-}X = \\partial(\\partial_{+}X)$.\n \\item The induced maps $(i_{\\pm})_{*}: H_{*}(\\partial_{\\pm} X) \\rightarrow H_{*}(X)$ are isomorphisms, where $i_{\\pm}: \\partial_{\\pm}X \\rightarrow X$ is the inclusions.\\\\\n\\end{itemize}\n\nNote that the third condition is equivalent to the condition that $X$ is connected and $i_{\\pm}$ induce isomorphisms on $H_{1}(\\cdot)$. \nMoreover by using the Poincar$\\rm{\\acute{e}}$ duality, we see that inducing isomorphism of one of $(i_{+})_{*}$ and $(i_{-})_{*}$ is sufficient. \\\\\nBy pasting $\\partial_{+}X$ and $\\partial_{-}X$ using any boundary-fixing homeomorphism, we get a closed 3-manifold $M$ and a surface $F$ in $M$, which is the image of $\\partial_{+}X$. In this situation, we say $M$ is representable as a closure of a homology cobordism over $\\Sigma_{g,n}$. \nThe link $\\partial F$ (usually together with $F$) is called a $homologically\\ fibered\\ link$ in $M$ and we call $F$ a $homologially\\ fibered\\ surface$. \\\\\nBy the above observations, for a link with a Seifert surface $F$ in a closed 3-manifold $M$ \nbeing a homologically fibered link with homologically fibered surface, it is enough to check that push-ups (or push-downs) of simple closed curves on $F$ which form a basis of $H_{1}(F)$ also form a basis of $H_{1}(M\\setminus Int(F\\times[0,1]))$. \n\n\n \\subsection{Mixing diagrams}\n$L(p,q)$ is obtained by the $\\left( -\\frac{p}{q}\\right)$--Dehn surgery along the unknot $U$ in $S^3$. \nIn $L(p,q)$, we see the knot $U'$, which is the image of $U$ under the surgery. \nFor any graph (1--dimensional subcomplex of a trianglation of $L(p,q)$) $G'$, \nwe can isotope $G'$ so that this is disjoint from $U'$. \nTherefore we get a graph $G$ in $S^3$, which becomes $G'$ after the surgery. \nWe call $U \\cup G$ in $S^3$ a $\\it {mixing \\ diagram}$ of $G'$ in $L(p,q)$.\n\n\n\\section{Proof of Proposition \\ref{prop}} \\label{sec3}\nWe fix a lens space $L(p,q)$. \nSuppose there exists a homologically fibered link in $L(p,q)$ with a Seifert surface $F$ which is homomorphic to $\\Sigma_{0,n+1}$. \nWe take a spine of $F$, $S'=K'_{1} \\cup v'_{1} \\cup K'_{2} \\cup \\cdots \\cup v'_{n-1} \\cup K'_{n}$ as in Figure \\ref{fig:one}. When $n=1$, there are no $v'$-arcs. \nNote that $\\{K'_{1}, K'_{2},\\ldots ,K'_{n}\\}$ forms a basis of $H_{1}(F)$. \nAccurately, $F$ is knotted $\\Sigma_{0,n+1}$ in $L(p,q)$, thus we fix an embedding. \nCutting $L(p,q)$ along $F$ is equivalent to remove $N(S)$: a small neighborhood of $S$, except for the information about a partition of the boundary. \nBy choosing a longitude of $K'_i$ and a twist number of the band corresponding to $v'_{j}$, we can assign the information about a partition of the boundary. \n\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width=100mm]{planar.eps}\n \\end{center}\n \\caption{$S'$, a spine of $F$}\n \\label{fig:one}\n\\end{figure}\n\nWe take a mixing diagram $U\\cup S$ of $S'$. \nWe denote by $K_{i}$ the loop of $S$ corresponding to $K'_{i}$, and by $v_{i}$ the arc corresponding to $v'_{i}$. \nWe take meridian curves of $N(S)$, $m_{i}$ which is dual to $K_{i}$, and $x_{i}$ which is dual to $v_{i}$ as in Figure \\ref{fig:two}. \nNote that $x_i$ bounds a surface homeomorphic to $\\Sigma_{i,1}$ in $S^3\\setminus (N(U)\\cup N(S))$, which lies on the boundary. \nWe also take the meridian curve $m$ of $U$. \n\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width=100mm]{meridional.eps}\n \\end{center}\n \\caption{meridian curves}\n \\label{fig:two}\n\\end{figure}\n\nSince $x_{i}$ is null-homologous in $S^3 \\setminus (N(U) \\cup N(S))$, by using Mayer--Vietoris exact sequence, we can see $H_{1}(S^3 \\setminus (N(U) \\cup N(S)))$ is isomorphic to $H_{1}(S^3 \\setminus (N(U)\\cup N(\\cup^{n}_{i=1}K_i )))$, which is isomorphic to $\\mathbb{Z}^{n+1}$ and generated by $\\{m,m_1,\\ldots ,m_n\\}$. \nAfter $\\left( -\\frac{p}{q} \\right)$--surgery along $U$, we see $H_{1}(L(p,q)\\setminus N(S'))$ is isomorphic to $H_{1}(L(p,q) \\setminus N(\\cup^{n}_{i=1}K'_i))$. \\\\\n\nWe will compute $H_{1}(L(p,q) \\setminus N(\\cup^{n}_{i=1}K'_i))$ from $H_{1}(S^3 \\setminus (N(U)\\cup N(\\cup^{n}_{i=1}K_i )))$. \nWe denote by $a_{i}$ the linking number of $U$ and $K_{i}$. \nThen, the canonical longitude of $U$ is represented by $\\Sigma^{n}_{i=1} a_{i}m_{i}$ in $H_{1}(S^3 \\setminus (N(U)\\cup N(\\cup^{n}_{i=1}K_i )))$. \nTherefore, $H_{1}(L(p,q) \\setminus N(\\cup^{n}_{i=1}K'_i))$ is isomorphic to $\\langle m,m_1,\\cdots ,m_n \\rangle \/(pm-q\\Sigma^{n}_{i=1} a_{i}m_{i})$, which is isomorphic to $\\mathbb{Z}^{n}\\bigoplus \\mathbb{Z}\/\\gcd(p,qa_1,\\cdots ,qa_n)\\mathbb{Z}$, where $\\gcd$ represents the greatest common divisor. \nWe fix this presentation of $H_{1}(L(p,q) \\setminus N(\\cup^{n}_{i=1}K'_i))$.\\\\\n\nSince $L(p,q\\setminus (F\\times[0,1]))$ is a homology cobordism over $\\Sigma_{0,n+1}$, $H_{1}(L(p,q)\\setminus (F\\times [0,1])) \\cong H_{1}(L(p,q)\\setminus N(S')) \\cong H_{1}(L(p,q) \\setminus N(\\cup^{n}_{i=1}K'_i))$ must be isomorphic to $\\mathbb{Z}^n$. \nHence, $\\gcd(p,qa_1,\\cdots ,qa_n)=1$.\\\\\n\nMoreover, being a homology cobordism over $\\Sigma_{0,n+1}$, there must be a longitude $\\bar{K'_{i}}$ for each $K'_{i}$ such that $\\{\\bar{K'_{1}},\\cdots ,\\bar{K'_n}\\}$ forms a basis of $H_{1}(L(p,q)\\setminus N(S'))$. \nIn $S^3$, the longitude $\\bar{K'_{i}}$ corresponds to a longitude $\\bar{K_i}$ for $K_i$. \nBy setting $l_{i,j}$ for $i\\neq j$ be the linking number of $K_i$ and $K_{j}$ ($l_{i,j}=l_{j,i}$), \nthis $\\bar{K_i}$ is represented by using some integer $t_{i}$ as $t_{i}m_{i}+a_{i}m+\\Sigma_{j\\neq i}l_{i,j}m_{j}$ in $H_{1}(S^3\\setminus (N(U)\\cup N(\\cup^{n}_{i=1}K_i ))) \\cong \\mathbb{Z}^{n+1}$. \nFor \n\\begin{eqnarray*}\n\\{pm-q\\Sigma^{n}_{i=1}a_{i}m_i, t_{1}m_{1}+a_{1}m+\\Sigma_{j\\neq 1}l_{1,j}m_{j}, t_{2}m_{2}+a_{2}m+\\Sigma_{j\\neq 2}l_{2,j}m_{j},\\cdots ,t_{n}m_{n}+a_{n}m+\\Sigma_{j\\neq n}l_{n,j}m_{j}\\}\n\\end{eqnarray*}\n forming a basis of $H_{1}(L(p,q) \\setminus N(\\cup^{n}_{i=1}K'_i)) \\cong \\mathbb{Z}^{n}\\bigoplus \\mathbb{Z}\/\\gcd(p,qa_1,\\cdots ,qa_n)\\mathbb{Z}$, the following equation must be satisfied:\\\\\n\n\\begin{eqnarray}\n\\left|\n\\begin{array}{ccccc}\n p & -qa_{1} & -qa_{2} & \\cdots & -qa_{n}\\\\\n a_{1} & t_{1} & l_{1,2} & \\cdots & l_{1,n} \\\\\n a_{2} & l_{2,1} & t_{2} & \\cdots & l_{2,n}\\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n a_{n} & l_{n,1} & l_{n,2} & \\cdots & t_{n}\\\\\n\\end{array}\n\\right| = \\pm 1 \\nonumber \n\\end{eqnarray}\n\nBy using the relations $l_{i,j}=l_{j,i}$ for $i\\neq j$, we obtain equation (1). \\\\\n\nConversely, if we have integers $a_{h}$, $l_{i,j}$ and $t_k$($h,i,j,k=1,2,\\dots,n$, $i\\lneq j$) satisfying the equation (1), \nwe can construct framed knots $K_{1}, K_{2},\\cdots ,K_{n}$ in $S^3$ which are disjoint from $U$ satisfying the following:\\\\\n\\begin{itemize}\n \\item the linking number of $U$ and $K_i$ is $a_{i}$.\n \\item the linking number of $K_i$ and $K_j$ for $i\\lneq j$ is $l_{i,j}$.\n \\item the framing of $K_i$ is the $t_i$-times twisting of the canonical longitude of $K_i$ along the meridian disk.\n\\end{itemize}\nReversing the above computation, we can construct a homologically fibered link with Seifert surface which is homeomorphic to $\\Sigma_{0,n+1}$. \\\\\n\n\n\n\\section{Proof of theorems } \\label{sec4}\n\\subsection{Proof of Theorem \\ref{thm1}}\n \nBy Proposition \\ref{prop}, that $L(p,q)$ is realizable as a closure of a homology cobordism over $\\Sigma_{0,2}$ is equivalent to that there exist integers $a$ and $t$ satisfying \n$\n\\left|\n\\begin{array}{cc}\n p & -qa \\\\\n a & t\\\\\n\\end{array}\n\\right| = \\pm 1\n$. \nThis equation is equivalent to $tp+qa^{2}=\\pm 1$. Note that this implies $p$ and $a^2$ are coprime and therefore $p$ and $a$ are coprime. \nThis $a$ satisfies $qa^{2} \\equiv \\pm 1$ mod $p$. \nSince $p$ and $a$ are coprime, this is equivalent to $q \\equiv \\pm {(a^{-1})}^2$ mod $p$. \nTherefore the condition for $L(p,q)$ being realizable as a closure of a homology cobordism over $\\Sigma_{0,2}$ is equivalent to that $q$ or $-q$ is quadratic residue modulo $p$.\n\n\n\\subsection{Proof of Theorem \\ref{thm2}}\nWe fix a lens space $L(p,q)$ with $p,q > 0$ (every lens space has such a representation). \nWe fix a positive integers $s$ and $r$ such that $ps-qr=1$. \\\\\\\\\n\nBy Proposition \\ref{prop}, that $L(p,q)$ is realizable as a closure of a homology cobordism over $\\Sigma_{0,2}$ is equivalent to that there exist integers $a_1,a_2,l_{1,2},t_1$, and $t_2$ satisfying \n$\n\\left|\n\\begin{array}{ccc}\n p & -qa_1 & -qa_2 \\\\\n a_1 & t_1 & l_{1,2}\\\\\n a_2 & l_{1,2}& t_2 \\\\\n\\end{array}\n\\right| = \\pm 1\n$. \nThis equation is equivalent to $p(t_{1}t_{2}-l^{2}_{1,2})-q(2l_{1,2}a_{1}a_{2}-t_{2}a^{2}_{1}-t_{1}a^{2}_{2})=\\pm1$, and also equivalent to \n \\begin{eqnarray}\n \\begin{cases}\n t_{2}a^2_{1}-2l_{1,2}a_{1}a_{2}+t_{1}a^{2}_{2}=-\\epsilon (r+kp) &\\\\\n \\left|\n \\begin{array}{cc}\n t_2 & -l_{1,2} \\\\\n -l_{1,2} & t_{1} \\\\\n \\end{array}\n \\right| = \\epsilon (s+kq) & \n \n \\end{cases}\n \\end{eqnarray}\\\\\n\nWe use the following lemma (see Section 5.3 of \\cite{quadra} for example). \nHere the determinant of $ax^{2}+2bxy+cy^{2}$ is $ac-b^2$.\n\\begin{lem} \\label{lem1}\nFor $n,D\\in \\mathbb{Z}$, there is a binary quadratic form $f(x,y)\\in \\mathbb{Z}[x,y]$ with determinant $D$ such that $f(x,y)=n$ has a primitive solution if and only if the congruence equation $-z^{2} \\equiv D$ mod $n$ has a solution. \n\\end{lem}\n\n\\begin{proof}\n(if part)\nWe set $z_{0}, C_{0}\\in \\mathbb{Z}$ to be $D=-z^{2}_{0}+C_{0}n$. \nThen $f_{0}(x,y)=nx^{2}+2z_{0}xy+C_{0}y^2$, whose determinant is $D$ satisfies $f_{0}(1,0)=n$.\\\\\n(only if part)\nSuppose $f(x,y)=Ax^{2}+2Bxy+Cy^{2}$ has primitive solution $(x,y)=(x_0,y_0)$ for $f(x,y)=n$. \nFix integers $\\bar{x}, \\bar{y}$ such that $x_{0}\\bar{x}-y_{0}\\bar{y}=1$. \nThen $g(x,y)=f\\left( U\\left( \\begin{array}{c} x\\\\ y \\end{array} \\right) \\right) =f(x_0,y_0)x^{2}+2B'xy+C'y^2$, where $U=\\left( \\begin{array}{cc} x_0 & \\bar{y} \\\\ y_0 & \\bar{x} \\end{array} \\right)$, and $B', C'$ are some integers, has the same determinant $D$ as $f(x,y)$. \nThus $D=C'n-B'^{2}$. This implies $-B'^{2} \\equiv D$ mod $n$. \n\\end{proof}\n\nBy the above lemma, the existence of a solution of (2) is equivalent to the existence of integers $k$ and $z'$, non-zero integer $w$ and $\\epsilon {'}$, which is $+1$ or $-1$ satisfying the following equation:\n\\begin{eqnarray*}\n \\begin{cases}\n-z'^2 \\equiv \\epsilon' (s+kq) \\ \\ \\ \\rm{mod} \\ \\ -\\epsilon' \\frac{r+kp}{w^2} &\\\\\nw^{2}\\mid r+kp&\n \\end{cases}\n\\end{eqnarray*}\nNote that since $(s+kq)$ and $(r+kp)$ are coprime, $z'^2$ and therefore $z'$ are prime to $(r+kp)$. \nThis equation is equivalent to the following\n by setting $z=z'^{-1}$ and $\\epsilon = \\epsilon'$:\n\\begin{eqnarray}\n \\begin{cases}\np \\equiv \\epsilon z^2 \\ \\ \\ \\rm{mod} \\ \\ \\frac{r+kp}{w^2} &\\\\\nw^{2}\\mid r+kp& \n \\end{cases}\n\\end{eqnarray}\n\\\\\\\\\nSince $L(p,q) \\cong L(p,r)$, we can replace $r$ in (3) with $q$. \\\\\nWe will show that for all $(p,q)$, there exist integers satisfying (3) by using the following two facts, which and whose use are fully explained in Section $3$ of \\cite{nozaki}:\n\\\\\nPut $K_1 = \\mathbb{Q}(\\zeta_p)$ and $K_2 = \\mathbb{Q}(\\sqrt{-1})$, where $\\zeta_p = \\exp(\\frac{2\\pi}{p} \\sqrt{-1})$.\n\n\\begin{fact} \\label{fact1}\n(A special case of Chebotarev density theorem, for example Theorem $10$ of \\rm{ \\cite{density}} )\\\\\nIf there exists $\\sigma \\in Gal(K_{1}K_{2}\/\\mathbb{Q})$ satisfying:\\\\\n\\begin{itemize}\n \\item $\\sigma |_{K_1}$ is $[\\zeta_{p} \\longmapsto \\zeta^{m}_{p}]$, where $m$ is prime to $p$.\n \\item $\\sigma |_{K_2}$ is $[\\sqrt{-1} \\longmapsto -\\sqrt{-1}]$.\n\\end{itemize}\nthen there exist infinitely many integers $k$ satisfying:\\\\\n\\begin{itemize}\n \\item $m+kp$ is prime.\\\\\n \\item $m+kp \\equiv -1$ mod $4$\n\\end{itemize}\n\\end{fact}\n\n\\begin{fact} \\label{fact2}\n(See Corollary $4.5.4$ of {\\rm{ \\cite{cycle}} }for example)\\ \n$\\sqrt{-1} \\in \\mathbb{Q}(\\zeta_{p})$ if and only if $4\\mid p$.\n\\end{fact}\n\\vspace{0.5cm}\nFrom these we have the following:\n\\begin{lem} \\label{make_prime}\nThere exists an integer $k$ satisfying at least one of the following :\\\\\n\\begin{itemize}\n \\item $q+kp$ is prime and $q+kp \\equiv -1$ \\rm{mod} $4$.\\\\\n \\item $r+kp$ \\it{is prime and} $r+kp \\equiv -1$ \\rm{mod} $4$.\n\\end{itemize}\n\\end{lem}\n\n\\begin{proof}\n(i) the case when $4\\nmid p$\\\\\nIn this case, $K_{1}\\cap K_{2}=\\mathbb{Q}$ by Fact \\ref{fact2}. \nThus in $Gal(K_{1}K_{2}\/\\mathbb{Q})$, $[\\zeta_{p} \\longmapsto \\zeta^{q}_{p}]$ and $[\\sqrt{-1} \\longmapsto -\\sqrt{-1}]$ are coexistable. \nBy Fact \\ref{fact1}, we have an integer $k$ such that $q+kp$ is prime and $q+kp \\equiv -1$ mod $4$.\\\\\n(ii) the case when $4 \\mid p$.\\\\\nWe write $p=4p'$. In this case, $K_{1}K_{2}=K_{1}$ by Fact \\ref{fact2}. \nNote that $\\zeta^{p'}_p = \\sqrt{-1}$. \nIf $\\zeta_p$ is mapped to $\\zeta^{m}_p$ ($m$ is $q$ or $r$), then $\\sqrt{-1}$ is mapped to $\\zeta^{mp'}_p$, this is $\\sqrt{-1}$ when $mp' \\equiv p'$ mod $p$ and $-\\sqrt{-1}$ when $mp' \\equiv -p'$ mod $p$. \nNote that $mp' \\equiv \\pm p'$ mod $p$ is equivalent to $m \\equiv \\pm 1$ mod $4$. \nSince $ps-qr=1$, one of $q$ and $r$ is congruent to $-1$ modulo $4$. \nTherefore, one of $[\\zeta_{p} \\longmapsto \\zeta^{q}_{p}]$ and $[\\zeta_{p} \\longmapsto \\zeta^{r}_{p}]$ maps $[\\sqrt{-1} \\longmapsto -\\sqrt{-1}]$. \nBy Fact \\ref{fact1}, there exists an integer $k$ s.t. \n$q+kp$ is prime and $q+kp \\equiv -1$ mod $4$, or $r+kp$ is prime and $r+kp \\equiv -1$ mod $4$.\n \n\\end{proof}\n\n\nFor $L(p,q)$, there exists an integer $q'$, which is $q+kp$ or $r+kp$ s.t. $L(p,q) \\cong L(p,q')$, $q'$ is prime, and $q' \\equiv -1$ mod $4$ by Lemma \\ref{make_prime}. \nBy using the Legendre symbol and Euler's criterion,\\\\\n$\\left( \\frac{-p}{q'} \\right) \\equiv \\left( \\frac{-1}{q'} \\right) \\left( \\frac{p}{q'} \\right) \\equiv (-1)^{\\frac{q'-1}{2}} \\left( \\frac{p}{q'} \\right) \\equiv -\\left( \\frac{p}{q'} \\right) $ mod $q'$\\\\\nThis implies there exists $\\epsilon \\in \\{\\pm1\\}$ such that $\\left( \\frac{\\epsilon p}{q'} \\right) =1$, i.e. \n$\\epsilon p$ is quadratic residue modulo $q'$. \nTherefore, the condition (3) is satisfied.\n\n\n\\section{Applications for hc($\\cdot$)} \\label{sec5}\n\\begin{defini} \\rm{(Remark 6.10 in \\cite{hc})}\\ \nFor a connected orientable closed 3-manifold $M$, the integer hc($M$) is defined as the minimal number $g$ such that $M$ is representable as a closure of a homology cobordism over $\\Sigma_{g,1}$.\n\\end{defini}\n\nIn \\cite{nozaki}, it was proved that for any 3-manifold $M$ whose first homology group is non-trivial and generated by one element, hc($M$)$=1$. \nHowever in general for a 3-manifold $M$ whose first homology group has a non-trivial torsion, the computation of hc($M$) becomes difficult.\\\\\nSince it is known that under plumbing operations, pairs of a 3-manifold and a homologically fibered link with Seifert surface in it are closed, we get some results about hc($\\cdot$) as follows.\n\n\\begin{cor}\n(1)\\ $\\rm{hc}\\left( \\it{L(p_1,q_1) \\# L(p_2,q_2)}\\right) \\leq 1$ if $q_1$ is a quadratic residue modulo $p_1$ and $q_2$ is a quadratic residue modulo $p_2$.\\\\\n(2)\\ $\\rm{hc}\\left( \\it{L(p,q) \\# L(p_1,q_1) \\# L(p_2,q_2)}\\right) \\leq 2$ if $q_1$ is a quadratic residue modulo $p_1$ and $q_2$ is a quadratic residue modulo $p_2$.\n\\end{cor}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nMore than 4000 exoplanets have been discovered so far via detection techniques such as the transit and radial velocity methods. A large number of them falls into the category of hot Jupiters and are normally found around around A, F and G type stars \\citep{Zellem2017}. In recent times, a new class of ultra-hot Jupiters has also been identified. Due to their close proximity to their host star, these ultra-hot Jupiters show distinct characteristics compared with hot Jupiters, such as a higher rate of transit and eclipse. Ultra-hot Jupiters with an orbital separation of less than 0.05 AU face the highest level of irradiation from their host stars. With large atmospheric scale heights they display dayside temperatures $\\geq$\\,2200 K. Their short orbital periods and high-temperature result in a high planet-to-star contrast ratio, particularly in the mid-infrared. This makes them ideal candidates to probe their atmospheres \\citep{Burrows2004}. \n\nThe atmospheres of hot Jupiters are predominantly composed of molecular hydrogen and atomic helium. Other significant molecules possibly found in their atmospheres are CO$_2$, C$_2$H$_2$, and HCN \\citep{Madhusudhan2012}. Recent study by \\cite{Lothringer2018} showed that various atomic and molecular species such as O, C, N, Fe, Mg, H$_2$O, CO, CH$_4$, N$_2$, NH$_3$ including SiO and metal hydrides are also present at high temperature in the atmospheres of hot Jupiters.\n\nWavelength dependence of the transit radii observations provides a measure of a planet's atmospheric composition \\citep{Fortney2003}. For example, atomic sodium is detected from the transit radii observations of hot-Jupiter HD209458b \\citep{Charbonneau2002} whereas H$_2$O has been detected in the atmosphere of both HD189733b \\citep{Tinetti2007} and HD209458b \\citep{Barman2007}. Observations of ultra-hot Jupiters have also revealed information about the thermal structure of their atmospheres \\citep{Madhusudhan2014}.\n\nDepending on the presence of inverted or non-inverted thermal structure profile of their atmospheres, ultra-hot Jupiters can display emission or absorption features in their spectra. They may also show a blackbody spectrum in presence of an isothermal profile \\citep{Fortney2008, Line2016}. The nature of the absorbers responsible for the thermal inversion in their atmospheres is not yet known, though the presence of vanadium oxide (VO) and titanium oxide (TiO) has been proposed to be cause for this \\citep{Burrows2008, Fortney2008}. The observed absorption broadband features in their optical transmission spectrum between 0.62 and 0.8 $\\mu$m can possibly be explained by these absorbers \\citep{Desert2008}. Processes such as vertical mixing that prevent the temperature inversion from occurring in their atmosphere are described by \\cite{Spiegel2009} and \\cite{Parmentier2013, Parmentier2016}, whereas the role of stellar activity for the thermal inversion in the atmosphere of hot-Jupiters is studied by \\cite{Knutson2010}. \n\nEvidences for thermal inversion and diverse characteristics of the atmospheres of ultra-hot Jupiters are also imprinted in their near-infrared (NIR) observations. For example, TiO has been detected as an emission feature in WASP-33b \\citep{Haynes2015, Nugroho2017} and as an absorption feature in WASP-19b \\citep{Sedaghati2017}. Emission features due to H$_2$O and VO have also been detected in WASP-121b \\citep{Evans2017}. The presence of thermal inversion without H$_2$O, TiO, and VO has been noticed in the low resolution secondary eclipse spectra of WASP-18b \\citep{Sheppard2017, Arcangeli2018}. Their study provides evidence for CO emission at 4.5 $\\mu$m, implying an inverted thermal structure. On the other hand, no clear evidence for emission or absorption features of H$_2$O is found in HAT-P-7b \\citep{Mansfield2018}, WASP-12b \\citep{Swain2013}, and WASP-103b \\citep{Kreidberg2018}. These planets show instead a thermal blackbody spectra. A handful of extremely hot-Jupiters also do not show any hint for thermal inversion such as HD209458b, KELT-1b, and Kepler-13Ab \\citep{Diamond-Lowe2014, Beatty2017a, Beatty2017b}.\n\nRetrieval techniques are commonly used to constrain the chemical composition, thermal structure, and to explain the featureless spectra of ultra-hot Jupiters \\citep{Madhusudhan2009, Line2013,Stevenson2014}. Such methods allowed to detect the presence of CO with high C\/O ratio in WASP-18b \\citep{Sheppard2017} and to derive a sub-solar metallicity associated with an oxygen-rich composition in WASP-33b \\citep{Haynes2015}. The forward modelling approach by \\cite{Arcangeli2018}, \\cite{Lothringer2018}, and \\cite{Parmentier2018} explains the lack of strong H$_2$O spectral features in the ultra-hot Jupiters due to molecular dissociation. Considering H-- opacity and the thermal dissociation of H$_2$O, \\cite{Arcangeli2018}, \\cite{Kreidberg2018}, \\cite{Kitzmann2018}, and \\cite{Mansfield2018} explained the featureless spectra in some ultra-hot Jupiters for example HAT-P-7b, KELT-9b, WASP-12b, WASP-121b, and WASP-18b. Although the retrieval techniques are useful to quantify the atmospheric abundances of ultra-hot Jupiters, the results demonstrate a large diversity in their chemical composition and thermal structures, which could lead to biased conclusions. \n\nIn this paper, we investigate the atmospheric properties of the ultra-hot Jupiter WASP-19b. WASP-19b is among the objects that exhibit weaker spectral features compared to their cooler counterparts. WASP-19b has a shorter orbital period and, thus, receives extreme irradiation from its host star. Its atmosphere is then of particular interest to study. We argue that proper chemistry and opacity play a key role in understanding its extreme hot atmosphere without invoking unusual abundances.\n\n\\section{Model Construction}\n\nTo model the atmosphere of WASP-19b, we have used the BT-Settl model atmosphere described in \\cite{Allard2012} and \\cite{Rajpurohit2012a}. BT-Settl is a state-of-the-art model atmosphere code which has been extremely successful to reproduce the properties of stellar to sub-stellar objects including cool brown dwarfs \\citep{Rajpurohit2012a, Rajpurohit2013}. The current version of the BT-Settl model uses H$_2$O along with other molecular line lists such as FeH, CrH, TiO, VO, CaH, NH$_3$, Mg, CO$_2$ and CO. Collision induced absorption (CIA) from H$_2$ collisions with H$_2$, He, CH$_4$, N$_2$, and Ar along with CO$_2$-CO$_2$, Ar-CH$_4$, and CH$_4$-CH$_4$ are also included (for more detail see \\cite{Allard2012}). The BT-Settl model accounts for many continuous opacity sources, including bound-free opacity from H, H--, He, C, N, O, Na, Mg, Al, Si, S, Ca, and Fe, free-free opacity from H, Mg, and Si, and scattering from e-- , H, He, and H$_2$. The model includes the opacities of more than 100 molecular species, including many molecules, isotopes, atomic species and their ionised states. Detailed profiles for the alkali lines as described in \\cite{Unsold1968}, \\cite{Valenti1996}, and \\cite{Allard2007} and approximation is utilised for the atomic damping constants with a correction factor to the widths of 2.5 for the non-hydrogenic atoms. Several important atomic transitions, such as the alkali, Ca I, and Ca II resonance lines along with more accurate broadening data for neutral hydrogen collisions by \\cite{Barklem2000} have been included. \n\nThe BT-Settl model also accounts for disequilibrium chemistry for C\/O and N \\citep{Graven1954, Visscher2010, Allard2012}. The reference solar elemental abundances are derived from \\cite{Caffau2011}. These models are computed with the version 15.5 of {\\tt PHOENIX}, a multi-purpose atmosphere code \\citep{Allard2001}. {\\tt PHOENIX} solves the radiative transfer in 1D spherical symmetry with irradiation such that flux is conserved at each layer. The basic assumption in {\\tt PHOENIX} while solving radiative transfer is hydrostatic equilibrium with convection using the mixing length theory, and a sampling treatment of the opacities \\citep[see][]{Allard2013}.\n\nThe atmosphere model for WASP-19b irradiated by the host star of spectral type G5 with an orbital separation of 0.0163 AU has been computed by considering boundary conditions as described in \\cite{Barman2001}. A pre-computed converged non-irradiated model structure at 400 K is chosen from \\cite{Allard2012} as the initial model structure to start with. The temperature is decreased down to 100 K to achieve the model structure. A temperature change of less than 1 K at every depth point in the thermal structure is considered as the criterion for the converged model. Effective temperature ($T_\\mathrm{eff}$) of 5500 K, surface gravity ($\\mathrm{log}\\,g$) of 4.5, and [M\/H]) = 0 have been taken as input parameters for the host star.\n \n \\begin{figure}[!htbp]\n \\centering\n \\includegraphics[width=0.50\\textwidth]{fig_1.pdf}\n \\vspace{-0.3cm}\n \\caption{Comparison of observed thermal spectra of WASP-19b with the modelled spectra calculated using {\\tt PHOENIX} (solid line) at different metallicity with an average dayside heat redistribution ($f$ = 0.5). The corresponding temperature-pressure profile is shown in the inset image (the grey shaded region corresponds to the structure inversion). The observed thermal spectrum of WASP-19b presented in this paper is adopted from \\cite{Anderson2010, Anderson2013} (filled square and filled circles), \\cite{Gibson2010} (filled upward triangle), \\cite{Bean2013} (open circles), \\cite{Burton2012} (open diamond), and \\cite{Lendl2013} (filled downward triangles). The emission features at 4.5 $\\mu$m are due to CO in presence of thermal inversion.}\n\\label{fig1}\n\\end{figure}\n\nTo explore and investigate the atmospheric properties of WASP-19b, models with uniform heat redistribution with $f$ = 0.5 across the entire dayside have been calculated with slanted rays for metallicity at --2.0, 0.0, and +2.0 dex. Here the factor $f$ measures the uniform heat redistribution where $f$ = 0.5 indicates heat redistribution across the entire dayside while $f$ = 1 indicates full heat redistribution. The model is calculated on an optical depth grid of 250 layers in log-space for $\\tau$ = $10^{-9}$ to $10^{3}$ at the peak of the spectral energy distribution which corresponds to pressures of $10^{-12}$ to $\\sim$10 bars. In the computation of the atmospheric model, some non-local thermal equilibrium (NLTE) processes have also been included for a small set of elements and level numbers (H I, He I, Li I, C I, Ni I, O I, Ne I, Na I, Na II, Mg I, Mg II, Si I, S I, K I, K II, Ca I, Ca II, Rb I), which affect the entire atmosphere. {\\tt PHOENIX} has been used to calculate both the planetary and stellar spectra from 10 to $10^{6}$ \\AA. The planetary, star and orbital parameters used for the computation irradiated model of WASP-19b are summarised in Table~\\ref{tab}. \n\n\\begin{table}[!htbp]\n \\center\n \\caption{\\label{tab} Planetary, stellar and orbital parameters used in the model are from \\cite{Butler2006}, \\cite{Southworth2007}, \\cite{Hebb2010} and \\cite{Hellier2011}.} \n \\begin{tabular}{cc}\n \\hline\n \\hline\n \\noalign{\\smallskip}\n Parameter & Value \\\\\n \\noalign{\\smallskip}\n \\hline\n \\noalign{\\smallskip}\n $a$ & 0.0163 AU \\\\[1mm]\n \\noalign{\\smallskip}\n R\\textsubscript{[p]} \t\t\t& 1.386 ({R\\textsubscript{Jupiter}}) \\\\[1mm]\n \\noalign{\\smallskip}\n $\\mathrm{log}\\,g$\\textsubscript{[p]} \t\t& 3.17 [cm\/sec$^2$]\\\\[1mm]\n \\noalign{\\smallskip}\n $T_\\mathrm{int}$\\textsubscript{[p]} \t\t& 100 [K]\\\\[1mm]\n \\noalign{\\smallskip}\n $T_\\mathrm{eff}$\\textsubscript{[$\\star$]} \t& 5500 [K]\\\\[1mm]\n \\noalign{\\smallskip}\n $\\mathrm{log}\\,g$\\textsubscript{[$\\star$]}\t\t& 4.5 [cm\/sec$^2$]\\\\[1mm]\n \\noalign{\\smallskip}\n R\\textsubscript{[$\\star$]} \t\t& 0.990 ({R\\textsubscript{$\\sun$}}) \\\\[1mm]\n \\noalign{\\smallskip}\n [M\/H]\\textsubscript{[$\\star$]} \t& 0.0 \\\\[1mm]\n \\hline\n \\end{tabular}\n\\end{table}\n\n\\section{Results and Discussion}\n\nA thermal spectrum of WASP-19b, from \\cite{Anderson2010}, \\cite{Gibson2010}, \\cite{Burton2012}, \\cite{Anderson2013}, \\cite{Bean2013}, and \\cite{Lendl2013}, is shown in Fig~\\ref{fig1}. The planet-to-star flux ratio at 4.5 $\\mu$m is higher than the one at 3.6 $\\mu$m {\\tt Spitzer}\/IRAC channel. Moreover, at higher wavelengths, namely at 5.8 and 8.0 $\\mu$m, the planet-to-star flux ratio is even higher than 4.5 $\\mu$m. This clearly indicates the presence of excess emission in some photometric bands over others. \n\nThe atmosphere of WASP-19b has been modelled with the parameters given in Table~\\ref{tab} using {\\tt PHOENIX}. It shows the presence of thermal inversion (see inset diagram, Fig.~\\ref{fig1}) and the equilibrium temperature is found to be $\\sim$ 2700 K. The comparison between observational data and our theoretical modelling (Fig~\\ref{fig1}) indicates that the excess planet-to-star flux ratio at 4.5 $\\mu$m is due to the presence of CO, which in turn is caused by thermal inversion that exists in the atmosphere. The corresponding pressure range, as probed by the secondary eclipse, is $10^{-2}$ to $10^{-3}$ bar. Previous studies on WASP-19b did not show any evidence of thermal inversion in its atmosphere \\citep{Anderson2013, Bean2013}. However, these studies were mainly based on retrieval techniques where abundances and thermal structure were fitted to the data.\n\n\n \\begin{figure*}[!htbp]\n \\centering\n \\includegraphics[height=6cm,width=17cm]{fig_2a_1.pdf}\n \\includegraphics[height=6cm,width=17cm]{fig_2b_1.pdf}\n \\includegraphics[height=6cm,width=17cm]{fig_2c_1.pdf}\n \\vspace{-0.2cm}\n \\caption{Concentration of important absorbing molecules and neutral atoms as a function of optical depth and temperature at [M\/H] = +2.0 (top), [M\/H] = 0.0 (middle), and [M\/H] = -2.0 (bottom). These species deplete at the given pressure range for the irradiated hot Jupiter WASP-19b (see main text). The grey shaded region (left panels) corresponds to the structure inversion and shows the window of the observable atmosphere from which most of the spectrum emerges (in the CO, H$_2$O, TiO molecular bands pseudo continuum). We notice that in the upper atmosphere ($\\tau$ $<$ $10^{-2}$), H$_2$O, TiO and VO dissociate whereas neutral atomic oxygen remains constant. We find that CO remain constant throughout the atmosphere of WASP-19b. The temperature inversion (right panels) causes the wiggles along the structure causing CO bands to emerge in emission.}.\n \\label{fig2}\n\\end{figure*}\n\nWe investigate the effect of metallicity on the atmosphere of WASP-19b by changing all the heavy-elemental abundances by a factor of two. As shown by the inset image of Fig.\\,\\ref{fig1}, the thermal inversion becomes stronger as the metallicity increases. This is mainly due to the temperature dependence of the thermal dissociation of various molecules, which changes the photospheric abundances ratio of TiO and H$_2$O at different metallicities \\citep{Parmentier2018}. At higher metallicities, the increase in the abundances of various absorbers such as TiO warms the upper atmosphere. This causes the thermal inversion to be even stronger as compared to solar metallicity which leads to strong emission features of CO and shallower features of H$_2$O in their thermal spectra. We find that models with sub-solar metallicities show signature of a very weak or no thermal inversion (see Fig.\\,\\ref{fig1}). This is a result of the decrease of TiO and VO abundances. In most of the hot Jupiters or massive planets, the metallicity is believed to approach the metallicity of the host star \\citep{Torres2012, Arcangeli2018}. We also investigate this by comparing the observed thermal spectra of the dayside atmosphere of WASP-19b with the models at [M\/H] = -2.0, 0.0 and +2.0. Our results show that thermal spectra of WASP-19b can be explained with a model at solar metallicity with thermal inversion without invoking unusual abundances.\n\nIn case of ultra-hot Jupiters, the molecular species required to explain the featureless observations are physically plausible. At extremely high temperature, the various molecules responsible for the radiative cooling in their atmosphere do not exist. In Fig.\\,\\ref{fig2}, we show the mixing ratio of the most important atomic and molecular species in the atmosphere of WASP-19b at different metallicity as a function of optical depth and temperature. We see that H$_2$ dissociates in the upper atmosphere due to impinging radiation from about $\\tau$ = $10^{-2}$, but not enough to prevent the molecular hydrogen atmosphere. Due to the impinging radiation, free electrons are available and constant over most of the upper photosphere. Neutral atomic oxygen is fully locked to the very stable CO molecule in the deep atmosphere ($P$ $>$ $10^{-2}$ bar), but becomes as abundant as CO in the upper atmosphere. This is the result of partial dissociation of CO, and other oxygen--bearing molecules. While CO is quasi-constant throughout the atmosphere. We find that H$_2$ and CO are the most abundant molecules, while CO, TiO and VO are the most important molecular opacity sources in the atmosphere of WASP-19b. \n\nAs shown Fig.\\,\\ref{fig2} (left panels), we find that the variation of atomic and molecular abundances, together with the strong impinging radiation, contribute to the thermal inversion in the atmosphere of WASP-19b. At 0.0163 AU, the thermal structure of WASP-19b is too hot for the formation of the BT-Settl clouds, whereas the presence of CO\/CO$_2$ reveals the disequilibrium chemistry. We find the formation of a limited amount of CO$_2$ in disequilibrium chemistry, but not abundant enough to participate significantly in the CO and H$_2$O balance. We also show that TiO and alkali doublets, seen in early to mid-type brown dwarfs, are the main opacities in the optical to near-IR spectrum of WASP-19b. It is evident from Fig.\\,\\ref{fig2} (right panels) that the temperature inversion causes the wiggles in the concentrations along the structure causing CO bands to appear in emission. \n\nFigure\\,\\ref{fig3} shows the synthetic spectra of WASP-19b in the optical at different metallicities. We see that despite extremely low abundances, similar to brown dwarfs and irradiated hot-Jupiter atmospheres \\citep{Allard2001, Barman2001, Barman2002}, the TiO cross-sections are strong enough to preserve TiO as the main optical to near-IR opacity, along with alkali atomic fundamental transitions. These are the most important opacity sources, together with CO in the infrared, that survive the immense heat impinging of the planet atmosphere. The thermal inversion potentially hides pseudo-continuum opacities such as H$_2$O, and the high temperatures do not allow triatomic to remain stable. Also at given resolution and at such high temperature conditions, H$_2$O has a much flatter opacity profile making it more difficult to recognise, especially at those spectral resolutions.\n\n\n \\begin{figure}[!htbp]\n \\centering\n \\includegraphics[width=0.50\\textwidth]{fig_3.pdf}\n \\vspace{-0.7cm}\n \\caption{Synthetic spectra of WASP-19b in the optical range calculated using {\\tt PHOENIX} at different metallicities with an average dayside heat redistribution (f = 0.5). Labels of the most prominent atomic and molecular features remaining important opacity sources, even at low concentrations, are indicated.\n}\n\\label{fig3}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nWe have used the state-of-the-art 1D NLTE opacity sampling model atmosphere code {\\tt PHOENIX} to study the atmosphere of WASP-19b. This model has been successfully used to study the atmosphere of cool stars, brown dwarfs and extrasolar planets. The temperature-pressure profile of WASP-19b computed using {\\tt PHOENIX} shows the presence of thermal inversion. Our model computed at solar metallicity successfully reproduces the observed photometry of WASP-19b without the need for non-solar composition. The secondary eclipse of WASP-19b shows evidence for CO emission features at 4.5 $\\mu$m, but however no sign of H$_2$O. We find that these features are the results of thermal dissociation and thermal inversion due to the strong impinging radiation. Also, the H$_2$O pseudo-continuum is much smoother at the concerned temperatures and densities, making it more difficult to identify at those extremely coarse spectral resolutions. Our results further strengthen the fact that the family of ultra-hot Jupiters commonly exhibit thermal inversions. \n\nAt longer wavelengths (above 1.4 \\,$\\mu$m) and at extremely hot temperatures (2200--2800 K), a significant amount of H$_2$O gets thermally dissociated at a pressure below $10^{-2}$ bar. At a temperature above 2200 K, TiO and VO do remain important opacity sources, even at low concentrations. At longer wavelengths, CO is the only molecule with its strong triple bond to be abundant below $10^{-2}$ bar. This molecule does not dissociate and its emission features appear at 4.5 $\\mu$m. We suggest that the actual reason for the drop in H$_2$O in ultra-hot Jupiters is the partial thermal dissociation of this molecule and the resulting thermal inversion which shapes the thermal spectra of ultra-hot Jupiters. This makes H$_2$O a poor diagnosis of the C\/O ratio. Also at such high dayside temperature ($>$ 2200 K), and at that resolution, the opacity profile of H$_2$O is spectrally much flatter and difficult to recognise.\n\n\\begin{acknowledgements}\nWe thank the anonymous referee for providing comments and suggestions which helped improve the clarity and conciseness of the paper. AR are especially grateful to Mudit Kumar Srivastava from Physical Research Laboratory (PRL) for providing feedback on the manuscript. The computations were performed on the HPC resources at the Physical Research Laboratory (PRL). The research leading to these results has received funding from the French \"Programme National de Physique Stellaire\" and the Programme National de Planetologie of CNRS (INSU). The computations were performed at the {\\sl P\\^ole Scientifique de Mod\\'elisation Num\\'erique} (PSMN) at the {\\sl \\'Ecole Normale Sup\\'erieure} (ENS) in Lyon, and at the {\\sl Gesellschaft f{\\\"u}r Wissenschaftliche Datenverarbeitung G{\\\"o}ttingen} in collaboration with the Institut f{\\\"u}r Astrophysik G{\\\"o}ttingen. O.M. acknowledges support from CNES. \n\\end{acknowledgements}\n\n \n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nNonlocal diffusion equations come up in a number of contexts. These range from\nbiology \\cite{Hutson2003,Bates2007,Maly2001,Weichsel2010}, to materials science\n\\cite{Chen2011}, and graph theory. Much of the analysis to this point has been\ndone on bounded domains \\cite{Coville2010}. The results presented here\nrepresent a first step in analyzing nonlocal diffusion equations on unbounded\ndomains. The theory of positive operators on Banach lattices is employed to\ngive useful conditions to show existence of a principal eigenvalue in the\nabsence of compact domains. Many more results are likely to come from applying\nknown results about positive operators to nonlocal diffusion equations. In\nparticular, in Section \\ref{maximum}, we use the existence of a particular type\nof maximum principal implies exponential convergence to zero.\n\nThe existence of a principal eigenvalue has been used to study many nonlocal\ndiffusion equations, both linear and nonlinear \\cite{Bates2007,\n Garcia-Melian2009}. Energy methods and Fourier analysis have been used to\nprovide polynomial bounds on the decay of nonlocal diffusion equations\n\\cite{Chasseigne2006, Ignat2009}, with exponential decay shown in some special\ncases \\cite{Ignat2009}. However, estimates of the principal eigenvalue would be\nable to definitely show exponential decay. While much work has been done on the\nexistence of principal eigenvalues on bounded domains, few general results\nexist for unbounded domains \\cite{Ignat2012, Coville2010}. It is important to\nnote that, even on bounded domains, existence of a principal eigenvalue is\nstill not guaranteed. See, for example, the counter example in section\n\\ref{sec:exist}. The work here uses the theory of positive operators to find a\nsufficient condition for the existence of a principal eigenvalue without any\nassumptions on the boundary conditions. The method also gives a technique for\nestimating the value of the eigenvalue. Further work is needed to characterize\nthe multiplicity of the eigenvalue or the existence of a spectral gap.\n\nThe general equation of interest is:\n\\begin{equation}\\label{dyna}\n \\dot{u}(x,t)=\\int\\limits_{\\Omega}J(x,y)u(y,t)\\,\\mathrm{d} y-a(x)u(x,t)\n\\end{equation}\nwhere $\\Omega$ is some connected, possibly unbounded subset of the real line\n$\\mathbb{R}$ and $u\\in\\mathnormal{L}^p(\\Omega)$ for $1\\leq p<\\infty$. This equation is often\ninterpreted as modeling some form population dispersal. Individuals propogate\nfrom point $y$ to point $x$ at a rate $J(x, y)$ and individuals die off at a\nrate $a(x)$ depending on their location. Note that the proofs here\nimmediately generalizes to $\\mathbb{R}^n$. $a$ is assumed to be continuous and\nbounded in the sense that there exists $c$, $c^\\prime$ such that:\n\\begin{equation}\n 0-\\inf a(x)$. The two operators\nare related in the sense that $\\op{L}$ has $\\lambda>-\\inf a$ as an eigenvalue\nif and only if $\\op{A}_\\lambda$ has an eigenvalue equal to one.\n\nThe theorem can be stated as such:\n\\begin{thm}\\label{condition}\n If:\n \\begin{equation}\n \\lim_{\\lambda\\to(-\\inf a)^+}\\spr(\\op{A}_\\lambda)>1\n \\end{equation}\n where $\\spr(\\op{A}_\\lambda)$ is the spectral radius and one of the conditions\n in Lemma \\ref{K-R-eigen} holds, then $\\op{L}$ has a principal eigenvalue,\n $\\lambda_0\\in\\mathbb{R}$, with positive eigenfunction such that for any element\n $\\lambda\\in\\sigma(\\op{L})$, we have the inequality $\\Re\\lambda\\leq\\lambda_0$.\n\\end{thm}\nThe condition in Theorem \\ref{condition} holds for any boundary conditions and\nis not dependent upon the domain being bounded.\n\nA quick lemma that gives a few conditions necessary for the existence of an\neigenvalue:\n\\begin{lem}\\label{K-R-eigen}\n $\\op{A}_\\lambda$ has a positive eigenvalue equal to its spectral radius with\n a positive eigenfunction in $\\mathnormal{L}^1$ whenever $\\lambda>-\\inf a$ if any\n of the following conditions hold:\n \\begin{enumerate}\n \\item Range of $\\op{J}^n\\subset \\mathnormal{L}^p$ for $11}\n \\item $J(x,y)=K(x-y)$ is of convolution type with $K$ absolutely bounded.\n \\label{convolution}\n \\item $\\exists c(x)\\in\\mathnormal{L}^1$ such that $J(x,y)\\leq c(x)$ \n \\label{boundedness}\n \\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n $\\op{A}_\\lambda$ is obviously a positive operator on the appropriate lattice\n in the sense that it preserves the positive cone. Because of the boundedness\n of $a+\\lambda$, we know that $\\op{A}_\\lambda$ has range in the same function\n space(s) as $\\op{J}$. For that reason, the rest of the proof will only be\n concerned with $\\op{J}$. Also, recall that we are assuming $\\op{J}$ has \n non-zero spectrum for the entirety of this article. \n\n For condition \\ref{p>1}, we know that all of the $\\mathnormal{L}^p$ spaces where\n $1\\mu$, we have that\n $\\spr(\\op{A}_\\lambda)$ is a monotonically decreasing, upper-continuous\n function. \n\n It remains to show that the spectral radius function is lower-continuous.\n Define the function $c(\\lambda,\\lambda^\\prime)$ for $\\lambda$,\n $\\lambda^\\prime>-\\inf a$:\n \\begin{equation}\n c(\\lambda,\\lambda^\\prime)\n =\\sup_{x\\in\\Omega}\\frac{\\lambda+a(x)}{\\lambda^\\prime+a(x)}\n \\end{equation}\n Observe that $c$ is a continuous function and $c(\\lambda,\\lambda)=1$ for all\n $\\lambda>-\\inf a(x)$. Assume $\\lambda>\\lambda^\\prime$. Then,\n $c(\\lambda,\\lambda^\\prime)>1$. We also have the two inequalities:\n \\begin{equation}\n \\op{A}_\\lambda\\leq\\op{A}_{\\lambda^\\prime}\n \\qquad\n c(\\lambda,\\lambda^\\prime)\\op{A}_\\lambda\\geq\\op{A}_{\\lambda^\\prime}\n \\end{equation}\n We can now directly calculate:\n \\begin{equation}\n \\lim_{\\lambda\\,\\downarrow\\,\\lambda^\\prime}\\spr(\\op{A}_\\lambda)\n \\geq\\lim_{\\lambda\\,\\downarrow\\,\\lambda^\\prime}c(\\lambda,\\lambda^\\prime)\n \\spr\\op{A}_{\\lambda^\\prime}=\\spr(\\op{A}_{\\lambda^\\prime}).\n \\end{equation}\n which completes the proof of lower-continuity.\n\\end{proof}\nNote that the result of the above lemma holds even withtout any of the\nnecessary conditions for Lemma \\ref{K-R-eigen}. That fact becomes important for\nthe result in Section \\ref{maximum}. We can now move on to the rest of the\nproof.\n\n\\section{Proof of the Theorem}\n\nFirst, we will show that the operator $\\op{L}$ is positive resolvent. Choose\n$\\lambda\\in\\mathbb{R}$ so large that $\\mu\\geq\\lambda$ implies $\\mu\\in\\rho(\\op{L})$ the\nresolvent of $\\op{L}$ and that $\\spr(\\op{A}_\\mu)<1$. Assume $f\\geq0$,\n$f\\in\\mathnormal{L}^1$, is in the range of $\\mu-\\op{L}$:\n\\begin{equation}\n \\mu v-\\op{L}v=f\n\\end{equation}\nfor some $v\\in\\mathnormal{L}^1$. Expanding the above equation gives:\n\\begin{align}\\nonumber\n \\mu v-\\op{J}v+a v &= f & \\iff \\\\\\nonumber\n v-\\frac{\\op{J}v}{\\mu+a} &= \\frac{f}{\\mu+a} & \\iff \\\\\\nonumber\n v-\\op{A}_\\lambda v &= \\frac{f}{\\mu+a} & \\iff \\\\\n v &= \\sum_{j=0}^\\infty\\op{A}_\\lambda^j\\frac{f}{\\mu+a}\\geq0\n\\end{align}\nSince we have assumed that $\\spr(\\op{A}_\\mu)<1$, the above series converges \nand $(\\mu-\\op{L})^{-1}\\geq0$. By \\cite{Nussbaum1984}, we know that there exists\n$\\lambda_0$ such that:\n\\begin{equation}\n \\spr\\Big((\\mu-\\op{L})^{-1}\\Big)=\\frac{1}{\\mu-\\lambda_0}\n\\end{equation}\nfor all $\\mu>\\lambda_0$. We also have the characterization:\n\\begin{equation}\n \\lambda_0=\\inf\\{\\lambda\\in\\mathbb{R}|(\\lambda-\\op{L})^{-1}\\geq0\\}\n\\end{equation}\nAssume $\\lambda_0>-\\inf a$. By \\cite{Top-Vect}, we know that:\n\\begin{equation}\n (\\op{I}-\\op{A}_\\lambda)^{-1}\\geq0 \\iff \\spr(\\op{A}_\\lambda)<1\n\\end{equation}\nThe above condition implies that $\\spr(\\op{A}_\\lambda)<1$ for all\n$\\lambda>\\lambda_0$ and $\\spr(\\op{A}_{\\lambda_0})\\geq1$. The continuity of\n$\\spr(\\op{A}_\\lambda)$ implies that $\\spr(\\op{A}_{\\lambda_0})=1$. From our\napplication of the Krein-Rutman theorem, we know there exists $u\\geq0$ such\nthat:\n\\begin{equation}\n \\op{A}_{\\lambda_0}u=u\n\\end{equation}\nFrom our definition of $\\op{A}_\\lambda$, the above implies:\n\\begin{equation}\n \\op{L}u=\\lambda_0u\n\\end{equation}\n$\\lambda_0$ is therefore our principal eigenvalue and $u$ its associated\npositive eigenfunction.\n\nAll that remains to be shown is that $\\lambda_0>-\\inf a$. Recall the hypothesis\non the spectral radius limit:\n\\begin{equation}\n \\lim_{\\lambda\\to(-\\inf a)^+}\\spr(\\op{A}_\\lambda)>1\n\\end{equation}\nThat inequality implies there exists a largest $\\lambda_0>-\\inf a$ such that\n$\\spr(\\op{A}_{\\lambda_0})=1$, and we are finished.\n\nNote that in the proof of the theorem, the structure of the integral equation\nonly comes up in Lemma \\ref{K-R-eigen}. Define the more general operator\n$\\op{B}$ as:\n\\begin{equation}\n \\op{B}f = \\op{K}f - af\n\\end{equation}\nwhere $\\op{K}$ is some positive operator. The only conditions we need for\nTheorem \\ref{condition} to hold are that $\\op{K}$ has non-zero spectrum and the\nrange of $\\op{K}^n$ is inside a compact topological space for some finite\n$n\\in\\mathbb{N}$.\\footnote{See Theorem 6.6 in Appendix V. and the following example in\n \\cite{Top-Vect} for details.} In particular, if $\\op{K}^n\\rightarrow\nU\\subset\\mathnormal{L}^p$ for $1c$ for all $(x,y)\\in U$. Then, $\\op{L}$ has a principal eigenvalue.\n\\end{prop}\n\n\\begin{proof}\n Without loss of generality, assume $x^\\star=0$. From \\cite{Drnovsek2000}, we\n can get a lower bound on the spectral radius by observing that:\n \\begin{equation}\n \\op{A}_\\lambda u\\geq c u \n \\quad \n \\implies \n \\quad\n \\spr(\\op{A}_\\lambda)\\geq c\n \\end{equation}\n We can then explicitly borrow the test function from \\cite{Hutson2003}. Fix\n $\\delta$, $\\varepsilon$ such that $J(x,y)\\geq\\varepsilon$ for all $|x|$, $|y|\\leq\\delta$.\n For every $\\gamma>0$ write $u_\\gamma$:\n \\begin{equation}\n u_\\gamma(x)=\n \\begin{cases}\n \\frac{1}{\\gamma+a(x)-a(0)} & \\text{if } |x|<\\delta \\\\\n 0 & \\text{else}\n \\end{cases}\n \\end{equation}\n We have the inequalities:\n \\begin{align}\\nonumber\n \\op{J}u_\\gamma(x)&=\\int\\limits_{-\\delta}^\\delta \n \\frac{J(x,y)}{\\gamma+a(y)-a(0)}\\,\\mathrm{d} y \\\\\n &\\geq \\varepsilon\\int\\limits_{-\\delta}^\\delta \\frac{1}{\\gamma+C|y|}\n \\geq \\frac{\\varepsilon}{C}\\ln\\left[\\frac{C\\delta+\\gamma}{\\gamma}\\right]\n \\end{align}\n Choose $\\gamma>0$ such that the last term above is greater than \n one and $-a(0)<\\lambda<\\gamma-a(0)$. Using the above inequality gives:\n \\begin{align}\\nonumber\n \\op{A}_\\lambda u_\\gamma(x)&=\\frac{\\op{J}}{\\lambda+a(x)} \\\\\\nonumber\n &\\geq\\frac{1}{\\gamma+a(x)-a(0)}\n \\frac{\\varepsilon}{C}\\ln\\left[\\frac{C\\delta+\\gamma}{\\gamma}\\right] \\\\\n &\\geq u_\\gamma(x)\n \\end{align}\n We have now shown that $\\spr(\\op{A}_\\lambda)\\geq1$ and can invoke Theorem \n \\ref{condition}.\n\\end{proof}\n\n\\begin{prop}\\label{converge-inf}\n Assume that $a(x)$ converges to its infimum at either $\\pm\\infty$, and there\n exists $c$, $c^\\prime$ such that $J(x,y)>\\varepsilon$ for $|y-x|<\\delta$ for all\n $x\\in\\Omega$. Then, $\\op{L}$ has a principal eigenvalue.\n\\end{prop}\n\\begin{proof}\n Without loss of generality, assume $a(x)\\to\\inf a$ as $x\\to\\infty$. We can\n construct almost the same inequality as in the proof of Proposition \n \\ref{achieves-inf}. Choose $\\lambda$, $z$ such that:\n \\begin{equation}\n \\frac{\\varepsilon}{\\lambda+a(x)}>1 \n \\quad\n \\text{for }|x-z|<\\delta\n \\end{equation}\n Choosing $u=\\mathbf{1}_{(z-\\delta, z+\\delta)}$ combined with the inequality in\n \\cite{Drnovsek2000} completes the proof. \n\\end{proof}\n\nNote that the conditions on J for Proposition \\ref{converge-inf} are easily\nsatisfied if $J(x,y) = f(x-y)$ and $f>c>0$ on some neighborhood of zero.\nThe proof of that proposition could also be easily generalized \nWithout enumerating all possible cases, it is obvious that other conditions on\n$J$ can be used for other similar cases, e.g. $a(x) =\n\\frac{1}{1+|x|}+\\cos(\\theta)$ and that weaker conditions on $J$ can be found\nfor the cases listed above.\n\n\\begin{prop}\n Assume $a(x)$ is locally Lipschitz and reaches its infimum at $x^\\star$ and\n $J(x,y)=J(y,x)$. Also, assume the range of $\\op{J}$ is in $\\mathnormal{L}^2$ and there\n exists $\\delta$, $\\varepsilon>0$ and a bounded set $U$ such that:\n \\begin{equation}\n \\int_U J(x,y)\\,\\mathrm{d} x > \\varepsilon\n \\quad\n \\text{for } |y-x^\\star|<\\delta \\text{ a.e.}\n \\end{equation}\nThen, $\\op{L}$ has a principal eigenvalue.\n\\end{prop}\n\\begin{proof}\n Again, we will assume $x^\\star=0$ for the sake of notation. Define the\n family of inner-products $\\langle\\cdot,\\cdot\\rangle_{\\lambda+a}$ by:\n \\begin{equation}\n \\langle f, g\\rangle_{\\lambda+a} = \n \\int_\\Omega f(x)\\bar{g}(x)\\Big(\\lambda+a(x)\\Big)\\,\\mathrm{d} x\n \\end{equation}\n It is easy to observe that $\\op{A}_\\lambda$ is self-adjoint with respect to\n the inner-product $\\langle\\cdot, \\cdot\\rangle_{\\lambda+a}$ and that\n $\\langle\\cdot,\\cdot\\rangle_{\\lambda+a}$ is topologically equivalent to the\n usual $\\mathnormal{L}^2$ inner-product for all $\\lambda>-\\inf a$.\n\n Need to show there exists $v$ and $\\lambda$ such that:\n \\begin{equation}\n \\frac{\\langle\\op{A}_\\lambda v,v\\rangle_{\\lambda+a}}\n {\\langle v,v\\rangle_{\\lambda+a}}>1\n \\end{equation}\n Use:\n \\begin{equation}\n v_\\lambda = \\frac{\\mathbf{1}_{(-\\delta,\\delta)}}{\\lambda+a(x)}+\\mathbf{1}_U\n \\end{equation}\n where $U$ and $\\delta$ are as in the statement of the proposition. Choose\n some $\\lambda^\\star>-\\inf a$ and observe that:\n \\begin{equation}\n \\langle v_\\lambda,v_\\lambda\\rangle_{\\lambda+a}\n \\leq \\langle v_{\\lambda^\\star}, v_{\\lambda^\\star}\\rangle_{\\lambda^\\star+a}=C\n \\end{equation}\n for all $\\lambda\\leq\\lambda^\\star$. \n\n Define $\\gamma=\\lambda-\\inf a$. We can now construct the sequence of\n inequalities:\n \\begin{align}\\nonumber\n \\langle \\op{A}_\\lambda v_\\lambda, v_\\lambda\\rangle_{\\lambda+a} &=\n \\left\\langle\\frac{\\op{J} v_\\lambda}\n {\\lambda+a},v_\\lambda\\right\\rangle_{\\lambda+a} \n \\\\\\nonumber\n &=\\Big\\langle\\op{J}v_\\lambda,v_\\lambda\\Big\\rangle \\\\\\nonumber\n &\\geq\\int_\\Omega\\frac{\\op{J}\\mathbf{1}_U(x)}{\\lambda+a(x)}\\,\\mathrm{d} x \\\\\n &\\geq\\int\\limits_{-\\delta}^\\delta\\frac{\\varepsilon}{\\gamma+c|x|}\\,\\mathrm{d} x\n \\geq \\frac{\\varepsilon}{c}\\ln\\left[\\frac{c\\delta+\\gamma}{\\gamma}\\right]\n \\end{align}\n Choosing $\\lambda$ sufficiently small gives $\\langle\\op{A}_\\lambda\n v_\\lambda,v_\\lambda\\rangle_{\\lambda+a}>C$, which gives the desired result.\n\\end{proof}\n\nFinally, a counter-example to show that the Lipschitz hypothesis is necessary\nwhen $a(x)$ reaches its global minimum and is bounded away from that minimum\nelsewhere. For simplicity, we will take $\\Omega=S^1$. Take the family of\nsystems where $J=\\frac{1}{2\\pi}$ and $a(x)=c|x|^\\alpha+c^\\prime$ where\n$c,\\,c^\\prime>0$ and $0<\\alpha<1$. Obviously, $a$ is $\\alpha$-H\\\"{o}lder\ncontinuous. Assume we have an eigenpair $\\mu$, $v(x)$. Dividing through\nwherever $a+\\mu\\neq0$, we have:\n\\begin{equation}\n v =\\frac{\\int v(\\omega)\\,\\mathrm{d}\\omega}{c|\\theta|^\\alpha+c^\\prime+\\mu}\n\\end{equation}\nThat means $\\int v(\\omega)\\,\\mathrm{d}\\omega=1$ and\n$v=\\frac{1}{c|\\theta|^\\alpha+c^\\prime+\\mu}$. However, if we set:\n\\begin{equation}\n c=2\\int\\limits_{S^1}|\\theta|^{-\\alpha}\\,\\mathrm{d}\\theta\n\\end{equation}\nthe above equations are not solvable for any pair $c,\\,\\mu$ since $\\int\nv(\\omega)\\,\\mathrm{d}\\omega\\leq\\frac{1}{2}$ for all $\\mu\\geq-c^\\prime$. If\n$\\mu\\frac{(r^2+a^2)^2}{a^2\\Delta}-\\frac{2Mr\\lambda}{a\\Delta}\\equiv \\sin^2\\theta_{c.v.}\n\\end{equation}\nwhere the subscript $(c.v.)$ stands for {\\it chronology violation}. \nClearly function \n$\\sin^2\\theta_{c.v.}(r;\\lambda)$\nidentifies a region in the $(\\sin^2\\theta-r)-$plane whose very existence and {\\it size} depend \non the orbit itself. In particular it is \neasy to see that the smaller is $\\lambda$ in the range $0<\\lambda0$ universe after a sufficient recovering \nof the lost (coordinate) time.\n\nA vortical orbit is confined between two values of the latitudinal angle $\\theta$; \nin the case of photon \norbits these angles are given by\n\\begin{equation}\n\\label{theta} \n\\sin^2\\theta_\\pm=\\frac{\\Lambda+a^2\\pm[\\Lambda+a^2)^2-4\\lambda^2a^2]^{1\/2}}{2a^2}\n\\end{equation}\nhence condition (\\ref{cv}) will be satisfied only if at least one of the hyperboloids \n$\\theta=\\theta_\\pm$ \n as in (\\ref{theta}) crosses the corresponding $(c.v.)-$region. \nThis circumstance takes place if \n\\begin{equation}\n\\label{Ltheta}\n\\Lambda=-a^2-\\frac{\\lambda^2a\\Delta}{\\Psi}-\\frac{a\\Psi}{\\Delta}\\equiv\\Lambda_\\theta\n\\end{equation}\nwhere\n\\begin{equation}\n\\Psi\\equiv 2M\\lambda r-\\frac{(r^2+a^2)^2}{a}.\n\\end{equation}\nTurning points are met where $\\dot r=0$ and this is assured when \n\\begin{equation}\n\\label{Lr}\n\\Lambda=\\frac{(2Mr-a\\lambda)^2}{\\Delta}+r^2+2Mr\\equiv\\Lambda_r.\n \\end{equation}\nA comparison of (\\ref{Ltheta}) with (\\ref{Lr}) shows that (see figure 1)\n\\begin{equation}\n\\Lambda_\\theta\\leq \\Lambda_r,\\qquad (r\\leq 0)\n\\end{equation}\nwhere the equality sign holds identically when $\\lambda=0$ and only at $r=0$ when $\\lambda^2=\\Lambda$. \n\n\\begin{figure}\n\\typeout{*** EPS figure 1}\n\\begin{center}\n\\includegraphics[scale=0.4]{de1.eps}\n\\end{center}\n\\caption{Plot of the functions $\\Lambda_r$ (solid line) and $\\Lambda_\\theta$ (dotted line)\nfor a general value of $\\lambda$ with $0<\\lambda\\Lambda\\approx \\Lambda_{r_{min}}\n\\end{equation}\nwill move on time reversed, spatially open and almost stationary loops at an average distance $r_{min}$ \nfrom the singularity \n(in the $r<0$ sheet; see figure 1) before moving back to positive infinity again. \nThese are the prerequisites of a Cosmic Time Machine. \n\n\\section{The time trap}\n\nFor a light signal to move on a time reversed trajectory, the light cone must be {\\it deformed}\nin such a way that its future pointing generators propagate light signals into the \nlocal coordinate past, namely with the coordinate time decreasing. The light cone structure can \nbe seen explicitely in the Kerr naked singularity solution. From (\\ref{geodeqt}) \n and (\\ref{geodeqr}) it follows that the light cone generators in the ($ct-r$)-plane satisfy the \nequation:\n \\begin{equation}\n \\label{generator}\n \\frac{dt}{dr}=\\pm\\frac{a^2(\\sin^2\\theta_{c.v.}-\\sin^2\\theta)}{\\left[(2Mr-a\\lambda)^2+\n\\Delta(r^2+2Mr-\\Lambda)\\right]^{1\/2}}\n\\end{equation}\nwhere $\\sin^2\\theta_{c.v.}$ is given by (\\ref{cv}). As $\\theta\\to\\theta_{c.v.}$ from below, \nnamely with $\\theta<\\theta_{c.v.}$,\n $dt\/dr$ decreases untill it vanishes for {\\it both} outgoing and ingoing generators.\nAt this moment the light cone is {\\it fully} open with respect to the coordinate time axis; \nas $\\theta$ increases further so that\n $\\theta>\\theta_{c.v.}$, the light cone shrinks again but with the local future reversed with respect \nto the coordinate time (see figure 2). \nAs it will be discussed in a separate paper (de Felice and Preti, 2006) the opening of the light cone \nis a manifestation of the repulsive character \nof the space-time; indeed this is the property of Kerr space-time nearby the ring singularity.\n\n\n\n\\begin{figure}\n\\typeout{*** EPS figure 2}\n\\begin{center}\n\\includegraphics[scale=0.4]{de2.eps}\n\\end{center}\n\\caption{ The light cone structure approaching the ring singularity in the $r<0$-sheet of the metric}\n\\label{grbfig2}\n\\end{figure}\n\nThe light trajectories which have the necessary prerequisites to become time-reversed \nact as time-traps since they force light signals to travel back in the coordinate time \n(see figure 2).\nEventually the light signals are bounced back at a tunrning point ($\\dot r=0$) and that happens, as stated,\n if $\\Lambda\\geq \\Lambda_{r_{min}}$; the coordinate time that a light signal recovers before \ngoing back to positive infinity depends on how close $\\Lambda$ is to $\\Lambda_{r_{min}}$. If \n$\\Lambda\\approx \\Lambda_{r_{min}}$ \n the light signal may loop on a spatially quasi-circular orbit \n($r\\sim r_{min}$) untill the coordinate time reaches reaches the value when the singularity first formed. \nAt this moment the singularity is of the most general type; we expect, in fact, \nthat a naked singularity becomes of a Kerr type only when it is about to \ndecay to a Kerr black hole. Although it is difficult to model the light cone \nstructure nearby a singularity at its onset, the occurrence of time inversion in its vicinity \nis assured by a general theorem as we said and will better illustrate next.\n\n\\section{A cosmic burst}\nThe connection between a naked singularity and a Cosmic Time Machine \nhas been established in general by Clarke and de Felice (1984) with \na theorem (theorem II of that paper). The main result of that theorem states: \n{\\it if there is a naked \nsingularity which satisfies Newman's strong curvature condition (Newman, 1983) and \nexists arbitrarily far into the future of a set of initial regular data,\nthen violation of strong causality occurs arbitrarily close to future null infinity.\nThus a Cosmic Time Machine is naturally implied}.\nThis {\\it pathology}\ncannot be cured by any quantum correction to the classical theory of relativity \nbefore the singularity forms, because the very source of this peculiar behaviour is not the \nsingularity itself but rather the space-time nearby it. Here, in fact, \nlight cones permit non space-like trajectories to\nrun backwards with respect to the coordinate time causing the\nlocal causal future to overlap with what would have been the causal past \nin a flat space-time.\nThis effect, which is induced by gravity, occurs (whenever it does) in a finite \ndomain surrounding the singularity; this region will be termed {\\it kernel} \nof a Cosmic Time Machine.\nA space-time which is also a CTM must satisfy basic physical requirements. \nFirst the matter source which eventually evolves to a singularity must \nsatisfy the energy conditions so to avoid \nquite arbitrary geometries as possible space-times. Then\nthe space-time solution must admit a regular flat (past and future) infinity \nfor the definition of a CTM to make sense.\nWe shall now illustrate what are the observational implications of a CTM did it \narise somewhere in the Cosmos. \n\nLet a coordinate time $t$ be chosen so to coincide \nwith the proper-time of an observer at a positive infinity. \nConsider two events in a CTM-kernel being one to \nthe (causal) future of the other (two subsequent flashes from the same light gun, \nsay); then, being in a CTM-kernel, there exist \nlight rays from these events which propagate backwards with respect\nto the local time coordinate untill they leave the kernel and escape to \npositive null infinity. If we allow for the existence of photon orbits which {\\it spatially} \nloop around the singularity before leaving the CTM-kernel, it may well happen that these light \nrays leave the kernel at about the same value of the $t$ coordinate and therefore reach infinity \nat about the same value of $t$ as well. But at flat infinity, the $t$ coordinate \nis also the proper-time of a stationary observer hence the latter would see the two events almost \nsimultaneously on her (his) clock. If we extrapolate this example to all the \nevents which are to the future\nof any given one in a CTM-kernel, we infer that in a Cosmic Time Machine\nthe entire causal future development of a given domain within its kernel \nmay be seen by a distant observer at the same time. \nEvidently this property makes a CTM potentially a source of an arbitrary strong burst. \n\nImpulsive cosmic events combine two main puzzling features, \nnamely an extremely short time of emission (order of a second) and a very high energy fluence.\nThe main challenge therefore is to find a unique mechanism which allows at once for both properties.\nThe most impressive examples of the above type of events are the Gamma Ray Bursts \n(Kluzniak and Ruderman, 1998; van Putten, 2001; Piran, 2004 and references therein). \nThe total energy emitted can be as \nhigh as $10^{54}\\, ergs$, mostly concentrated in a pulse as short as a second. \nThis amount of energy appears much more stunning if we think to it as being the \nenergy emitted in a second-long pulse by $10^{10}$ galaxies each made of $10^{11}$ \nSun-like stars, each emitting at a rate of $\\sim 10^{33}\\, ergs\/sec.$, concentrated \nin a region probably smaller than a galactic core!\n\nThere are several models which provide reasonable explanations for these high \nenergy events; most of them however suffer of some kind of incompleteness due to \nthe rich and complicated morphology of those sources. \nNevertheless, whatever process is considered, the common starting point is \ngravitational collapse; this may trigger \na supernova explosion or just provide a black hole which will be the main engine for the \nburst production.\n\nHere I envisage a completely new scenario based on the hypothesis that what we \nbelieve to be a black hole is on the contrary a generic strong curvature naked singularity sitting \ninside a CTM-kernel.\nSince Cosmic Time Machines involve astronomical objects, they allow one to make\npredictions which could in principle be confronted {\\it here-and-now} \nwith observations. \nWhile in the kernel, in fact, the coordinate time decreases and, as we said, it may \n reach the value when the singularity formed. \nAt this time the conditions for a time trap did not yet develop and therefore all the photons\nwould only propagate to the coordinate future again (coordinate time $t$ increasing) leaving the \nregion nearby the singularity just formed\nand leading to a burst of radiation as seen at far distance.\nAs illustrated beforehand, the light cone goes from a complete inversion with respect to the \ncoordinate time inside the kernel\nwhen the local future corresponds to a decreasing $t$, to a marginal time inversion at the\nkernel boundary where the future pointing light cone generators have an almost stationary $t$. \nBecause of this the internal past and future are mixed \nup and can be seen at infinity over the whole time interval during which the singularity is visible. \n\nWe can plausibly think of a situation where an accretion disk sits around a (spinning)\nnaked singularity. Let a substantial part of the emitted radiation enter the kernel and \nbe funneled, at least part of it, into spatially quasi-circular orbits along which light \ncones allow for local time reversed time-like or null trajectories. \nFurthermore let accretion cause an energy output of about\n $10^{40} ergs\/sec$ corresponding to a moderate quasar-like object \nshining for \nsome $10^{9}$ years ($\\sim 10^{16}$ seconds) untill the naked singularity decays \nclose to a black hole state becoming invisible to distant observers \n\\footnote{ Indeed this phenomenon may repeat itself and so will do \nthe effects which are here discussed.}. \nIf a thiny fraction of the emitted radiation, $\\zeta = 1\\%$\n say, propagates to the local future along the time-reversed orbits that we have shown to exist, \nit will likely reach the {\\it bottom} of the kernel and leave it at the same value of the \n$t$ coordinate as result of the local light cone opening.\nThen an observer at infinity would see the integrated energy of $10^{54} \nergs$ almost at the same time. \n\nOne can argue that the amount of radiation which was driven by the time-trap to the bottom \nof the kernel would give rise, before flowing out, to an energy condensate\ncapable to alter the background geometry. Even if it is only a small fraction ($\\zeta$ as said) of \nthe total radiation \nemitted by the cosmic source in its life-time, \nthe curvature produced by this energy concentration may be such to turn the naked singularity \ninto a black hole hiding the phenomenon on the start. This may certainly be a possibility, however \nan amount of radiation of $10^{54}ergs$ as in the previous example corresponds to a gravitational source of \n$\\sim 10^{33} g$, namely about one solar mass. This may be negligible if we think to a main source\nsingularity of $10^{8}-10^{9} M_\\odot$. Moreover since we conceive a situation where most of the time-trapped\nradiation is confined on spatially quasi-circular orbits\nin the innermost part of the kernel, then the contribution to the geometric curvature would come \nfrom a rotating energy current; this would strenghten rather than weaken \nthe naked singularity condition. \n\nEvidently the longer a naked singularity lasts as such the more luminous will be the burst because \nlonger is the future development which will be \"compressed\" by the time inversion and therefore\nmore are the photons which will contribute to the prompt emission. This {\\it mechanism} \nmay lead to undesirable bursts of infinite intensity! Naked singularities however appear to prevent \nthis circumstance.\nIt is well established that\n a naked singularity decays to a black hole. In this case, the instability of a \nKerr black hole inner horizon \nleads to the insurgence of a \"larger\" singularity which will cancel any kernel structure around the \ncentral spinning singularity.\nEven if the transitin to a black hole takes place only asymptotically, we know (Calvani and de Felice, 1978) \nthat \na Kerr naked singularity has a \"memory of the last horizon\"; this manifests itself with a rather peculiar\nconcentration of stable spherical \nnull orbits around the spatial surface $r=M$ which is the spatial location of the {\\it extreme} \nKerr black hole horizon. This concentration increases as one approaches the black hole state untill \nit creates a sort of radiation layer which will effectively prevent the reach of the CTM-kernel \n stopping any time machine activity. \n\n\nThe opening of the light cone generators inside a CTM-kernel ranges from a complete reversal \nwith respect to the local time axis to a marginally complete opening nearby the kernel boundary. \nIt is reasonable to expect that part of the radiation emitted by the source will leave \nthe kernel before it reaches its bottom and therefore at a value of the coordinate time larger \nthan that when the bulk of radiation is emitted from the bottom of the kernel. \nThe radiation which leaks out from the kernel boundary will then reach the distant observer \nat a later time with respect to the main burst. This may account for the afterglows observed \nin some of the impulsive sources. \nThe latters, like \nGamma Ray Bursts for example, have a reach phenomenology and their emission properties show correlations\n(Ghisellini, 2004; Ghirlanda et al., 2004a, 2004b, 2005; Piran, 2004 and references therein). \nAlthough the proposed scenario does not allow for definite predictions yet, we can expect obvious \ncorrelations.\n \n As an example consider a \nKerr naked singularity of total mass $M$ and rotation \nparameter $a=M(1+\\beta)$ where $\\beta\\ll 1$; because of accretion, this singularity \nwill have a life time, before decaying \nto a black hole, given approximately by (de Felice, 1975):\n\\begin{equation}\n T\\approx 1.5\\times 10^4\\frac{M_\\odot}{M}\\frac 1{\\rho(1+\\beta)}\\,years\n \\end{equation}\nHere $\\rho$ is the density of matter which accretes on the singularity. \nThe life time then critically depends on the factor $(M\\rho)^{-1}$.\nFor sake of illustration, a $10^9M_\\odot$ naked singularity will last for $10^8$ years, \nas in the previous example, if it is surrounded by accreting material of density\n$\\rho\\approx 10^{-13}\\, g\\,cm^{-3}$, a value which appears compatible with \nwhat one could have in active galactic nuclei. \nIn this scenario then we expect that the total \nluminosity of the impulsive emission goes as $L\\sim \\zeta(M\\rho)^{-1}$.\nMoreover, the longer a naked singularity is visible to distant observers the longer one expects the \nafterglows to last. Hence a correlation such as more luminous burst being followed by longer afterglows \nis expected.\n\nEvidently the survival of the above conjecture about the nature of impulsive sources depends\non the possibility to be falsified by more definite observational constraints; this however is a \nchallenge for the future. \n\n\\section{Conclusions}\nIf naked singularities exist in the Cosmos as predicted by general relativity\nthen they may give rise to a Cosmic Time Machine. In this case, in fact, the \nsingularity could likely be surrounded by a space time region where the local causal future is \ntime-inverted with respect to infinity. Naked singularities however \nwill likely evolve close to a black-hole state in a finite interval of coordinate time \nhence if all that happens, then we could directly observe \n astrophysical phenomena which are observationally constrained by the \npeculiarities of a \ntime machine. This may be the case of the most energetic Gamma Ray Bursts whose \nimpulsive emission may just be the time integration over a finite interval of the \nlocal proper-time \nof emission processes taking place in some CTM-kernel.The latter then will be sources of \nthe most powerful bursts in the Universe. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection*{Acknowledgement}\nThe financial support of the ARC, including funds to support the visit\nof G.~Olshanski whose lectures benefitted the present work, are\nacknowledged. Also, the remarks of T.H.~Baker on the original manuscript\nare appreciated. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdjiy b/data_all_eng_slimpj/shuffled/split2/finalzzdjiy new file mode 100644 index 0000000000000000000000000000000000000000..235debdcc9569bb693468108f6492da9023bf207 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzdjiy @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe observations presented in this paper are part of the \n{\\sl Hubble Space Telescope} Key Project on the Extragalactic Distance Scale,\na detailed description of which can be found in\n\\markcite{kfm1995}Kennicutt, Freedman \\& Mould (1995). \nThe main goal of the Key Project\nis to measure the Hubble Constant, H$_{0}$, to an accuracy of 10\\%, by\nusing Cepheids to calibrate several secondary distance indicators such as\nthe planetary nebula luminosity\nfunction, the Tully--Fisher relation, surface brightness fluctuations,\nand methods using supernovae.\nThese methods will then be used to\ndetermine the distances to more distant galaxies whereby the global\nHubble Constant can be measured.\nPublished results from other Key Project galaxies\ninclude M81 (\\markcite{free1994}Freedman {\\it et al.} 1994), \nM100 (\\markcite{lff1996}Ferrarese {\\it et al.} 1996),\nM101 (\\markcite{kelson1996}Kelson {\\it et al.} 1996), \n\\ngc{925} (\\markcite{silb1996}Silbermann {\\it et al.} 1996),\n\\ngc{3351} (\\markcite{grah1997}Graham {\\it et al.} 1997), \n\\ngc{3621} (\\markcite{raw1997}Rawson {\\it et al.} 1997),\n\\ngc{2090} (\\markcite{phel1998}Phelps {\\it et al.} 1998),\n\\ngc{4414} (\\markcite{turn1998}Turner {\\it et al.} 1998),\n\\ngc{7331} (\\markcite{hughes1998}Hughes {\\it et al.} 1998), and\n\\ngc{2541} (\\markcite{lff1998a}Ferrarese {\\it et al.} 1998).\n \nNGC~1365 ($\\alpha_{1950} = 3^{\\rm h} 31^{\\rm m}$, $\\delta_{1950} = -36\\arcdeg\n18\\arcmin$) is a large, symmetric, barred spiral galaxy with a measured\nheliocentric velocity of 1652 km~s$^{-1}$ \\markcite{st1981} \n(Sandage \\& Tammann 1981) located in the Fornax cluster of galaxies.\nIt is classified as an SBb(s)I galaxy by \\markcite{st1981}\nSandage \\& Tammann and as an SBs(b) galaxy by\n\\markcite{dev1991}de Vaucouleurs {\\it et al.} (1991). \nWork by Veron {\\it et al.} (1980) found that \\ngc{1365} contains a\nhidden Seyfert 1 nucleus. \\ngc{1365} has been extensively mapped in HI\nby Ondrechen \\& van der Hulst (1989) and more recently by\n\\markcite{jvm1995}J\\\"{o}rs\\\"{a}ter \\& van Moorsel (1995). In particular,\n\\markcite{jvm1995}J\\\"{o}rs\\\"{a}ter \\& van Moorsel found that the \ninner disk of \\ngc{1365} has a significantly different inclination\nangle (40\\arcdeg) compared to previous work using optical isophotes (55\\arcdeg)\nby Linblad (1978). Other inclination angles found in the literature are\n56\\arcdeg (Bartunov {\\it et al.} 1994), 61\\arcdeg (Schoniger \\& Sofue 1994,\nAaronson {\\it et al.} 1981),\n44\\arcdeg~$\\pm$ 5\\arcdeg~(Bureau, Mould, \\& Staveley-Smith 1996),\n46\\arcdeg $\\pm$ 8\\arcdeg~(Ondrechen \\& van der Hulst 1989)\nand 63\\arcdeg~(Tully 1988). This scatter is a result of \nthe warped nature of the \\ngc{1365} disk\n(\\markcite{jvm1995}J\\\"{o}rs\\\"{a}ter \\& van Moorsel 1995) combined with the \nvarious observational methods the authors used to determine the\nthe major and minor axis diameters.\nThe true inclination angle of \\ngc{1365} is not vital for our Cepheid\nwork but is critial for photometric and HI line-width corrections\nfor the Tully-Fisher (TF) relation. If one uses infrared ($H$ band) absolute \nmagnitudes the effect of using the various inclination angles between\n40\\arcdeg~to 63\\arcdeg~is minimal as the extinction correction in the infrared\nis small. \nHowever, the correction to the line-width, (sin $i$)$^{-1}$, is not small.\nThe corrected HI line width will decrease by\n40\\% if one uses 63\\arcdeg~instead of 40\\arcdeg. Detailed discussion of\nthe TF method is beyond the scope of this paper but we caution readers\nto accurately determine the inclination angle of \\ngc{1365} before using\nit as a TF calibrator.\n\nOverall, Fornax is a relatively compact cluster of about 350 galaxies\nwith a central, dense concentration of elliptical galaxies.\nThe center of the cluster is dominated by the large E0 galaxy\n\\ngc{1399}, with \\ngc{1365} lying $\\sim$0.5 Mpc from \\ngc{1399} as projected\non the sky. The heliocentric radial velocity of the Fornax cluster is \n1450 km~s$^{-1}$ with a dispersion\nof 330 km~s$^{-1}$ (\\markcite{hm1994}Held \\& Mould 1994). \nThe Fornax\ncluster has been the target of many studies involving secondary distance \nindicators, as reviewed by Bureau, Mould, \\& Staveley-Smith (1996). \nA Cepheid distance to the cluster will be an important step\nforward in calibrating the extragalactic distance scale.\n\nThis is the first of two papers on the Cepheid distance to \\ngc{1365}\nand the Fornax cluster. This paper describes the HST observations of\nCepheid variable stars in \\ngc{1365} and the distance derived to the\ngalaxy. A companion paper by \\markcite{mad1998a} Madore {\\it et al.} (1998a) \ndiscusses the\nimplications for the distance to the Fornax cluster and the calibration\nof the extragalactic distance scale.\nThe location of \\ngc{1365} within the Fornax cluster and the geometry\nof the local universe is discussed in \\markcite{mad1998b}Madore {\\it et al.}\n(1998b).\nWe note that the Key Project has recently observed two other galaxies within \nthe Fornax cluster, \\ngc{1425} (Mould {\\it et al.} 1998) and \\ngc{1326A} \n(Prosser {\\it et al.} 1998).\n\n\\section{Observations}\n\nNGC~1365 was imaged using the Hubble Space Telescope's Wide Field and\nPlanetary Camera 2 (WFPC2). A description of\nWFPC2 instrument is given in the HST WFPC2 Instrument Handbook\n(\\markcite{burr1994}Burrows {\\it et al.} 1994). The camera\nconsists of four 800$\\times$800 pixel CCDs.\nChip 1 is the Planetary Camera with 0.046 arcsec pixels and an illuminated\n$\\sim 33\\times31$ arcsec field of view. The three other CCDs make up\nthe Wide Field Camera (Chips 2-4), each with 0.10 arcsec pixels and a\n$\\sim 1.25\\times1.25$ arcmin illuminated field of view. Each CCD has a readout\nnoise of about 7 $e^{-}$. A gain setting of 7 $e^{-}$\/ADU was used for all\nof the \\ngc{1365} observations.\n\nWFPC2 imaged the eastern part of \\ngc{1365},\nas seen in Figure 1, from 1995 August through September. As with all the \ngalaxies chosen for the Key Project,\nthe dates of observation were selected using a power-law time series\nto minimize period aliasing and maximize uniformity of phase coverage\nfor the expected range of Cepheid periods from 10 to 60 days\n(\\markcite{free1994}Freedman {\\it et al.} 1994). Twelve epochs in\n$V$ (F555W) and four in $I$ (F814W) were obtained. Each \nepoch consisted of two exposures, taken on successive orbits,\nwith typical integration times of 2700 seconds. All observations\nwere made with the camera at an operating temperature of\n$-$88$\\arcdeg$ C. Table 1 lists the image identification code,\nHeliocentric Julian Date of observation (mid-exposure), exposure time, \nand filter, of each observation. On 1995 August 28 the focus of HST\nwas changed, occurring between the 5th and 6th epoch of \\ngc{1365}\nobservations. The effect of this refocus is discussed in Section 3.1.\n\n\\section{Photometric Reductions}\n\nAll observations were preprocessed through the standard Space\nTelescope Science Institute (STScI) pipeline as described by\n\\markcite{holt1995b}Holtzman {\\it et al.} (1995b). The images were\ncalibrated with the most up-to-date version of the routine reference\nfiles provided by the Institute at the time the images were taken.\nOur post-pipeline processing included masking out the vignetted edges, \nbad columns, and bad pixels.\nThe images were then multiplied by a pixel area map to correct for the\nWFPC2 geometric distortion (\\markcite{hill1998}Hill {\\it et al.} 1998), \nmultiplied by 4, and converted to integer values. Then the ALLFRAME and\nDoPHOT photometry packages were used to obtain profile-fitting photometry\nof all the stars in the HST images.\nDetails of each method are described below.\n\n\\subsection{DAOPHOT II\/ALLFRAME Photometry}\n\n\nThe extraction of stellar photometry from CCD images using the\nALLFRAME (\\markcite{pbs1994}Stetson 1994) package first requires a \nrobust list of stars in the\nimages. The long exposures of \\ngc{1365} contain significant numbers of \ncosmic ray events which can be misidentified as stars by automated\nstar-finding programs. To solve this problem, images\nfor each chip were median averaged to produce a\nclean cosmic ray free image. DAOPHOT II and ALLSTAR\n(\\markcite{pbs1987}Stetson 1987, \\markcite{sdc1990}Stetson {\\it et al.} 1990, \nStetson 1991, 1992, 1994) were then used to identify the stars\nin the deep, cosmic ray free images for each chip.\nThe star lists were subsequently used by ALLFRAME to\nobtain profile-fitting photometry of the stars in the original\nimages. The point spread functions were derived from\npublic domain HST WFPC2 observations of the globular clusters\nPal 4, and \\ngc{2419}.\n\nTen to twenty fairly isolated stars in each WFPC2 chip were chosen to\ncorrect the profile-fitting ALLFRAME photometry out to the 0.5 arcsec\nsystem of \\markcite{holt1995a}Holtzman {\\it et al.} (1995a). These\nstars are listed in the appendix as secondary standards to our photometric\nreductions.\nTo remove the effects of nearby neighbors on the aperture photometry\nall the other stars were first subtracted from the images. Growth curves for\nthe isolated stars were constructed using DAOGROW \n(\\markcite{pbs1990}Stetson 1990). \nBest estimates\nof the aperture corrections were determined by running the program COLLECT\n(\\markcite{pbs1993}Stetson 1993), which uses the profile-fitting photometry\nand the growth-curve photometry to determine the best photometric correction \nout to the largest aperture, 0.5 arcsec in our case. \n\nAs with the previous Key Project galaxy \\ngc{925} \n(\\markcite{silb1996}Silbermann {\\it et al.} 1996),\nthe aperture corrections (ACs) for all the frames in each filter were averaged \nto produce mean ACs that were then applied to each frame to shift the\nCepheid photometry onto the \\markcite{holt1995a}Holtzman {\\it et al.} \n0.5 arcsec system.\nFor nonvariable stars, mean instrumental $V$ and $I$ magnitudes averaged over\nall epochs were calculated using DAOMASTER (\\markcite{pbs1993}Stetson 1993) \nand then shifted to \nthe 0.5 arcsec system using the mean ACs. However, for \\ngc{1365} we\nnoticed there were relatively large variations in the individual frame\nACs, typically on the order of $\\pm$ 0.07 mag, but for one case as large as\n$-$0.2 mag, from the mean values. \nThese variations are due to random photometric scatter in the small number of\nisolated, bright stars available to us for AC determination, and the\nrealization that on any given image, a few of these stars may be corrupted by\ncomic ray events or chip defects further reducing the number of usable AC\nstars for a particular image.\nThere was also a focus change between the\n5th and 6th epoch, which resulted in an obvious shift of about +0.1 mag in\nthe individual ACs for epochs obtained after the focus change.\nAs a result, we also calculated mean magnitudes and fit period-luminosity \nrelations for the Cepheids using photometry corrected by individual frame ACs.\nThe typical difference between\nmean $V$ and $I$ magnitudes for the Cepheids (mean ACs $-$ individual ACs) \nwas +0.01 mag, and for some\nof the Cepheids was as large as +0.02 mag.\nDue to the refocusing event and the relatively large variation in the \nindividual ACs we feel that the individual photometric measurements of the\nCepheids at each epoch are best represented using the individual ACs. The\neffect on the distance modulus to \\ngc{1365} is minimal, +0.01 mag.\nThroughout the rest of this paper we will\nuse the individual AC corrected Cepheid photometry.\n\nThe final form of the conversion equations for the ALLFRAME photometry is: \n\n\\begin{equation}\nM = m + 2.5\\log t - C_{1}(V-I) + C_{2}(V-I)^{2} + AC + ZP\n\\end{equation}\n\n\\noindent \nwhere M the standard magnitude, m is the \ninstrumental magnitude, $t$ is the exposure time, $C_{1}$ and $C_{2}$ are\nthe color coefficients, AC is the aperture \ncorrection (different for each frame for the Cepheids), and ZP is the zero \npoint.\nSince we have shifted our photometry to the 0.5 arcsec system of\n\\markcite{holt1995a}Holtzman {\\it et al.} (1995a) we use their color\ncoefficients from their Table 7: $C_{1} = -0.052$ for $V$ and $-$0.063 for $I$,\nand $C_{2} = 0.027$ for $V$ and 0.025 for $I$.\nThe ZP term includes the \\markcite{hill1998}Hill {\\it et al.} (1998) long \nexposure zero point, \nthe ALLFRAME zero point of $-$25.0 mag, and a correction for multiplying the \nimages by four before converting them to integers (2.5log(4.0)). For $V$ band \nthe ZP terms are $-$0.984 $\\pm$ 0.02, $-$0.973 $\\pm$ 0.01, \n$-$0.965 $\\pm$ 0.01, and $-$0.989 $\\pm$ 0.02 for Chips 1$-$4 respectively,\nand for\n$I$ band are $-$1.879 $\\pm$ 0.04, $-$1.838 $\\pm$ 0.02, $-$1.857 $\\pm$ 0.02, \nand $-$1.886 $\\pm$ 0.01 for Chips 1$-$4 respectively.\nFor each epoch, an initial guess of each star's color, $V-I$ = 0.0, was used\nand the $V$ and $I$ equations were then solved iteratively. \n\nFor the nonvariable stars, an average magnitude over all epochs was calculated\nusing DAOMASTER, which corrects for frame-by-frame differences due to \nvarying exposure times and focus, and the equations above were used to \ncalculate the final standard $V$ and $I$ magnitudes using mean aperture \ncorrections for each chip and filter combination. \n\n\\subsection{DoPHOT Photometry}\n\nAs a double-blind check on our reduction procedures, we\nseparately reduced the \\ngc{1365} data using a variation \nof DoPHOT (\\markcite{schec1993}Schechter et al. 1993) described\nby \\markcite{saha1996}Saha {\\it et al.} (1996).\nThe DoPHOT reductions followed the procedure described by\n\\markcite{saha1996}Saha {\\it et al.},\n\\markcite{lff1996} Ferrarese {\\it et al.} (1996) and \n\\markcite{silb1996}Silbermann {\\it et al.} (1996). Here we \nonly mention aspects of the reductions that were unique to NGC 1365.\nThe DoPHOT reduction procedure identifies cosmic rays when\ncombining the single epoch exposures, prior to running DoPHOT\n(\\markcite{saha1994}Saha {\\it et al.} 1994). \nDue to the unusually long exposure times in\nthe \\ngc{1365} images, however, the number of pixels affected by cosmic\nrays was so large that significant\nnumbers of residual cosmic ray artifacts remained in the combined\nimages. To overcome this problem, the combined images for each epoch\nwere further combined in pairs, to create a single clean master image.\nThe master image was then compared with each original image to identify\nand flag the cosmic rays. The process was then repeated, but with\nthe pixels affected by cosmic ray events in both single epoch image pairs\nflagged. This process was iterated several times to ensure both a\nclean master image, and that tips of bright stars were not incorrectly\nidentified as cosmic rays and removed. These master images were then used to\ngenerate a coordinate list for the DoPHOT photometry.\n\nCalibration of the DoPHOT photometry was carried out as follows. \nAperture corrections were determined from the \\ngc{1365} frames\nand applied to the raw magnitudes.\nThe magnitudes were then corrected to an exposure\ntime of 1 second, and a zero point calibration was applied to\nbring them to the 0.5 arcsec system of \\markcite{holt1995a}Holtzman \n{\\it et al.} (1995a). These zero point corrections are as\ngiven in \\markcite{holt1995a}Holtzman {\\it et al.}, but with a \nsmall correction applied to account for differences in star and sky \napertures (\\markcite{hill1998}Hill {\\it et al.} 1998).\nThe prescription of \\markcite{holt1995a}Holtzman {\\it et al.}\nwas then used to convert the instrumental magnitudes\nto standard Johnson $V$ and Cousins $I$ magnitudes. \n\n\\subsection{Comparison of DAOPHOT and DoPHOT Photometric Systems}\n\nThe independent data reductions using ALLFRAME and DoPHOT provide a\nrobust external test for the accuracy of the profile-fitting photometry of\nthese crowded fields. \nFigure 2 shows the comparison between the DoPHOT and ALLFRAME final photometry.\nAs expected we see increased scatter as one goes to fainter magnitudes.\nWe also see that for some of the filter and chip combinations (i.e. Chip 1 $V$\nand $I$ bands, and Chip 2 $V$ band) there are small scale errors, on the order\nof 0.01 mag mag$^{-1}$. The nature of these scale errors is still not \ncompletely\nclear at this time, but it is most likely due to small differences in the sky \ndetermination, which translate into correlations between the photometric\nerror and the star magnitudes. Artificial star simulations (discussed below) \nshow that DoPHOT is\nsomewhat more robust than ALLFRAME in separating close companions, which will\ntherefore be measured brighter by ALLFRAME than DoPHOT. This explains the \nlarger number of outlyers found with positive DoPHOT$-$ALLFRAME residuals for \nsome filter\/chip combinations.\n\n\nThe horizontal line in each panel in Figure 2 marks the average difference\nbetween DoPHOT and ALLFRAME,\nusing stars brighter than 25 mag and removing wildly discrepant stars.\nThe differences are listed in Table 2. The errors are the rms of the means.\nIn general the differences are on the order of\n$\\pm$ 0.1 mag, which is slightly larger than DoPHOT\/ALLFRAME comparisons\nfor other Key Project galaxies ($\\sim 0.07$ mag or less for\nM101, \\ngc{925}, \\ngc{3351}, \\ngc{2090}, and \\ngc{3621}). \nAn extensive set of simulations was carried out adding artificial stars to the\nNGC 1365 frames, with the intent of understanding the nature of the observed\ndifferences. The main result of these observations, discussed in detail in a\nsubsequent paper (\\markcite{lff1998b}Ferrarese {\\it et al.} 1998b), \nis that the dominant cause of\nuncertainty lies in the aperture corrections, while errors in sky \ndeterminations\nand ability to resolve close companions play only a second order role. The\npaucity of bright isolated stars in crowded fields makes the determination of\naperture corrections problematic at best. As an extreme example, the ALLFRAME \naperture corrections derived for the first and second exposure of the first \n$I$ epoch differ by 0.3 mag for the PC. \nSince focus\/jitter changes between consecutive orbit exposures are irrelevant, \n0.3 mag can be taken as a reasonable\nestimate of the uncertainty in the aperture corrections for that particular\nchip\/filter\/epoch combination (more typical variations in the aperture \ncorrections are $\\pm$ 0.07 mag for both the DoPHOT and ALLFRAME datasets). \nSimilar considerations hold for the DoPHOT\naperture corrections. \nThe DoPHOT$-$ALLFRAME differences observed for the bright\nNGC 1365 stars are therefore found to be not significant when compared to the\nuncertainty in the aperture corrections, and simply reflect the limit to which\nphotometric reduction can be pushed in these rather extreme fields. \n\n \nWe made a similar DoPHOT-ALLFRAME comparison\nfor the 34 Cepheids used to fit the period-luminosity relations (see Section 6).\nThe results of those comparisons are shown in Figure 3, and listed at the\nbottom of Table 2. The DoPHOT-ALLFRAME differences for the Cepheids show more \nscatter, as expected, since they are $\\sim$2 mag fainter than the brighter\nnonvariable stars in Figure 2, but overall, the Cepheids mirror the nonvariable\nstar DoPHOT-ALLFRAME differences.\n\n\\section{Identification of Variable Stars}\n\nTwo methods were used to search for variable stars using the ALLFRAME \ndataset. In both cases, the search for variables was done using only \nthe $V$ photometry. The $I$ photometry was used to help confirm variability and \nto determine $V-I$ colors. \nThe first method was a search for stars with unusually high dispersion in \ntheir mean $V$ magnitudes. A few candidate variables were found this way. \nThe more fruitful method employed a variation on the correlated variability \ntest suggested by \\markcite{ws1993}Welch \\& Stetson (1993). \nFirst, the average magnitude and standard deviation \nover all epochs was calculated for each star. Any magnitude\nmore than 2 standard deviations from the average was discarded. This removed \nmany of the cosmic ray events. As another filter, if the magnitude \ndifference between two single epoch observations was greater than 2.75 mag\nthe epoch was thrown out. Note that cosmic ray events that lead to\nreasonably measured magnitudes (i.e. about the same as the average magnitude)\nslip through these filters. \nFor each pair of observations in an epoch, the difference between\neach observation and the average magnitude is then calculated. The two\ndifferences are multiplied together and summed over all epochs. \nThe net result is that true variables will consistently have both\nsingle epoch observations brighter or fainter than the average magnitude,\nincreasing the sum over all epochs. Random high or low observations will\ntend to scatter around the average but not systematically within a single\nepoch, so nonvariable stars will have lower sum values. \nTypical values of the sum were $\\ge$ 1.0 for the variable candidates while\nthe nonvariable stars scattered around $\\le$ 0.2, with increased scatter as\none went to the faintest magnitudes. Note that we discarded bad data only to\nderive a variability index.\n\nAfter obtaining a set of variable candidates from\nthe above procedure, the photometry for each candidate was plotted against \ndate of observation. We were then able to note an approximate period for\nthe candidate as well as any observations affected by cosmic-ray events.\nAt the faint end, where we expected more contamination by nonvariable stars, \nwe were able to exclude obvious nonvariables immediately. For candidate \nvariables that passed this stage, any cosmic-ray event or otherwise\ncorrupted observations were removed in anticipation of determining a period \nof variation. Periods for the candidate variables were found using a\nphase-dispersion minimization routine as described by\n\\markcite{stell1978}Stellingwerf (1978). The resulting light curves\nwere checked by eye to verify the best period for each candidate.\nErrors\non the periods were determined by examining changes in the light curve as \nvarious periods were used. When the light curve became visibly degraded\n(i.e. photometry points out of phase) an upper\/lower limit to the period\ncould safely be assigned.\nThese errors are subjective but provide the reader with a guide to how\nwell the light curves are sampled. As a final step, the local environment\nof each candidate was inspected to check for severe crowding.\n\nThe search for variables in the DoPHOT reductions followed closely the\nprocedure described in \\markcite{sh1990}Saha \\& Hoessel (1990) and in\n\\markcite{lff1996}Ferrarese {\\it et al.} (1996). Candidates that\nwere classed as having a $\\geq$ 99\\% confidence of being variables\n(based on a reduced chi-squared test) were then checked for\nperiodicity using a variant of the method of \\markcite{lk1965}Lafler\n\\& Kinman (1965). The number of spurious variables was minimized by\nrequiring that the reduced chi-squared statistic be greater than 2.0\nwhen the minimum and maximum values were removed from the calculation.\nThe light curves of each variable candidate were then inspected\nindividually and any alternate minima in the phase dispersion relation\nwere checked to see which produced the best Cepheid light\ncurve. Generally the minimum in the phase dispersion plot produced the\nbest light curve. The image of each Cepheid candidate was also\ninspected at a number of epochs. Those falling in severely crowded\nregions or in areas dominated by CCD defects were also excluded.\n\nThe two lists of candidate variables, one from the ALLFRAME photometry\nand one from the DoPHOT photometry, were then compared. Any candidates found\nin only one dataset were located in the other dataset and checked for\nvariability. \nCandidates that appeared to be real variables in {\\it both} the\nALLFRAME and DoPHOT datasets make up our final sample of \n52 Cepheids in \\ngc{1365}. \nFinder charts for the 52 Cepheids are shown in Figures 4 and 5.\nThe Cepheid astrometry, periods, and period errors are given in Table 3.\nThe variables have been labeled V1 through V52 in order of descending period.\nColumn 1 in Table 3 identifies the Cepheids. Column 2 lists the CCD chip the\nCepheid is on. Chip 1 is the Planetary Camera and Chips 2-4 are the\nWide Field Camera chips. Columns 3 and 4 list the pixel position of\neach Cepheid, as found on image u2s70202t (see Table 1). Columns\n5 and 6 give the right ascension and declination in J2000 coordinates\nfor each Cepheid. Column 7 lists the Cepheid period in days. Column 8\ngives the period error in days and column 9 lists the logarithm of the\nperiod. In addition to these 52 variables, \nthere are several stars that appear to be definitely variable but for \none reason or another did not make it into our list of definite Cepheids.\nReasons include uncertain periods, questionable environment (very crowded)\nor variability seen in either the ALLFRAME or DoPHOT dataset but not both.\nPositions, mean $V$ and $I$ magnitudes, and possible periods for these stars \nare given in Table A2 of the appendix.\n\n\\section{Variable Light Curves and Parameters}\n \nTo construct the ALLFRAME light curves, magnitudes obtained from images \ntaken within a single epoch were averaged, with the resulting mean magnitude \nthen plotted. Light curves for the Cepheids are shown in Figure 6 and the\nALLFRAME photometry is listed \nin Tables 4 and 5. The error bars in Figure 6 are \naverages of the two single exposure errors as reported by ALLFRAME that make up \neach epoch. In cases where one of the two single exposure magnitudes is\nbad (i.e. cosmic ray event), the magnitude and error are from the surviving \nmeasurement. As with previous Key Project galaxies, ALLFRAME \noverestimates the error for a given photometric measurement in these \nundersampled WFPC2 images. This effect is discussed in the analysis of WFPC2 \ndata from M101 by \\markcite{pbs1998a}Stetson {\\it et al.} (1998a).\nBriefly, when we\ncompare photometric measurements within a given epoch for \\ngc{1365}\nwe find that the difference between them is significantly less than\nthe errors quoted by ALLFRAME. Typical real photometric differences\nare $\\pm$ 0.06 mag, at the magnitude level of the Cepheids,\nfor both $V$ and $I$, while ALLFRAME\nreports errors significantly larger than $\\pm$ 0.10 mag.\n\nMean $V$ magnitudes for the Cepheids were determined two different ways.\nFirst, as in other papers in this series, since the observations were \npreselected to evenly sample a\ntypical 10$-$60 day Cepheid light curve \n(\\markcite{free1994}Freedman {\\it et al.} 1994), \nunweighted intensity averaged\nmean magnitudes were calculated. Second, phase-weighted mean intensity \nmagnitudes $$ were also calculated using\n\n\\begin{equation}\n = -2.5\\log[\\sum_{i}^{N} 0.5(\\phi_{i+1} - \n\\phi_{i-1})10^{-0.4m_{i}}]\n\\end{equation}\n\n\\noindent \nwhere $\\phi$ is the phase, and the sum is over the entire light\ncycle. The average difference between the unweighted and\nphase-weighted intensity averaged mean $V$ magnitudes is only $-$0.027\n$\\pm$ 0.002 mag for the 52 Cepheids. This difference is quite small,\nas expected, since most of the Cepheids have nearly uniformly sampled\nlight curves.\n\nWith only four $I$ observations, total mean $I$ magnitudes were calculated\nas follows. Using the $V$ and $I$ magnitudes at the four $I$ epochs,\naverage $V$ and average $I$ magnitudes were calculated. Then, the\ndifference between the four-epoch $V$ average ($_{4}$) and 12-epoch\n$V$ mean magnitude ($_{12}$) was calculated for each Cepheid.\nSince the amplitude of Cepheids in $V$ is almost exactly twice the\namplitude in $I$ (V:I = 1.00:0.51, \\markcite{free1988} Freedman 1988),\nthe four-epoch $I$ magnitude was corrected to obtain the full\n12-epoch $I$ magnitude, as follows:\n\n\\begin{equation}\n_{12}~=~_{4} + 0.51(_{12} - _{4}).\n\\end{equation}\n\n\\noindent \nCosmic-ray corrupted data were removed before determining mean\nmagnitudes. Figure 7 shows the $I$ vs\n($V-I$) color-magnitude diagram for the HST field of \\ngc{1365},\nhighlighting the 52 Cepheids. The Cepheids fall neatly between\nthe well-defined blue plume and weak red plume of supergiants.\n\nTable 6 lists derived ALLFRAME photometric parameters for the Cepheids.\nColumn 1 identifies the Cepheids. \nColumns 2-5 list the intensity-average and phase-weighted mean $V$\nmagnitudes and errors.\nColumns 6-9 list the intensity-average and phase-weighted mean $I$\nmagnitudes and errors. \nAll of the errors listed in Table 6 are mean magnitude dispersions.\nThey reflect the uncertainty due to the star's variability \n(the amplitude of variation)\nand are not derived from the overestimated ALLFRAME errors.\nColumn 10 and 11 lists the intensity-averaged and phase-weighted $V-I$ \ncolor of each Cepheid. \nA symbol in column 12 \nindicates the Cepheid was used in the Cepheid period-luminosity relation\nfit to determine the distance to \\ngc{1365} based on the criteria listed\nin the next section. \n\n\\section{Period-Luminosity Relations and the Distance to \\ngc{1365}}\n\nStandard period-luminosity (PL) relations for the LMC Cepheids are\nadopted from \\markcite{mf1991} Madore \\& Freedman (1991) which\nassume a true LMC distance modulus of 18.50 mag and total\nline-of-sight LMC Cepheid reddening of E($B-V$) = 0.10 mag:\n\n\\begin{equation}\nM_{V} = -2.76\\log {\\rm P} - 1.40\n~~~{\\rm and}~~~ M_{I} = -3.06\\log {\\rm P} - 1.81 .\n\\end{equation}\n\nTo determine the \ndistance to \\ngc{1365} a subset of the 52 Cepheids were chosen based on the \nfollowing\ncriteria. The period of the Cepheid had to be between 10 and 47 days.\nThe lower limit of 10 days is common for all of the Key Project galaxies\nand was chosen to avoid first overtone pulsators which have periods less \nthan $\\sim$ 10 days (Madore \\& Freedman 1991). The longest period of 47 days \nwas estimated from our observing window of 49 days and our actual sequence\nof observations throughout that window. \nNone of our Cepheids have periods under\n10 days but 5 Cepheids have periods $> 47$ days, so they are\nexcluded from the fit.\nEach variable also had to have a Cepheid-like light curve and\nthe same period, to within 10\\%, in the ALLFRAME and DoPHOT datasets.\nNext, to exclude Cepheids that were too crowded, each Cepheid had to \ncontribute more than 50\\% of\nthe light within a 2 pixel box surrounding it.\nOur last criterion was that each Cepheid \nhad to have a typical Cepheid-like color, 0.5 $\\leq V-I \\leq$ 1.5.\nThere are a total of 34 Cepheids that satisfied all of the above criteria and\nthey are indicated in column 11 of Table 6. These 34 Cepheids were used to\nfit the PL relations and determine the distance to \\ngc{1365}.\n\nIn order to avoid incompleteness bias in fitting a slope to the\n\\ngc{1365} data, only the zero point of the regression was fitted,\nwith the slope of the fit fixed to the LMC values.\nThe phase-weighted $V$ and $I$ PL relations are shown in Figure 8. The filled \ncircles are the 34 Cepheids used to fit the PL relations, while the open \ncircles are the other Cepheids. The solid line in each\nfigure represents the best fit to each dataset. The dashed lines drawn\nat $\\pm$ 0.54 mag for the $V$ PL relation and $\\pm$ 0.36 mag for the\n$I$ PL relation represent 2-sigma deviations from the mean PL\nrelations. In the absence of significant differential reddening the intrinsic\nwidth of the Cepheid instability strip is expected to place the\n\\ngc{1365} Cepheids within these limits. \nWe note that the full sample of 52 Cepheids are well contained within\nthe $V$ and $I$ instability strips in Figure 8.\nFrom the PL fits, the apparent distance moduli to \\ngc{1365}\nare $\\mu_{V}$ = 31.70 $\\pm$ 0.05 mag\\, and $\\mu_{I}$ = 31.54 $\\pm$ 0.06 mag\\,,\nwhere the errors are\ncalculated from the observed scatter in the \\ngc{1365} PL data\nthemselves, appropriately reduced by the sample size of Cepheids.\n\nThe observed difference in the apparent distance moduli for \\ngc{1365}\ngives $\\mu_{V} - \\mu_{I}$ = $E(\\hbox{\\it V--I)}$ = 0.16 $\\pm$ 0.08 mag\\,. \nThe Key Project has adopted a reddening law of $R_{V} = A_{V}\/E(V-I) = 2.45$\nwhich is consistent with the work of Dean, Warren \\& Cousins (1978),\n\\markcite{card1989}Cardelli {\\it et al.} (1989) and Stanek (1996).\nWe therefore obtain A$_{V}$ = 0.40\\,~ for this region of \\ngc{1365}.\nThe true distance modulus to \\ngc{1365} is then\n31.31 $\\pm$ 0.08 mag\\,, corresponding to a\nlinear distance of 18.3 $\\pm$ 0.7 Mpc\\, (internal errors only). \nTo test how robust our calculated distance to \\ngc{1365} was,\nwe also calculated the distance modulus using intensity averaged mean\nmagnitudes for the 34 Cepheids.\nThe resulting true distance modulus, 31.34 mag, is just slightly larger \nthan our result using phase-weighted mean magnitudes. \nAs another test we used the full compliment of \n52 Cepheids to fit the PL relations, using phase-weighted mean magnitudes,\nand obtained a distance modulus of 31.35 mag. As expected we do see a\nslight variation in the distance modulus depending on which mean magnitudes\nare used or the sample size of Cepheids but the net result is that all of\nthese values are contained within our 31.31 $\\pm$ 0.08 mag\\, distance modulus and error.\n\nThe distance to \\ngc{1365} was derived independently using the Cepheid\nparameters derived from the DoPHOT reductions. \nThe resulting apparent distance moduli\nare $\\mu_{V}$ = 31.64 $\\pm$ 0.07 mag\\, and $\\mu_{I}$ = 31.49 $\\pm$ 0.07 mag\\,,\nwith a true modulus $\\mu_{0}$ = 31.26 $\\pm$ 0.10 mag\\,. These differ from\nthe ALLFRAME moduli by $-$0.06, $-$0.05, and $-$0.05 mag,\nrespectively. Despite the differences in DoPHOT and ALLFRAME photometry\n(Figures 2 and 3) the true distance moduli agree very well.\nTo better understand why this is so\nwe fit PL relations separately\nfor each WFPC2 chip, as seen in Table 7. The scatter in individual chip\napparent distance moduli is quite small for the ALLFRAME photometry and\nsomewhat more scattered for the DoPHOT photometry. \nAlso, DoPHOT consistently measures a smaller \nreddening compared\nto ALLFRAME for each chip and filter, except for Chip 2. Since over one-third\nof the Cepheids used to fit the PL relations are in Chip 2, this reduces the\noverall discrepancy in reddening when combining the Cepheids in all four chips.\nAs a test, we fit PL relations excluding the Chip 2 Cepheids. \nThe resulting true\ndistance moduli are then 31.31 mag (no change) for ALLFRAME and 31.37 mag \n(change of +0.11 mag) for DoPHOT. The net effect of this comparison is\nthat the ALLFRAME and DoPHOT true distance moduli still agree within\ntheir errors. Table 7 summarizes all of these PL fit tests. Column 1 in Table 7\nindicates if the dataset used was the phase-weighted or intensity averaged\nmean $V$ and $I$ magnitudes. Column 2 lists which chip subset was used (1,2,3\nor 4) or all 4 chips (1-4). Column 3 gives the number of Cepheids used in\nthe fit. Columns 4 and 5 give the apparent distance moduli. Column 6 gives\nthe extinction in magnitudes. Column 7 lists the true distance moduli for\neach PL fit test. The ALLFRAME results are on top, and the corresponding\nDoPHOT results are at the bottom.\n\nTo check for any effects due to incompleteness at the faintest magnitudes\nwe split the final sample of 34 Cepheids into two sets of 17 Cepheids,\na bright and a faint sample, sorted using the $V$ phase-weighted ALLFRAME\nphotometry. The true distance modulus is\n31.36 $\\pm$ 0.11 mag for the bright sample and 31.28 $\\pm$ 0.12 mag for\nthe faint sample. The slight changes in distance moduli seen by excluding\nthe brightest or faintest Cepheids are not significant compared to the\nerrors and we conclude that we\nare not affected by incompleteness at the faintest magnitudes.\n\n\\subsection{Error Budget}\n\nTable 8 presents the error budget for the distance to \\ngc{1365}. \nThe errors are classified as either random or systematic based on how we\nwill use \n\\ngc{1365} to determine the Hubble Constant. For example, the LMC distance\nmodulus uncertainty and PL relation zero point uncertainties\nare systematic errors because the Key Project is using the same LMC distance\nmodulus and Cepheid PL relation slopes for all of our\ngalaxies. We now discuss each source of error in Table 8 in detail.\n\nThe Key Project has adopted an LMC distance modulus of 18.50 $\\pm$ 0.10 mag.\nThe review of published distances to the LMC, via various techniques,\nby \\markcite{west1997}Westerlund (1997) (his Table 2.8) indicates that the \ndistance modulus\nto the LMC is still uncertain at the 0.10 mag level. \nMore recently, \nGould \\& Uza (1998), using observations of a light echo from\nSupernova 1987A, suggest an LMC distance modulus no larger than \n18.37 $\\pm$ 0.04 mag \nor 18.44 $\\pm$ 0.05 mag for a circular or elliptical ring respectively. \nPanagia {\\it et al.} (1998) used HST observations of the ring\naround SN 1987A to obtain an LMC distance modulus of 18.58 $\\pm$ 0.05 mag.\nWork by Wood, Arnold, \\& Sebo (1997) using models of the LMC Cepheid HV 905 \nproduced an LMC distance modulus of 18.51 $\\pm$ 0.05 mag. Recent results from \nthe MACHO Project (Alcock {\\it et al.} 1997)\nusing double mode RR Lyrae stars give an LMC distance modulus of \n18.48 $\\pm$ 0.19 mag.\nIn light of these recent results we feel that our adopted LMC distance modulus\nand error are still valid.\n\n\\markcite{mf1991}Madore \\& Freedman (1991) fit Cepheid PL\nrelations\nto 32 LMC Cepheids using {\\it BVRI} photoelectric photometry. The dispersions\nfor a single point about the PL relations are $\\pm$ 0.27 mag for $V$ and \n$\\pm$ 0.18 mag for $I$. The PL zero point errors are obtained \nby dividing by the square root of the number of Cepheids used in the fit,\ngiving us \n$\\pm$ 0.05 mag for $V$ and $\\pm$ 0.03 for $I$, items (2) and (3) in Table 8.\n\nThe LMC distance modulus and PL relation zero point uncertainties\nare sources of systematic error within the Key Project, since we use\nthe Madore \\& Freedman equations for all of the galaxies, and all galaxy\ndeterminations will change systematically as improvements to the zero point\nbecome available. \nCombining items (1), (2), and (3) in Table 8 in quadrature we obtain a PL \nrelation uncertainty of $\\pm$ 0.12 mag.\n\n\\markcite{hill1998}Hill {\\it et al.} (1998) estimated our WFPC2 zero point \nuncertainties to be \n$\\pm$ 0.02 mag for both $V$ and $I$ by comparing ground-based and HST \nobservations of M100. In addition we must include the uncertainties in the \naperture corrections. \nWhile there is one case where two aperture corrections within an epoch\ndiffered by 0.3 mag (Chip 1 $I$ band, first epoch), the scatter about the mean \nALLFRAME aperture corrections is \n$\\pm$ 0.07 mag for $V$ and $\\pm$ 0.06 mag for $I$.\nWe will take these to be the errors in zero point due to the aperture\ncorrections. We then combine the \\markcite{hill1998}Hill {\\it et al.} errors \nand the aperture\ncorrection errors in quadrature to obtain WFPC2 zero point errors of \n$\\pm$ 0.07 mag in $V$ and $\\pm$ 0.06 mag in $I$, items (4) and (5) in Table 8. \nSince the photometric errors in the two bands are uncorrelated\nwe combine items (4) and (5) in quadrature,\nbut we must weight these errors, $\\sigma_{V}$ and $\\sigma_{I}$, by the \ndiffering effects of reddening, as given by\n$[(1-R)^{2}(\\sigma_{V})^{2} + R^{2}(\\sigma_{I})^{2}]^{1\/2}$.\nAs stated previously, we have adopted a reddening law of\n$R_{V}= A_{V}\/E(V-I)$ = 2.45. Our photometric contribution to the error in\nthe distance modulus is therefore $\\pm$ 0.18 mag.\n\nEarly work by the Key Project discovered what appeared to be a long versus\nshort exposure zero point offset, such that stars in exposures \nlonger than approximately\n1000 seconds were systematically brighter, by approximately 0.05 mag, than\nstars in exposures of less than 1000 seconds duration. The effect is now\nthought to be a charge transfer efficiency effect in the WFPC2 CCDs. This\neffect is discussed in detail in \\markcite{hill1998}Hill {\\it et al.} (1998) \nand also in\n\\markcite{wh1997}Whitmore \\& Heyer (1997). For the Key Project\nwe have used the long exposure zero point for our photometric calibration.\nAt this time, an offset of +0.05 mag for both $V$ and $I$ is thought to be the \nbest estimate to correct for this effect. \nAdditional work on the zero points is currently\nbeing undertaken by \\markcite{pbs1998b}Stetson {\\it et al.} (1998b) \nbased on an \nextensive set of ground based and HST data.\nThis effect is included in Table 8 as a source of systematic error in the\ndistance to \\ngc{1365}.\n\nThere is an additional distance modulus error introduced from fitting the\nCepheid PL relations. \nFrom Section 5, these errors are $\\pm$ 0.05 mag\nin $V$ and $\\pm$ 0.06 mag in $I$, and include photometric errors,\ndifferential reddening and the apparent width of the instability strip in\nboth $V$ and $I$. These errors are combined in quadrature in item (R2)\nin Table 8.\n\nThere is concern that the Cepheid PL relation may have a metallicity\ndependence. The Cepheids in the calibrating LMC are thought to be\nrelatively metal poor compared Fornax and Virgo cluster galaxies\n(Kennicutt {\\it et al.} 1998),\nso a metallicity dependence in the Cepheid PL relation\nwould be a source of systematic error in our distance determination.\nKennicutt {\\it et al.} measured\na marginal metallicity dependence in one of the Key Project\ngalaxies, M101, leading to a\nshift in distance modulus of \n$\\delta(m-M)_0\/\\delta[O\/H] = -0.24 \\pm 0.16$ mag\/dex.\nWe can use this relation to estimate the systematic error in distance modulus \nto \\ngc{1365} due to a metallicity dependence on the Cepheid PL relations.\nZaritsky, Kennicutt \\& Huchra (1994)\nfound the oxygen abundance in \\ngc{1365} to be $12 + {\\rm log}(O\/H) = 9.0$\nwhile the measured oxygen\nabundance in the LMC on this scale is $12 + {\\rm log}(O\/H) = 8.50$ \n(Kennicutt {\\it et al.} 1998).\nThe 0.5 dex difference in oxygen abundance would, if applied, \nincrease the true distance modulus by +0.12 mag or a 6\\% \nincrease in the distance to \\ngc{1365}.\n\nThe total uncertainty in distance modulus to \\ngc{1365} due to random\nerrors is obtained by combining the distance modulus and PL fit uncertainties\nin quadrature:\n$[0.18^{2} + 0.08^{2}]^{1\/2}$ = $\\pm$ 0.20 mag. \n\n\nThe total systematic error in the distance to \\ngc{1365} is obtained by\ncombining the systematic errors, the LMC Cepheid PL relation, a possible \nmetallicity dependence, and the long versus short exposure zero point,\nin quadrature:\n$[0.12^{2} + 0.12^{2}+0.05^{2}]^{1\/2}$ = $\\pm$ 0.18 mag.\nThus our final distance modulus to \\ngc{1365} is \n31.31 $\\pm$ 0.20 (random) $\\pm$ 0.18 (systematic).\nTh implications of a Cepheid distance to \\ngc{1365} and the Fornax\ncluster are discussed in the companion paper Madore {\\it et al.} (1998).\n\n\\section{Conclusion}\nWe have used the HST WFPC2 instrument to obtain 12 epochs in $V$ and\n4 epochs in $I$ of the eastern part of \\ngc{1365} located in the Fornax Cluster.\nThe images were reduced separately using the ALLFRAME and DoPHOT \nphotometry packages. The raw photometry was transformed to standard\nJohnson $V$ and Cousins $I$ photometry via Hill {\\it et al.} (1998)\nzero points, Holtzman {\\it et al.} (1995a) color terms, and \naperture corrections\nderived from the \\ngc{1365} images. The $\\lesssim$ 0.1 mag difference\nbetween the final ALLFRAME and DoPHOT photometry is most likely due to\nthe relatively large uncertainties in both the ALLFRAME and DoPHOT\naperture corrections, which show a scatter of $\\pm$ 0.07 mag. \n\nA total of 52 Cepheids\nwere discovered in \\ngc{1365}, ranging in period from 14 to 60 days. \nA subset of 34 Cepheids was chosen based on period, light curve\nshape, color and relative crowding to fit Cepheid period-luminosity relations\nusing fixed slopes from \n\\markcite{mf1991} Madore \\& Freedman (1991). A distance modulus of\n31.31 $\\pm$ 0.20 (random) $\\pm$ 0.18 (systematic) mag, corresponding to \n18.3 $\\pm$ 1.7 (random) $\\pm$ 1.6 (systematic) Mpc is obtained from the \nALLFRAME photometry. A distance modulus of 31.26 $\\pm$ 0.10 mag\\, (internal errors only) \nwas obtained from the DoPHOT photometry.\nThe average reddening in this region of \\ngc{1365} was measured to be\n$E(\\hbox{\\it V--I)}$ = 0.16 $\\pm$ 0.08 mag\\, using ALLFRAME and 0.15 $\\pm$ 0.10 mag \nusing DoPHOT.\nThe apparent agreement in distance between ALLFRAME and DoPHOT despite the\nobserved $\\lesssim$ 0.1 mag photometric differences is due to DoPHOT measuring\na lower reddening to \\ngc{1365} in WFPC2 Chips 1, 3 and 4 and a higher\nreddening than ALLFRAME in Chip 2. Over one-third of the Cepheids in the\nPL fit sample are found in Chip 2, reducing the {\\it mean} difference in\nreddening and distance modulus between DoPHOT and ALLFRAME.\nThe companion paper by Madore {\\it et al.} (1998) discusses in detail\nthe implications of a Cepheid distance to \\ngc{1365}.\n\n\\acknowledgments\nThe work presented in this paper is based on observations made by the\nNASA\/ESA Hubble Space Telescope, obtained by the Space Telescope\nScience Institute, which is operated by AURA, Inc. under NASA contract\nNo. 5-26555. We gratefully acknowledge the support of the NASA and\nSTScI support staff, with special thanks our program coordinator, Doug\nVan Orsow. Support for this work was provided by NASA through grant\nGO-2227-87A from STScI. \nThis paper is based partially on data obtained at the Las Campanas\nObservatory 2.5m telescope in Chile, owned and operated by the Carnegie\nInstitution of Washington.\nSMGH and PBS are grateful to NATO for travel support\nvia a Collaborative Research Grant (960178).\nLF acknowledges support by NASA through Hubble Fellowship grant\nHF-01081.01-96A awarded by the Space Telescope Science Institute,\nwhich is operated by AURA, Inc., for NASA under contract NSA 5-26555.\nThe research described in this paper was partially carried out by the Jet\nPropulsion Laboratory, California Institute of Technology, under a\ncontract with the National Aeronautics and Space Administration. This\nresearch has made use of the NASA\/IPAC Extragalactic Database (NED)\nwhich is operated by the Jet Propulsion Laboratory, California\nInstitute of Technology, under a contract with the National\nAeronautics and Space Administration. \n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\vspace{-2mm}\n\nInherent uncertainties derived from different root causes have realized as serious hurdles to find effective solutions for real world problems. Critical safety concerns have been brought due to lack of considering diverse causes of uncertainties, resulting in high risk due to misinterpretation of uncertainties (e.g., misdetection or misclassification of an object by an autonomous vehicle). Graph neural networks (GNNs)~\\cite{kipf2017semi, velickovic2018graph} have received tremendous attention in the data science community. Despite their superior performance in semi-supervised node classification and regression, they didn't consider various types of uncertainties in the their decision process. Predictive uncertainty estimation~\\cite{kendall2017uncertainties} using Bayesian NNs (BNNs) has been explored for classification prediction and regression in the computer vision applications, based on aleatoric uncertainty (AU) and epistemic uncertainty (EU). AU refers to data uncertainty from statistical randomness (e.g., inherent noises in observations) while EU indicates model uncertainty due to limited knowledge (e.g., ignorance) in collected data. In the belief or evidence theory domain, Subjective Logic (SL)~\\cite{josang2018uncertainty} considered vacuity (or a lack of evidence or ignorance) as uncertainty in a subjective opinion. Recently other uncertainty types, such as dissonance, consonance, vagueness, and monosonance~\\cite{josang2018uncertainty}, have been discussed based on SL to measure them based on their different root causes. \n\n\\vspace{-1mm}\nWe first considered multidimensional uncertainty types in both deep learning (DL) and belief and evidence theory domains for node-level classification, misclassification detection, and out-of-distribution (OOD) detection tasks. By leveraging the learning capability of GNNs and considering multidimensional uncertainties, we propose a uncertainty-aware estimation framework by quantifying different uncertainty types associated with the predicted class probabilities. \nIn this work, we made the following {\\bf key contributions}:\n\\vspace{-2mm}\n\\begin{itemize}[leftmargin=*, noitemsep]\n\\item \\textbf{A multi-source uncertainty framework for GNNs}. Our proposed framework first provides the estimation of various types of uncertainty from both DL and evidence\/belief theory domains, such as dissonance (derived from conflicting evidence) and vacuity (derived from lack of evidence). In addition, we designed a Graph-based Kernel Dirichlet distribution Estimation (GKDE) method to reduce errors in quantifying predictive uncertainties.\n\\item \\textbf{Theoretical analysis}: Our work is the first that provides a theoretical analysis about the relationships between different types of uncertainties considered in this work. We demonstrate via a theoretical analysis that an OOD node may have a high predictive uncertainty under GKDE.\n\\item \\textbf{Comprehensive experiments for validating the performance of our proposed framework}: Based on the six real graph datasets, we compared the performance of our proposed framework with that of other competitive counterparts. We found that the dissonance-based detection yielded the best results in misclassification detection while vacuity-based detection best performed in OOD detection. \n\\end{itemize}\n\\vspace{-2mm}\nNote that we use the term `predictive uncertainty' in order to mean uncertainty estimated to solve prediction problems.\n\n\\vspace{-2mm}\n\\section{Related Work} \\label{sec:related-work}\n\\vspace{-2mm}\nDL research has mainly considered {\\it aleatoric} uncertainty (AU) and {\\it epistemic} uncertainty (EU) using BNNs for computer vision applications. AU consists of homoscedastic uncertainty (i.e., constant errors for different inputs) and heteroscedastic uncertainty (i.e., different errors for different inputs)~\\cite{gal2016uncertainty}. A Bayesian DL framework was presented to simultaneously estimate both AU and EU in regression (e.g., depth regression) and classification (e.g., semantic segmentation) tasks~\\cite{kendall2017uncertainties}. Later, {\\em distributional uncertainty} was defined based on distributional mismatch between testing and training data distributions~\\cite{malinin2018predictive}. {\\em Dropout variational inference}~\\cite{gal2016dropout} was used for an approximate inference in BNNs using epistemic uncertainty, similar to \\textit{DropEdge}~\\cite{rong2019dropedge}. Other algorithms have considered overall uncertainty in node classification~\\cite{eswaran2017power, liu2020uncertainty, zhang2019bayesian}. However, no prior work has considered uncertainty decomposition in GNNs. \n\nIn the belief (or evidence) theory domain, uncertainty reasoning has been substantially explored, such as Fuzzy Logic~\\cite{de1995intelligent}, Dempster-Shafer Theory (DST)~\\cite{sentz2002combination}, or Subjective Logic (SL)~\\cite{josang2016subjective}. Belief theory focuses on reasoning inherent uncertainty in information caused by unreliable, incomplete, deceptive, or conflicting evidence. SL considered predictive uncertainty in subjective opinions in terms of {\\em vacuity} (i.e., a lack of evidence) and {\\em vagueness} (i.e., failing in discriminating a belief state)~\\cite{josang2016subjective}. Recently, other uncertainty types have been studied, such as {\\em dissonance} caused by conflicting evidence\\cite{josang2018uncertainty}.\nIn the deep NNs, \\cite{sensoy2018evidential} proposed evidential deep learning (EDL) model, using SL to train a deterministic NN for supervised classification in computer vision based on the sum of squared loss. However, EDL didn't consider a general method of estimating multidimensional uncertainty or graph structure. \n\\vspace{-2mm}\n\\section{Multidimensional Uncertainty and Subjective Logic}\n\\vspace{-2mm}\n\nThis section provides the overview of SL and discusses SL-based multiple types of uncertainties, called {\\em evidential uncertainty}, with the measures of \\textit{vacuity} and \\textit{dissonance}. In addition, we give a brief overview of {\\em probabilistic uncertainty}, discussing the measures of \\textit{aleatoric} and \\textit{epistemic}.\n\n\\vspace{-2mm}\n\\subsection{Subjective Logic}\\label{SL}\n\\vspace{-2mm}\nSL offers the formulation of a subjective opinion based on both probabilistic logic (PL)~\\cite{nilsson1986probabilistic} and belief theory (BT)~\\cite{shafer1976mathematical} with two unique extensions. First, SL explicitly represents uncertainty by introducing vacuity of evidence (or uncertainty mass) in its opinion representation. This addresses the limitations of PL by modeling a lack of confidence in probabilities. Second, SL extends the traditional BT by incorporating base rates as the prior probabilities in Bayesian theory. The Bayesian nature of SL allows it to use second-order uncertainty to express and reason the uncertainty mass, where second-order uncertainty is represented by a probability density function (PDF) over first-order probabilities~\\cite{josang2016subjective}. For multi-class problems, we use a multinomial distribution (i.e., first-order uncertainty) to model class probabilities and a Dirichlet PDF (i.e., second-order uncertainty) to model the distribution of class probabilities. Second-order uncertainty enriches uncertainty representation with evidence information, playing a key role in \ndistinguish OOD from conflicting prediction as detailed later. \n\n\\vspace{-1mm}\nOpinions are the arguments in SL. In the multi-class setting, the multinomial opinion of a random variable $y$ in domain $\\mathbb{Y}=\\{1,...,K\\}$ is given by a triplet as: \n\\vspace{-1mm}\n\\begin{align}\n\\label{eq:so}\n \\omega = ({\\bm{b}},u,{\\bm{a}}), \\text{ with } \\sum\\nolimits_{k=1}^K b_k + u =1,\n\\end{align}\nwhere ${\\bm{b}} = (b_1, \\ldots,b_K)^T ,u$, and ${\\bm{a}}=(a_1, \\ldots, a_K)^T$ denote the belief mass distribution over $\\mathbb{Y}$, uncertainty mass representing vacuity of evidence, and base rate distribution over $\\mathbb{Y}$, respectively, and $\\forall k, a_k\\ge 0, b_k\\ge 0, u\\ge0$. The probability that $y$ is assigned to the $k$-class is given by $ P(y = k)= b_k + a_ku$,\nwhich combines the belief mass with the uncertain mass using the base rates. In the multi-class setting, $a_k$ can be regarded as the prior preference over the $k$-th class. When no specific preference is given, we assign all the base rates as $1\/K$.\n\\vspace{-1mm}\n\\subsection{Evidential Uncertainty}\n\\vspace{-1mm}\nIn this section, we explain how the second order uncertainty (evidential uncertainty) is derived from the first order uncertainty as a Dirichlet PDF. Given a set of random variables\n $\\textbf{p}=(p_1,...,p_K)^T$, where $\\textbf{p}$ is distributed on a simplex of dimensionality $K-1$, a conditional distribution $P(y = k|\\textbf{p})=p_k$ can be represented by the marginal distribution, $P(y)=\\int P(y|\\textbf{p})p(\\textbf{p})d\\textbf{p}$. We define $p(\\textbf{p})$ as a Dirichlet PDF over $\\textbf{p}$: $\\text{Dir}(\\textbf{p}|\\boldsymbol{\\alpha})$, where $\\boldsymbol{\\alpha}=(\\alpha_1, \\ldots, \\alpha_K)^T$ is a $K$-dimensional strength vector, with $\\alpha_k\\ge0$ denoting the effective number of observations of the $k$-th class. SL explicitly introduces uncertain evidence through a weight $W$ representing non-informative evidence and redefines the strength parameter as: \n $\\alpha_k=e_k+a_kW, \\text{ with } e_k \\ge 0, \\forall k \\in \\mathbb{Y}$,\nwhere $e_k$ is the amount of evidence (or the number of observations) to support the $k$-th class and $W$ is usually set to $K$, i.e., the number of classes. Given the new definition of the strength parameter, the expectation of the class probabilities $\\textbf{p}=(p_1, \\ldots, p_K)^T$ is given by: \n\\begin{align}\n \\mathbb{E}[p_k]=\\frac{\\alpha_k}{\\sum_{j=1}^K \\alpha_j}=\\frac{e_k+a_kW}{\\sum_{j=1}^K e_j + W}, \n\\end{align}\nwhere $a_k=1\/K$. By marginalizing out $\\textbf{p}$, we can derive an evidence-based expression of belief mass and uncertainty mass: \n\\begin{align}\n\\label{eq:belief}\n b_k= \\frac{e_k}{S} \\quad \\forall k \\in \\mathbb{Y}, \\quad u_v= \\frac{W}{S}, \\text{ with } S = \\sum_{k=1}^K \\alpha_k. \n\\end{align}\nSL categorizes uncertainty into two primary sources~\\cite{josang2016subjective}: (1) basic belief uncertainty derived from single belief masses, and (2) intra-belief uncertainty based on the relationships between different belief masses. These two sources of uncertainty can be boiled down to {\\em \\textbf{vacuity}} and {\\em \\textbf{dissonance}}, respectively, that correspond to vacuous belief and contradicting beliefs. In particular, vacuity of an opinion $\\omega$ is captured by uncertainty mass $u_v$ in~\\eqref{eq:belief} while dissonance of an opinion~\\cite{josang2018uncertainty} is formulated by:\n\\begin{align}\n\\label{eq:dis}\n diss(\\omega)=\\sum_{k=1}^{K}\\Big(\\frac{b_k \\sum_{j\\neq k}b_j \\text{Bal}(b_j,b_k)}{\\sum_{j\\neq k}b_j}\\Big), \\text{Bal}(b_j,b_k)=\n \\begin{cases} \n 1-\\frac{|b_j-b_k|}{b_j+b_k} & \\text{if $b_i b_j \\neq 0$}\\\\\n 0 & \\text{if $\\min(b_i,b_j)=0$}, \n \\end{cases}\n\\end{align}\nwhere $\\text{Bal}(b_j,b_k)$ is the relative mass balance function between two belief masses. The belief dissonance of an opinion is measured based on how much belief supports individual classes. Consider a binary classification example with a binomial opinion given by $(b_1,b_2,u,{\\bm{a}}) = (0.49, 0.49, 0.02, {\\bm{a}})$. Based on \\eqref{eq:dis}, it has a dissonance value of $0.98$. In this case, although the vacuity is close to zero, a high dissonance indicates that one cannot make a clear decision because both two classes have the same amount of supporting both beliefs, which reveals strong conflict within the opinion.\n\n\\vspace{-1mm}\n\\subsection{Probabilistic Uncertainty} \n\\label{sect:multi-dim uncertainty}\n\\vspace{-1mm}\nFor classification, the estimation of the probabilistic uncertainty relies on the design of an appropriate Bayesian DL model with parameters $\\bm{\\theta}$. Given input $x$ and dataset $\\mathcal{G}$, we estimate a class probability by $P(y|x) = \\int P(y|x;\\bm{\\theta}) P(\\bm{\\theta}|\\mathcal{G}) d\\bm{\\theta}$, and obtain \\textbf{\\textit{epistemic uncertainty}} estimated by mutual information~\\cite{depeweg2018decomposition, malinin2018predictive}:\n\\begin{eqnarray}\n\\footnotesize\n\\vspace{-2mm}\n\\underbrace{I(y, \\bm{\\theta}|x, \\mathcal{G})}_{\\text{\\textbf{\\textit{Epistemic}}}} =\\underbrace{\\mathcal{H}\\big[ \\mathbb{E}_{P(\\bm{\\theta}|\\mathcal{G})}[P(y|x;\\bm{\\theta})] \\big]}_{\\text{\\textbf{\\textit{Entropy}}}} - \\underbrace{\\mathbb{E}_{P(\\bm{\\theta}|\\mathcal{G})}\\big[\\mathcal{H}[P(y|x;\\bm{\\theta})] \\big]}_{\\text{\\textbf{\\textit{Aleatoric}}}}, \n\\vspace{-2mm} \\label{eq:epistemic}\n\\end{eqnarray}\nwhere $\\mathcal{H}(\\cdot)$ is Shannon's entropy of a probability distribution. The first term indicates {\\bf \\textit{entropy}} that represents the total uncertainty while the second term is {\\bf \\textit{aleatoric}} that indicates data uncertainty. By computing the difference between entropy and aleatoric uncertainties, we obtain epistemic uncertainty, which refers to uncertainty from model parameters. \n\n\\section{Relationships Between Multiple Uncertainties}\n\\begin{wrapfigure}{R}{0.45\\textwidth}\n \\vspace{-5mm} \n \\centering\n \\includegraphics[width=0.42\\textwidth]{fig\/fig_1.png}\n \\vspace{-2mm}\n \\caption{\\footnotesize{Multiple uncertainties of different prediction. Let ${\\bf u}=[u_v, u_{diss}, u_{alea}, u_{epis}, u_{en}]$.\\vspace{-5mm}}}\\label{fig:example}\n \\vspace{-1mm}\n\\end{wrapfigure}\nWe use the shorthand notations $u_{v}$, $u_{diss}$, $u_{alea}$, $u_{epis}$, and $u_{en}$ to represent vacuity, dissonance, aleatoric, epistemic, and entropy, respectively. \n\nTo interpret multiple types of uncertainty, we show three prediction scenarios of 3-class classification in Figure~\\ref{fig:example}, in each of which the strength parameters $\\alpha = [\\alpha_1, \\alpha_2, \\alpha_3]$ are known. To make a prediction with high confidence, the subjective multinomial opinion, following a Dirichlet distribution, will yield a sharp distribution on one corner of the simplex (see Figure~\\ref{fig:example} (a)). For a prediction with conflicting evidence, called a conflicting prediction (CP), the multinomial opinion should yield a central distribution, representing confidence to predict a flat categorical distribution over class labels (see Figure~\\ref{fig:example} (b)). For an OOD scenario with $\\alpha=[1, 1, 1]$, the multinomial opinion would yield a flat distribution over the simplex (Figure~\\ref{fig:example} (c)), indicating high uncertainty due to the lack of evidence. The first technical contribution of this work is as follows.\n\n\\begin{restatable}{theorem}{primetheorem}\nWe consider a simplified scenario, where a multinomial random variable $y$ follows a K-class categorical distribution: $y \\sim \\text{Cal}(\\textbf{p})$, the class probabilities $\\textbf{p}$ follow a Dirichlet distribution: $\\textbf{p}\\sim \\text{Dir}({\\bm \\alpha})$, and ${\\bm \\alpha}$ refer to the Dirichlet parameters. Given a total Dirichlet strength $S=\\sum_{i=1}^K \\alpha_i$, \nfor any opinion $\\omega$ on a multinomial random variable $y$, we have\n\\vspace{-2mm}\n\\begin{enumerate}\n\\item General relations on all prediction scenarios. \n \n(a) $u_v+ u_{diss} \\le 1$; (b) $u_v > u_{epis}$.\n \n \n \n \n \n \\item Special relations on the OOD and the CP.\n \\begin{enumerate}\n \\item For an OOD sample with a uniform prediction (i.e., $\\alpha=[1, \\ldots, 1]$), we have \n \\begin{eqnarray}\n 1= u_v = u_{en}> u_{alea} > u_{epis} > u_{diss} = 0 \\nonumber\n \\end{eqnarray}\n \\item For an in-distribution sample with a conflicting prediction (i.e., $\\alpha=[\\alpha_1, \\ldots, \\alpha_K]$ with $\\alpha_1 = \\alpha_2 =\\cdots = \\alpha_K$, if $S \\rightarrow \\infty$), we have \n \\begin{eqnarray}\n u_{en} = 1, \\lim_{S\\rightarrow \\infty} u_{diss} =\\lim_{S\\rightarrow \\infty} u_{alea} =1 , \\lim_{S\\rightarrow \\infty} u_{v} =\\lim_{S\\rightarrow \\infty} u_{epis} =0 \\nonumber\n \\end{eqnarray}\n \\text{with} $u_{en} > u_{alea}> u_{diss}> u_{v}>u_{epis} $.\n \\end{enumerate}\n\\end{enumerate}\n\\label{theorem1}\n\\end{restatable}\n\n\\vspace{-2mm}\nThe proof of Theorem~\\ref{theorem1} can be found in Appendix A.1. As demonstrated in Theorem~\\ref{theorem1} and Figure~\\ref{fig:example}, entropy cannot distinguish OOD (see Figure~\\ref{fig:example} (c)) and conflicting predictions (see Figure~\\ref{fig:example} (b)) because entropy is high for both cases. Similarly, neither aleatoric uncertainty nor epistemic uncertainty can distinguish OOD from conflicting predictions. In both cases, aleatoric uncertainty is high while epistemic uncertainty is low. On the other hand, vacuity and dissonance can clearly distinguish OOD from a conflicting prediction. For example, OOD objects typically show high vacuity with low dissonance while conflicting predictions exhibit low vacuity with high dissonance. This observation is confirmed through the empirical validation via our extensive experiments in terms of misclassification and OOD detection tasks.\n\\vspace{-2mm}\n\\section{Uncertainty-Aware Semi-Supervised Learning}\n\\vspace{-2mm}\nIn this section, we describe our proposed uncertainty framework based on semi-supervised node classification problem. The overall description of the framework is shown in Figure~\\ref{fig:framework}. \n\\vspace{-2mm}\n\\subsection{Problem Definition} \\label{subsec:problem-definition}\n\\vspace{-1mm}\nGiven an input graph $\\mathcal{G} = (\\mathbb{V}, \\mathbb{E}, {\\bf r}, {\\bf y}_\\mathbb{L})$, where $\\mathbb{V} = \\{1, \\ldots, N \\}$ is a ground set of nodes, $\\mathbb{E} \\subseteq \\mathbb{V}\\times \\mathbb{V}$ is a ground set of edges, $\\textbf{r} = [\\textbf{r}_1, \\cdots, \\textbf{r}_N]^T \\in \\mathbb{R}^{N\\times d}$ is a node-level feature matrix, $\\textbf{r}_i\\in \\mathbb{R}^d$ is the feature vector of node $i$, $\\textbf{y}_{\\mathbb{L}}=\\{y_i \\mid i \\in \\mathbb{L}\\}$ are the labels of the training nodes $\\mathbb{L} \\subset \\mathbb{V}$, and $y_i \\in \\{1, \\ldots, K\\}$ is the class label of node $i$. {\\bf We aim to predict}: (1) the \\textbf{class probabilities} of the testing nodes: $\\textbf{p}_{\\mathbb{V} \\setminus \\mathbb{L}} = \\{\\textbf{p}_i \\in [0, 1]^K \\mid i \\in \\mathbb{V} \\setminus \\mathbb{L}\\}$; and (2) the \\textbf{associated multidimensional uncertainty estimates} introduced by different root causes: $\\mathbf{u}_{\\mathbb{V} \\setminus \\mathbb{L}} = \\{\\mathbf{u}_i \\in [0, 1]^m \\mid i \\in \\mathbb{V} \\setminus \\mathbb{L}\\}$, where $p_{i, k}$ is the probability that the class label $y_i = k$ and $m$ is the total number of\nuncertainty types. \n\n\\begin{figure*}[t!]\n \\centering\n \\vspace{-1mm}\n \\includegraphics[width=0.75\\linewidth]{fig\/framework1.png}\n \\vspace{-2mm}\n \\scriptsize{\n \\caption{\\footnotesize \\textbf{Uncertainty Framework Overview.} Subjective Bayesian GNN (a) designed for estimating the different types of uncertainties (b).\n}\n \\label{fig:framework}\n\\vspace{-4mm}\n }\n\\end{figure*} \n\n\\vspace{-2mm}\n\\subsection{Proposed Uncertainty Framework} \\label{subsec:bay-dl}\n\\vspace{-2mm}\n\\textbf{Learning evidential uncertainty.}\nAs discussed in Section~\\ref{SL}, evidential uncertainty can be derived from multinomial opinions or equivalently Dirichlet distributions to model a probability distribution for the class probabilities. Therefore, we design a Subjective GNN (S-GNN) $f$ to form their multinomial opinions for the node-level Dirichlet distribution $\\text{Dir}(\\textbf{p}_i | {\\bm \\alpha}_i)$ of a given node $i$. Then, the conditional probability $P(\\textbf{p} |A, \\textbf{r}; \\bm{\\theta})$ can be obtained by:\n\\begin{eqnarray}\\small \nP(\\textbf{p} |A, \\textbf{r}; \\bm{\\theta} )=\\prod\\nolimits_{i=1}^N \\text{Dir}(\\textbf{p}_i|\\bm{\\alpha}_i), \\ \\bm{\\alpha}_i=f_i(A,\\textbf{r};\\bm{\\theta}), \\label{GCN_1}\n\\end{eqnarray}\nwhere $f_i$ is the output of S-GNN for node $i$, $\\bm{\\theta}$ is the model parameters, and $A$ is an adjacency matrix. The Dirichlet probability function $\\text{Dir}(\\textbf{p}_i | \\bm{\\alpha}_i)$ is defined by:\n\\begin{eqnarray}\\small \n \\text{Dir}(\\textbf{p}_i | \\bm{\\alpha}_i)=\\frac{\\Gamma(S_i)}{\\prod_{k=1}^K \\Gamma(\\alpha_{ik})}\\prod\\nolimits_{k=1}^K p_{ik}^{\\alpha_{ik}-1}.\n\\end{eqnarray}\nNote that S-GNN is similar to classical GNN, except that we use an activation layer (e.g., \\textit{ReLU}) instead of the \\textit{softmax} layer (only outputs class probabilities). This ensures that S-GNN would output non-negative values, which are taken as the parameters for the predicted Dirichlet distribution. \n\n\\textbf{Learning probabilistic uncertainty.}\nSince probabilistic uncertainty relies on a Bayesian framework, we proposed a Subjective Bayesian GNN (S-BGNN) that adapts S-GNN to a Bayesian framework, with the model parameters $\\bm{\\theta}$ following a prior distribution. The joint class probability of $\\textbf{y}$ can be estimated by: \n\\begin{eqnarray}\n\\vspace{-3mm}\n\\small \nP(\\textbf{y} |A, \\textbf{r}; \\mathcal{G}) &=& \\int \\int P(\\textbf{y} | \\textbf{p}) P(\\textbf{p} |A, \\textbf{r}; \\bm{\\theta} ) P(\\bm{\\theta} | \\mathcal{G}) d \\textbf{p} d\\bm{\\theta} \\nonumber \\\\\n&\\approx& \\frac{1}{M}\\sum_{m=1}^M \\sum_{i=1}^N \\int P(\\textbf{y}_i | \\textbf{p}_i) P(\\textbf{p}_i | A, \\textbf{r};\\bm{\\theta}^{(m)} ) d \\textbf{p}_i, \\quad \\bm{\\theta}^{(m)} \\sim q( \\bm{\\theta}) \n\\label{Baye_model}\n\\vspace{-2mm}\n\\end{eqnarray}\nwhere $P(\\bm{\\theta} | \\mathcal{G})$ is the posterior, estimated via dropout inference, that provides an approximate solution of posterior $q(\\bm{\\theta})$ and taking samples from the posterior distribution of models~\\cite{gal2016dropout}. Thanks to the benefit of dropout inference, training a DL model directly by minimizing the cross entropy (or square error) loss function can effectively minimize the KL-divergence between the approximated distribution and the full posterior (i.e., KL[$q(\\bm{\\theta})\\|P(\\theta|\\mathcal{G})$]) in variational inference~\\cite{gal2016dropout, kendall2015bayesian}. For interested readers, please refer to more detail in Appendix B.8.\n\nTherefore, training S-GNN with stochastic gradient descent enables learning of an approximated distribution of weights, which can provide good explainability of data and prevent overfitting. We use a {\\em loss function} to compute its Bayes risk with respect to the sum of squares loss $\\|\\textbf{y}-\\textbf{p}\\|^2_2$ by:\n\\begin{eqnarray}\\small \n\\mathcal{L}(\\bm{\\theta}) = \\sum\\nolimits_{i\\in \\mathbb{L}} \\int \\|\\textbf{y}_i-\\textbf{p}_i\\|^2_2 \\cdot P(\\textbf{p}_i |A, \\textbf{r}; \\bm{\\theta}) d \\textbf{p}_i \n= \\sum\\nolimits_{i\\in \\mathbb{L}} \\sum\\nolimits_{k=1}^K \\big(y_{ik}-\\mathbb{E}[p_{ik}]\\big)^2 + \\text{Var}(p_{ik}),\n\\label{loss}\n\\end{eqnarray}\nwhere $\\textbf{y}_i$ is an one-hot vector encoding the ground-truth class with $y_{ij} = 1$ and $y_{ik} \\neq $ for all $k \\neq j$ and $j$ is a class label. Eq.~\\eqref{loss} aims to minimize the prediction error and variance, leading to maximizing the classification accuracy of each training node by removing excessive misleading evidence.\n\\vspace{-1mm}\n\\subsection{Graph-based Kernel Dirichlet distribution Estimation (GKDE)}\n\\begin{wrapfigure}{R}{0.4\\textwidth}\n \\centering\n \\includegraphics[width=0.4\\textwidth]{fig\/GKDE.png}\n \\caption{\\small{Illustration of GKDE. Estimate prior Dirichlet distribution $\\text{Dir}(\\hat{\\alpha})$ for node $j$ (red) based on training nodes (blue) and graph structure information.}}\n \\label{fig:gkde}\n \\vspace{-2mm}\n\\end{wrapfigure} \n\\vspace{-1mm}\nThe loss function in Eq.~\\eqref{loss} is designed to measure the sum of squared loss based on class labels of training nodes. However, it does not directly measure the quality of the predicted node-level Dirichlet distributions. To address this limitation, we proposed \\textit{Graph-based Kernel Dirichlet distribution Estimation} (GKDE) to better estimate node-level Dirichlet distributions by using graph structure information. The key idea of the GKDE is to estimate prior Dirichlet distribution parameters for each node based on the class labels of training nodes (see Figure~\\ref{fig:gkde}). Then, we use the estimated prior Dirichlet distribution in the training process to learn the following patterns: (i) nodes with a high vacuity will be shown far from training nodes; and (ii) nodes with a high dissonance will be shown near the boundaries of classes.\n\n\\vspace{-1mm}\nBased on SL, let each training node represent one evidence for its class label. Denote the contribution of evidence estimation for node $j$ from training node $i$ by $\\mathbf{h}(y_i,d_{ij}) =[h_1, \\ldots, h_k, \\ldots, h_K] \\in[0, 1]^K$, where $h_k(y_i,d_{ij})$ is obtained by: \n\\vspace{-1mm}\n\\begin{eqnarray}\nh_k(y_i,d_{ij}) = \\begin{cases}0 & y_i \\neq k \\\\ g(d_{ij}) & y_i = k, \\end{cases}\n\\vspace{-2mm}\n\\end{eqnarray}\n $g(d_{ij}) = \\frac{1}{\\sigma \\sqrt{2\\pi}}\\exp({-\\frac{{d^2_{ij}}}{2\\sigma^2}})$ is the Gaussian kernel function used to estimate the distribution effect between nodes $i$ and $j$, and $d_{ij}$ means the \\textbf{node-level distance} (\\textbf{a shortest path between nodes $i$ and $j$}), and $\\sigma$ is the bandwidth parameter. The prior evidence is estimated based GKDE: $\\hat{\\bm{e}}_j = \\sum_{i\\in \\mathbb{L}} \\mathbf{h}(y_i,d_{ij})$, where $\\mathbb{L}$ is a set of training nodes and the prior Dirichlet distribution $\\hat{\\bm{\\alpha}}_j = \\hat{\\bm{e}}_j +\\bf 1$. \nDuring the training process, we minimize the KL-divergence between model predictions of Dirichlet distribution and prior distribution: $\\min \\text{KL}[\\text{Dir}(\\bm{\\alpha}) \\| \\text{Dir}(\\hat{\\bm{\\alpha}})]$.\nThis process can prioritize the extent of data relevance based on the estimated evidential uncertainty, which is proven effective based on the proposition below.\n\n\\begin{restatable}{proposition}{primeproposition}\nGiven $L$ training nodes, for any testing nodes $i$ and $j$, let ${\\bm d}_i = [d_{i1}, \\ldots, d_{iL}]$ be the vector of graph distances from nodes $i$ to training nodes and ${\\bm d}_j = [d_{j1}, \\ldots, d_{jL}]$ be the graph distances from nodes $j$ to training nodes, where $d_{il}$ is the node-level distance between nodes $i$ and $l$. If for all $l\\in \\{1, \\ldots, L\\}$, $d_{il} \\ge d_{jl}$, then we have\n\\begin{eqnarray}\n\\hat{u}_{v_i} \\ge \\hat{u}_{v_j}, \\nonumber\n\\end{eqnarray}\nwhere $ \\hat{u}_{v_i}$ and $\\hat{u}_{v_j}$ refer to vacuity uncertainties of nodes $i$ and $j$ estimated based on GKDE.\n\\label{theorem: vacuity}\n\\vspace{-1mm}\n\\end{restatable}\nThe proof for this proposition can be found in Appendix A.2. The above proposition shows that if a testing node is too far from training nodes, the vacuity will increase, implying that an OOD node is expected to have a high vacuity.\n\nIn addition, we designed a simple iterative knowledge distillation method~\\cite{hinton2015distilling} (i.e., Teacher Network) to refine the node-level classification probabilities. The key idea is to train our proposed model (Student) to imitate the outputs of a pre-train a vanilla GNN (Teacher) by adding a regularization term of KL-divergence. This leads to solving the following optimization problem: \n\\begin{eqnarray}\\small \n\\vspace{-1mm}\n\\min\\nolimits_{\\bm{\\theta}} \\mathcal{L}(\\bm{\\theta}) + \\lambda_1 \\text{KL}[\\text{Dir}({\\bm \\alpha}) \\| \\text{Dir}(\\hat{\\bm \\alpha})] + \\lambda_2 \\text{KL}[P(\\textbf{y} \\mid A,\\textbf{r};\\mathcal{G}) \\parallel P(\\textbf{y}|\\hat{\\textbf{p}})], \n\\label{joint loss}\n\\vspace{-1mm}\n\\end{eqnarray}\nwhere $\\hat{\\textbf{p}}$ is the vanilla GNN's (Teacher) output and $\\lambda_1$ and $\\lambda_2$ are trade-off parameters. \n\n\\section{Experiments} \\label{sec:exp-results-analysis}\n\\vspace{-1mm}\nIn this section, we conduct experiments on the tasks of misclassification and OOD detections to answer the following questions for semi-supervised node classification:\n\n\\vspace{-1mm}\n\\noindent {\\bf Q1. Misclassification Detection:} What type of uncertainty is the most promising indicator of high confidence in node classification predictions? \n\n\\vspace{-1mm}\n\\noindent {\\bf Q2. OOD Detection:} What type of uncertainty is a key indicator of accurate detection of OOD nodes? \n\n\\vspace{-1mm}\n\\noindent {\\bf Q3. GKDE with Uncertainty Estimates:} How can GKDE help enhance prediction tasks with what types of uncertainty estimates?\n\n\\vspace{-1mm}\nThrough extensive experiments, we found the following answers for the above questions:\n\n\\vspace{-1mm}\n\\noindent {\\bf A1.} Dissonance (i.e., uncertainty due to conflicting evidence) is more effective than other uncertainty estimates in misclassification detection. \n\\vspace{-1mm}\n\n\\noindent {\\bf A2.} Vacuity (i.e., uncertainty due to lack of confidence) is more effective than other uncertainty estimates in OOD detection.\n\n\\vspace{-1mm}\n\\noindent {\\bf A3.} GKDE can indeed help improve the estimation quality of node-level Dirichlet distributions, resulting in a higher OOD detection.\n\n\\vspace{-2mm}\n\\subsection{Experiment Setup} \n\\vspace{-2mm}\n\\textbf{Datasets}: We used six datasets, including three citation network datasets~\\cite{sen2008collective} (i.e., Cora, Citeseer, Pubmed) and three new datasets~\\cite{shchur2018pitfalls} (i.e., Coauthor Physics, Amazon Computer, and Amazon Photo). We summarized the description and experimental setup of the used datasets in Appendix B.2\\footnote{The source code and datasets are accessible at \\href{https:\/\/github.com\/zxj32\/uncertainty-GNN}{\\color{magenta}{https:\/\/github.com\/zxj32\/uncertainty-GNN}}}. \n\n\\vspace{-1mm}\n\\textbf{Comparing Schemes}: \nWe conducted the extensive comparative performance analysis based on our proposed models and several state-of-the-art competitive counterparts. We implemented all models based on the most popular GNN model, GCN~\\cite{kipf2017semi}. We compared our model (S-BGCN-T-K) against: (1) Softmax-based GCN~\\cite{kipf2017semi} with uncertainty measured based on entropy; and (2) Drop-GCN that adapts the Monte-Carlo Dropout~\\cite{gal2016dropout, ryu2019uncertainty} into the GCN model to learn probabilistic uncertainty; (3) EDL-GCN that adapts the EDL model~\\cite{sensoy2018evidential} with GCN to estimate evidential uncertainty; (4) DPN-GCN that adapts the DPN~\\cite{malinin2018predictive} method with GCN to estimate probabilistic uncertainty. We evaluated the performance of all models considered using the area under the ROC (AUROC) curve and area under the Precision-Recall (AUPR) curve in both experiments~\\cite{hendrycks2016baseline}.\n\n\\vspace{-1mm}\n\\subsection{Results}\n\\vspace{-1mm}\n\\noindent {\\bf Misclassification Detection.} The misclassification detection experiment involves detecting whether a given prediction is incorrect using an uncertainty estimate. Table~\\ref{AUPR:uncertainty} shows that S-BGCN-T-K outperforms all baseline models under the AUROC and AUPR for misclassification detection. The outperformance of dissonance-based detection is fairly impressive. This confirms that low dissonance (a small amount of conflicting evidence) is the key to maximize the accuracy of node classification prediction. We observe the following performance order: ${\\tt Dissonance} > {\\tt Entropy} \\approx {\\tt Aleatoric} > {\\tt Vacuity} \\approx {\\tt Epistemic}$, which is aligned with our conjecture: higher dissonance with conflicting prediction leads to higher misclassification detection. We also conducted experiments on additional three datasets and observed similar trends of the results, as demonstrated in Appendix C.\n\\vspace{-1mm}\n\\begin{table*}[th!]\n\\scriptsize\n\\caption{AUROC and AUPR for the Misclassification Detection.}\n\\vspace{-2mm}\n\\centering\n\\begin{tabular}{c||c|ccccc|ccccc|c}\n\\hline\n\\multirow{2}{*}{Data} & \\multirow{2}{*}{Model} & \\multicolumn{5}{c|}{AUROC} & \\multicolumn{5}{c|}{AUPR} & \\multirow{2}{*}{Acc} \\\\ \n & & Va.\\tnote{*}& Dis. & Al. & Ep. & En. & Va. & Dis. & Al. & Ep. &En. & \\\\ \\hline\n\\multirow{5}{*}{Cora} & S-BGCN-T-K & 70.6 & \\textbf{82.4} & 75.3 & 68.8 & 77.7& 90.3 & \\textbf{95.4} & 92.4 & 87.8 &93.4 & \\textbf{82.0} \\\\ \n & EDL-GCN & 70.2 & 81.5 & - & - & 76.9 &90.0 & 94.6 & - & - & 93.6 & 81.5 \\\\ \n & DPN-GCN & - & - & 78.3 & 75.5 & 77.3 & - & - & 92.4 & 92.0 &92.4 & 80.8 \\\\ \n & Drop-GCN & - & - & 73.9 & 66.7 & 76.9 & - & - & 92.7 & 90.0 &93.6 & 81.3 \\\\ \n &GCN & - & - & - & - & 79.6& - & - & - & - & 94.1 & 81.5 \\\\ \n \\hline\n\\multirow{5}{*}{Citeseer} & S-BGCN-T-K & 65.4& \\textbf{74.0}& 67.2& 60.7& 70.0& 79.8 & \\textbf{85.6} & 82.2 & 75.2 &83.5& \\textbf{71.0} \\\\ \n & EDL-GCN & 64.9 & 73.6 & - & - & 69.6 &79.2 & 84.6 & - & - & 82.9 & 70.2 \\\\ \n & DPN-GCN & - & - & 66.0 & 64.9 & 65.5 & - & - & 78.7 & 77.6 &78.1 & 68.1 \\\\ \n & Drop-GCN & - & - & 66.4 & 60.8 & 69.8 & - & -& 82.3 & 77.8 &83.7 & 70.9 \\\\ \n & GCN & - & - & - & - & 71.4 & - & - & - & - &83.2 & 70.3 \\\\ \\hline\n\\multirow{5}{*}{Pubmed} & S-BGCN-T-K & 64.1 & \\textbf{73.3} & 69.3 & 64.2 & 70.7& 85.6 & \\textbf{90.8} & 88.8& 86.1 &89.2 & \\textbf{79.3} \\\\\n & EDL-GCN & 62.6 & 69.0 & - & - & 67.2 &84.6 & 88.9 & - & - & 81.7 & 79.0 \\\\ \n & DPN-GCN & - & - & 72.7 & 69.2 & 72.5 & - & - & 87.8 & 86.8 &87.7 & 77.1 \\\\ \n & Drop-GCN & - & - & 67.3 & 66.1 & 67.2& - & -& 88.6 & 85.6 &89.0 & 79.0 \\\\ \n & GCN & - & - & - & - & 68.5& - & - & - & - &89.2 & 79.0 \\\\ \\hline\n\\end{tabular}\n\\begin{tablenotes}\\scriptsize\n\\centering\n\\item[*] Va.: Vacuity, Dis.: Dissonance, Al.: Aleatoric, Ep.: Epistemic, En.: Entropy \n\\end{tablenotes}\n\\vspace{2mm}\n\\label{AUPR:uncertainty}\n\\vspace{-5mm}\n\\end{table*}\n\n\n\n\n\\begin{table*}[th!]\n\\scriptsize\n\\caption{AUROC and AUPR for the OOD Detection.}\n\\vspace{-2mm}\n\\centering\n\\begin{tabular}{c||c|ccccc|ccccc}\n\\hline\n\\multirow{2}{*}{Data} & \\multirow{2}{*}{Model} & \\multicolumn{5}{c|}{AUROC} & \\multicolumn{5}{c}{AUPR} \\\\ \n & & Va.\\tnote{*}& Dis. & Al. & Ep. &En. & Va. & Dis. & Al. & Ep. & En. \\\\ \\hline\n\\multirow{4}{*}{Cora} & S-BGCN-T-K & \\textbf{87.6} & 75.5 & 85.5 & 70.8 & 84.8& \\textbf{78.4} & 49.0 & 75.3 & 44.5 & 73.1 \\\\ \n & EDL-GCN & 84.5 & 81.0 & & -& 83.3& 74.2 & 53.2 & - & -&71.4 \\\\ \n & DPN-GCN & - & - & 77.3 & 78.9 & 78.3 & - & - & 58.5 & 62.8 & 63.0 \\\\ \n & Drop-GCN & - & - & 81.9 & 70.5 & 80.9 & - & - & 69.7 & 44.2 & 67.2 \\\\ \n & GCN & - & - & - & - & 80.7& - & - & - & - & 66.9 \\\\ \\hline\n\\multirow{4}{*}{Citeseer} & S-BGCN-T-K & \\textbf{84.8} &55.2&78.4 & 55.1 & 74.0& \\textbf{86.8} & 54.1 & 80.8 & 55.8 & 74.0 \\\\ \n & EDL-GCN & 78.4 &59.4&- & - & 69.1& 79.8 & 57.3 & - & - & 69.0 \\\\ \n & DPN-GCN & - & - & 68.3 & 72.2 & 69.5 & - & -& 68.5 & 72.1 & 70.3 \\\\ \n & Drop-GCN & - & - & 72.3 & 61.4 & 70.6 & - & -& 73.5 & 60.8 & 70.0 \\\\ \n & GCN & - & - & - & - & 70.8 & - & - & - & - & 70.2 \\\\ \\hline\n\\multirow{4}{*}{Pubmed} & S-BGCN-T-K & \\textbf{74.6} &67.9& 71.8 & 59.2 & 72.2& \\textbf{69.6} & 52.9 & 63.6& 44.0 &56.5 \\\\\n & EDL-GCN & 71.5 &68.2& - & - & 70.5& 65.3 & 53.1 & -& - & 55.0 \\\\\n & DPN-GCN & - & - & 63.5 & 63.7 & 63.5& - & -& 50.7 & 53.9 & 51.1 \\\\ \n & Drop-GCN & - & - & 68.7 & 60.8 & 66.7& - & -& 59.7 & 46.7 & 54.8 \\\\ \n & GCN & - & - & - & - & 68.3& - & - & - & -&55.3 \\\\ \\hline\n\\multirow{4}{*}{Amazon Photo} & S-BGCN-T-K & \\textbf{93.4} & 76.4& 91.4& 32.2 & 91.4& \\textbf{ 94.8} & 68.0 & 92.3& 42.3 & 92.5 \\\\\n & EDL-GCN & 63.4 & 78.1& - & - & 79.2& 66.2 & 74.8 & -&- & 81.2 \\\\\n & DPN-GCN & - & - & 83.6 & 83.6 & 83.6& - & -& 82.6 & 82.4 & 82.5 \\\\ \n & Drop-GCN & - & - & 84.5 & 58.7 & 84.3& - & -& 87.0 & 57.7 &86.9 \\\\ \n & GCN & - & - & - & - & 84.4& - & - & - & -&87.0 \\\\ \\hline\n\\multirow{4}{*}{Amazon Computer} & S-BGCN-T-K & \\textbf{82.3} & 76.6& 80.9& 55.4 & 80.9& \\textbf{70.5} & 52.8 & 60.9& 35.9 & 60.6 \\\\\n & EDL-GCN & 53.2 & 70.1& - & - & 70.0& 33.2 & 43.9 & -& - & 45.7 \\\\\n & DPN-GCN & - & - & 77.6 & 77.7 & 77.7 & - & -& 50.8 & 51.2 & 51.0 \\\\ \n & Drop-GCN & - & - & 74.4 & 70.5 & 74.3& - & -& 50.0 & 46.7 & 49.8 \\\\ \n & GCN & - & - & - & - & 74.0& - & - & - & -&48.7 \\\\ \\hline\n\\multirow{4}{*}{Coauthor Physics} & S-BGCN-T-K & \\textbf{91.3} & 87.6& 89.7& 61.8 & 89.8& \\textbf{72.2} & 56.6 & 68.1& 25.9 & 67.9 \\\\\n & EDL-GCN & 88.2 & 85.8& - & - & 87.6& 67.1 & 51.2 & -&- & 62.1 \\\\\n & DPN-GCN & - & - & 85.5 &85.6 & 85.5& - & -& 59.8 & 60.2 & 59.8 \\\\ \n & Drop-GCN & - & - & 89.2 & 78.4 & 89.3& - & -& 66.6 & 37.1 &66.5 \\\\ \n & GCN & - & - & - & - & 89.1& - & - & - & -&64.0 \\\\ \\hline\n\\end{tabular}\n\\begin{tablenotes}\\scriptsize\n\\centering\n\\item[*] Va.: Vacuity, Dis.: Dissonance, Al.: Aleatoric, Ep.: Epistemic, En.: Entropy \n\\end{tablenotes}\n\\vspace{2mm}\n\\label{Table: AUROC_AUPR:ood}\n\\vspace{-7mm}\n\\end{table*}\n\n\\vspace{-0.3mm}\n\\noindent {\\bf OOD Detection.} This experiment involves detecting whether an input example is out-of-distribution (OOD) given an estimate of uncertainty. For semi-supervised node classification, we randomly selected one to four categories as OOD categories and trained the models based on training nodes of the other categories. Due to the space constraint, the experimental setup for the OOD detection is detailed in Appendix B.3. \n\n\\vspace{-1mm}\nIn Table~\\ref{Table: AUROC_AUPR:ood}, across six network datasets, our vacuity-based detection significantly outperformed the other competitive methods, exceeding the performance of the epistemic uncertainty and other type of uncertainties. This demonstrates that vacuity-based model is more effective than other uncertainty estimates-based counterparts in increasing OOD detection. We observed the following performance order: ${\\tt Vacuity} > {\\tt Entropy} \\approx {\\tt Aleatoric} > {\\tt Epistemic} \\approx {\\tt Dissonance}$, which is consistent with the theoretical results as shown in Theorem~\\ref{theorem1}.\n\n\\vspace{-1mm}\n\\noindent {\\bf Ablation Study.}\nWe conducted additional experiments (see Table~\\ref{Ablation}) in order to demonstrate the contributions of the key technical components, including GKDE, Teacher Network, and subjective Bayesian framework. The key findings obtained from this experiment are: (1) GKDE can enhance the OOD detection (i.e., 30\\% increase with vacuity), which is consistent with our theoretical proof about the outperformance of GKDE in uncertainty estimation, i.e., OOD nodes have a higher vacuity than other nodes; and (2) the Teacher Network can further improve the node classification accuracy.\n\\vspace{-2mm}\n\\subsection{Why is Epistemic Uncertainty Less Effective than Vacuity?}\n\\vspace{-2mm}\nAlthough epistemic uncertainty is known to be effective to improve OOD detection~\\cite{gal2016dropout, kendall2017uncertainties} in computer vision applications, our results demonstrate it is less effective than our vacuity-based approach. The first potential reason is that epistemic uncertainty is always smaller than vacuity (From Theorem~\\ref{theorem1}), which potentially indicates that epistemic may capture less information related to OOD. Another potential reason is that the previous success of epistemic uncertainty for OOD detection is limited to supervised learning in computer vision applications, but its effectiveness for OOD detection was not sufficiently validated in semi-supervised learning tasks. Recall that epistemic uncertainty (i.e., model uncertainty) is calculated based on mutual information (see Eq.~\\eqref{eq:epistemic}). In a semi-supervised setting, the features of unlabeled nodes are also fed to a model for training process to provide the model with a high confidence on its output. For example, the model output $P(\\textbf{y}|A, \\textbf{r};\\theta)$ would not change too much even with differently sampled parameters $\\bm{\\theta}$, i.e., $P(\\textbf{y}|A, \\textbf{r};\\theta^{(i)})\\approx P(\\textbf{y}|A, \\textbf{r};\\theta^{(j)})$, which result in a low epistemic uncertainty. We also designed a semi-supervised learning experiment for image classification and observed a consistent pattern with the results demonstrated in Appendix C.6. \n\n\\begin{table*}[th!]\n\\scriptsize\n\\caption{Ablation study of our proposed models: (1) {\\tt S-GCN}: Subjective GCN with vacuity and dissonance estimation; (2) {\\tt S-BGCN}: S-GCN with Bayesian framework; (3) {\\tt S-BGCN-T}: S-BGCN with a Teacher Network; (4) {\\tt S-BGCN-T-K}: S-BGCN-T with GKDE to improve uncertainty estimation.}\n\\vspace{1mm}\n\\centering\n\\vspace{-3mm}\n\\begin{tabular}{c||c|ccccc|ccccc|c}\n\\hline\n\\multirow{2}{*}{Data} & \\multirow{2}{*}{Model} & \\multicolumn{5}{c|}{AUROC (Misclassification Detection)} & \\multicolumn{5}{c|}{AUPR (Misclassification Detection)} & \\multirow{2}{*}{Acc} \\\\\n & & Va.\\tnote{*}& Dis. & Al. & Ep. & En. & Va. & Dis. & Al. & Ep. &En. & \\\\ \\hline\n\\multirow{4}{*}{Cora} & S-BGCN-T-K & 70.6 & 82.4 & 75.3 & 68.8 & 77.7& 90.3 & \\textbf{95.4} & 92.4 & 87.8 &93.4 & 82.0 \\\\ \n & S-BGCN-T & 70.8 & \\textbf{82.5} &75.3 & 68.9 & 77.8 &90.4 & \\textbf{95.4} & 92.6 & 88.0 &93.4 & \\textbf{82.2} \\\\\n & S-BGCN & 69.8 & 81.4 & 73.9 & 66.7 & 76.9 & 89.4 & 94.3 & 92.3 & 88.0 &93.1 & 81.2 \\\\ \n & S-GCN & 70.2 & 81.5 & - & - & 76.9 & 90.0 & 94.6 & - & - &93.6 & 81.5 \\\\\n \\hline\n & & \\multicolumn{5}{c|}{AUROC (OOD Detection)} & \\multicolumn{5}{c|}{AUPR (OOD Detection)} & \\\\ \\hline \n\\multirow{4}{*}{Amazon Photo} & S-BGCN-T-K & \\textbf{93.4} & 76.4& 91.4& 32.2 & 91.4& \\textbf{ 94.8} & 68.0 & 92.3& 42.3 & 92.5 &- \\\\\n & S-BGCN-T & 64.0 & 77.5& 79.9 & 52.6 & 79.8& 67.0 & 75.3 & 82.0& 53.7 & 81.9&- \\\\\n & S-BGCN & 63.0 & 76.6& 79.8 & 52.7 & 79.7& 66.5 & 75.1 & 82.1& 53.9 & 81.7&- \\\\ \n & S-GCN & 64.0 & 77.1& - & - & 79.6& 67.0 & 74.9 & -& - & 81.6&- \\\\ \\hline\n\\end{tabular}\n\\begin{tablenotes}\\scriptsize\n\\centering\n\\item[*] Va.: Vacuity, Dis.: Dissonance, Al.: Aleatoric, Ep.: Epistemic, En.: Entropy \n\\end{tablenotes}\n\\vspace{2mm}\n\\label{Ablation}\n\\vspace{-5mm}\n\\end{table*}\n\n\\vspace{-2mm}\n\\section{Conclusion} \\label{sec:conclusion}\n\\vspace{-2mm}\nIn this work, we proposed a multi-source uncertainty framework of GNNs for semi-supervised node classification. Our proposed framework provides an effective way of predicting node classification and out-of-distribution detection considering multiple types of uncertainty. We leveraged various types of uncertainty estimates from both DL and evidence\/belief theory domains. Through our extensive experiments, we found that dissonance-based detection yielded the best performance on misclassification detection while vacuity-based detection performed the best for OOD detection, compared to other competitive counterparts. In particular, it was noticeable that applying GKDE and the Teacher network further enhanced the accuracy in node classification and uncertainty estimates.\n\n\\section*{Acknowledgments}\nWe would like to thank Yuzhe Ou for providing proof suggestions. \nThis work is supported by the National Science Foundation\n(NSF) under Grant No \\#1815696 and \\#1750911.\n\n\n\\section*{Broader Impact}\nIn this paper, we propose a uncertainty-aware semi-supervised learning framework of GNN for predicting multi-dimensional uncertainties for the task of semi-supervised node classification. Our proposed framework can be applied to a wide range of applications, including computer vision, natural language processing, recommendation systems, traffic prediction, generative models and many more~\\cite{zhou2018graph}. Our proposed framework can be applied to predict multiple uncertainties of different roots for GNNs in these applications, improving the understanding of individual decisions, as well as the underlying models. While there will be important impacts resulting from the use of GNNs in general, our focus in this work is on investigating the impact of using our method to predict multi-source uncertainties for such systems. The additional benefits of this method include improvement of safety and transparency in decision-critical applications to avoid overconfident prediction, which can easily lead to misclassification. \n\nWe see promising research opportunities that can adopt our uncertainty framework, such as investigating whether this uncertainty framework can further enhance misclassification detection or OOD detection. To mitigate the risk from different types of uncertainties, we encourage future research to understand the impacts of this proposed uncertainty framework to solve other real world problems.\n\n\\newpage\n\\small\n\\medskip\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nMany of the advances of quantum computation based on superconducting qubits rely on the ability to readout the qubit state by measuring microwave photons leaking out of a superconducting resonator \\cite{wallraff-schoelkopf-nature2004}. Thanks to the development of near-quantum-limited Josephson parametric amplifiers (JPAs) \\cite{beltran-lehnert-apl2007,bergeal-devoret-nature2010,zhou-esteve-prb2014,eichler-wallraff-prl2014}, high-fidelity single-shot qubit readout is now possible \\cite{mallet-esteve-nphys2009,walter-wallraff-prappl2017}. These amplifiers are, moreover, finding use in a wide range of applications, from measuring quantum features in the radiation emitted by mesoscopic conductors \\cite{zakka-portier-prl2010,gasse-reulet-prl2013,stehlik-petta-prappl2015,westig-portier-prl2017,simoneau-reulet-prb2017}, to the detection of small ensembles of electronic spins \\cite{bienfait-bertet-prx2017}, and even to the search for dark matter \\cite{brubaker-carosi-prl2017}. JPAs are also versatile sources of single- and two-mode squeezed states \\cite{beltran-lehnert-nphys2008,eichler-wallraff-prl2014}, which have been used to confirm decade old predictions in quantum optics \\cite{murch-siddiqui-nature2013,toyli-siddiqi-prx2016}, and to improve electron-spin resonance spectroscopy \\cite{bienfait-bertet-prx2017}. On the theoretical side, squeezed states were proposed as a resource to improve qubit readout and to perform high-fidelity gates \\cite{didier-blais-clerk-prl2015,puri-blais-prl2016,royer-blais-quantum2017}, or as a basis for continuous variable quantum computing \\cite{braunstein-loock-rmp2005,grimsmo-blais-npjqi2017}.\n\nCurrent JPAs are able to amplify signals to more than 20 dB, and to squeeze vacuum fluctuations by $7$ dB ($12$ dB) in single- (two-) mode experiments \\cite{boutin-blais-prappl2017,eichler-wallraff-prl2014}. However, in this devices, the amplification bandwidth is limited to hundreds of megahertz \\cite{mutus-martinis-apl2014,roy-vijay-apl2015,westig-klapwijk-arxiv2017}. At the price of increasing device fabrication complexity, much larger amplification bandwidth, over $\\sim 3$ GHz, has been demonstrated with the recently developed Josephson traveling wave parametric amplifier \\cite{macklin-siddiqi-science2015}. The development of a simpler quantum-limited microwave amplifier, generating far-separated two-mode squeezed states and capable of amplifying signals over gigahertz bandwidths, is still needed to further advance quantum information processing science. It would also be an important tool to better characterize the radiation emitted by mesoscopic conductors, for which there is an increasing body of interesting predictions \\cite{beenakker-schomerus-prl2004,armour-rimberg-prl2013,leppakangas-johansson-prl2015,mendes-mora-njp2015,mendes-mora-prb2016}. Here, we propose such a simple broadband parametric amplifier, consisting of a single dc- and ac-voltage biased superconductor-insulator-superconductor (SIS) junction. The device can be operated in both phase-sensitive and phase-preserving modes and can be used for near-quantum-limited amplification and two-mode squeezing in few GHz bandwidth.\n\nThe proposed setup, illustrated in Fig.~\\ref{fig1}(a), operates as an amplifier in reflection mode. Parametric amplification is possible by taking advantage of the strong non-linearity of the transport characteristics of the junction. To this end, we consider a dc-voltage bias $V$ smaller than twice the superconducting gap $\\Delta$. This sets the junction as an open circuit, and the conduction of quasiparticles is enabled by applying a sinusoidal ac-voltage $V_\\text{ac}(t) =V_\\text{ac}\\cos(2\\omega_0 t)$, with $\\omega_0$ the measurement center frequency. This voltage combination gives rise to modulations of the admittance of the junction $Y_n(\\omega)$, with $n$ the $n$-th harmonic of the pump. As illustrated in Fig.~\\ref{fig1}(b), the finite-frequency admittance for dc-voltage approaching $(2\\Delta-\\hbar\\omega_0)\/e$ is dominated by a large non-linear susceptance Im$Y_1(\\omega_0)$ \\cite{barone-paterno-book,lee-apl1982}. On the other hand, the dissipative term, $\\textrm{Re}Y_0(\\omega_0)$, is only weakly affected by the ac-voltage and it is dominant for $eV > 2\\Delta-\\hbar\\omega_0$. Taking advantage of dc-voltage control, we demonstrate that parametric amplification and squeezing emerges when $Y_1(\\omega_0)$ is much larger than Re$Y_0(\\omega_0)$. As we show below, $Y_1(\\omega_0)$ is related to the coherent conversion process of $2\\hbar\\omega_0$ quanta of energy from the pump to two photons of frequency $\\omega_0$, while Re$Y_0(\\omega_0)$ is related to dissipative effects due to the tunneling of quasiparticles. It is surprising that even though SIS junctions are routinely used as high-frequency microwave quantum-limited mixers \\cite{trucker-feldman-rmp1985}, exploiting a very similar principle, their operation as parametric amplifiers have been mostly disregarded \\cite{lee-apl1982,devyatov-zorin-jap1986}. Here, we use the input-output formalism \\cite{yurke-denker-pra1984}, together with photon-assisted tunneling theory \\cite{tien1963} to compute the parametric amplification and squeezing properties of an ac- and dc-biased SIS junction \\cite{trucker-feldman-rmp1985}. The resulting Heisenberg-Langevin equations \\cite{gardiner-collet-pra1985} are numerically solved, allowing us to explore parametric amplification far from the small detuning limit considered previously \\cite{lee-apl1982,devyatov-zorin-jap1986}. For an Aluminum junction ($\\Delta = 180\\ \\mu$eV) pumped at $2\\omega_0 = 8$ GHz and operated at temperatures $T \\ll \\Delta\/k_B$, with $k_B$ the Boltzmann constant, we find that when operated in the phase-sensitive mode this device can produce more than 30 dB of gain and -20 dB of squeezing. On the other hand, operated in the phase preserving-mode, it achieves 15 dB of gain with an added noise close to the quantum limit and -14 dB of two-mode squeezing, in both cases over a large bandwidth exceeding 4 GHz. Moreover, the gain and the squeezing bandwidths are easily tuned by the dc-voltage and ac-pumping, allowing for instance to achieve higher gain in a reduced yet sizable bandwidth.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig1.pdf}\n\\caption{(a) Electrical scheme of the device: a SIS junction is dc-voltage biased close to the onset of quasiparticle transport $eV \\sim 2\\Delta$, while it is ac pumped with a single tone $V_\\text{ac} \\cos(2\\omega_0 t)$. Current conservation at the coupling mode ($x=0$) allows to relate the transmission line outgoing field $a_\\text{out}[\\omega]$ with its incoming field $a_\\text{in}[\\omega]$ and the current flowing through the junction $\\hat{I}_\\text{J}[\\omega]$. (b) Non-linear admittance as a function of the dc-voltage for an Aluminum junction ($\\Delta = 180\\ \\mu$eV) and $eV_\\text{ac} = 0.135 \\times 2\\hbar\\omega_0$. On one hand, as shown in Sec.~\\ref{model}, dissipation emerges from the absorption and emission of photons, which is illustrated by Re$Y_0(\\omega)$. Notably, Re$Y_0(\\omega)$ is only slightly perturbed by such small ac pumping. However, the pumping gives rise to a parametric interaction characterized by a sizable non-linear admittance $Y_1(\\omega_0)$. For dc-voltages in the range $2(\\Delta-\\hbar\\omega_0) 2\\omega_0$ are up- or down-converted to $\\omega$. This approximation is justified by the fact that the $R_TC$ time of the junction acts as a high-frequency cutoff. For the results presented in the next section, we take $\\omega_{RC} = 1\/R_T C = 2\\pi \\times 30$ GHz as a fixed parameter of the junction, which is obtained for a normal state resistance $R_T = 50 \\Omega$ and $C=100$ fF. In addition, a low-pass filter can be used to filter all frequencies above $2\\omega_0$. Since we are interested in signals of frequency $\\omega_0 \\ll \\Delta\/\\hbar$ amplified by operating the junction close to the onset of quasi-particle transport $eV \\approx 2\\Delta$, we also ignored current and noise terms originating from the tunneling of Cooper pairs whose Josephson frequency $\\sim 4\\Delta\/\\hbar \\gg 2\\omega_0$ would be efficiently filtered as well.\n\n\\section{Results \\label{results}}\n\nWe first present results for an ideal SIS junction, with transport response rising steeply for voltages $eV = 2\\Delta + n\\hbar \\omega_0$ as illustrated in Fig.~\\ref{fig1}(b). We then investigate how gain and squeezing properties are affected by low-frequency noise, which are smoothing out the transport response of the junction and, consequently, diminishing the strength of the parametric interaction.\n\n\\subsection{Ideal SIS junction \\label{resultsA}}\n\nBefore turning to the full amplification and squeezing frequency dependences, we first present results for $\\omega = 0$ which corresponds to the phase-sensitive mode. Figs.~\\ref{fig2}(a) and (b) illustrate, respectively, the phase-sensitive gain and single-mode squeezing as a function of $\\Delta\/\\hbar\\omega_0$, when optimizing $V$ and $V_\\text{ac}$ to maximize single-mode squeezing. Here, $\\Delta$ varies while $\\omega_0\/2\\pi$ is kept fixed and equal to 4 GHz. Already at small $\\Delta\/\\hbar\\omega_0 \\simeq 2$ we observe 15 dB of amplification and $-10$ dB of squeezing. As $\\Delta\/\\hbar\\omega_0$ increases, the strength of the nonlinearities giving rise to parametric interaction increases, thus enhancing gain and squeezing as illustrated in Fig.~\\ref{fig2}. This can be understood by the fact that $S_n(\\omega)$ and, consequently, $Y_n(\\omega)$ are proportional to $\\Delta\/R_T$ in the limit of $\\Delta\/\\hbar\\omega_0 \\rightarrow \\infty$. Thus, increasing $\\Delta$ leads to an increase of the parametric interaction strength $\\propto Y_1(\\omega)$. For $\\Delta\/\\hbar\\omega_0 = 30$, amplification and squeezing reach $43$ dB and $-25$ dB, respectively. In Fig.~\\ref{fig2}, the dashed-line corresponds to an Aluminum junction. As expected, the optimal dc-voltage is $eV^\\text{opt} \\lesssim 2\\Delta - \\hbar\\omega_0$ for all values of $\\Delta\/\\hbar\\omega_0$ (not shown). At this dc-voltage, the susceptance Im$Y_1$ is larger than the dissipative contributions $\\propto$ Re$Y_0$, thus making the parametric interaction the dominant process over dissipation, as illustrated in Fig.~\\ref{fig1}(b). It is interesting to note that $V^\\text{opt}$ is also the optimal operational point of SIS mixers \\cite{trucker-feldman-rmp1985}. On the other hand, the optimal ac-voltage $V_\\text{ac}^\\text{opt}$ decreases as $\\Delta\/\\hbar\\omega_0$ increases.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig2.pdf}\n\\caption{(a) Phase-sensitive gain and (b) single-mode squeezing as a function of $\\Delta\/\\hbar\\omega_0$ for $\\omega_0\/2\\pi = 4$ GHz. The dashed line marks the values for an Aluminum superconducting junction. For each value of $\\Delta\/\\hbar\\omega_0$ there are optimal ac- and dc-voltages that maximize single-mode squeezing. The optimal dc-voltage is always equal to $eV^\\text{opt} \\lesssim 2\\Delta - \\hbar \\omega_0$, while the optimal ac-voltage amplitude diminishes as $\\Delta\/\\hbar\\omega_0$ increases (not shown here). \\label{fig2}} \n\\end{center}\n\\end{figure}\n\nFor the remainder of this article, we consider an Aluminum SIS junction with $\\Delta\/\\hbar\\omega_0\\approx 10.9$ for $\\omega_0\/2\\pi = 4$ GHz. In the phase-sensitive mode ($\\omega=0$), gain and squeezing are respectively $\\sim 34$ dB and $\\sim -20$ dB at the optimal operational point ($eV^\\text{opt} \\approx 20.762 \\times \\hbar\\omega_0$ and $eV_\\text{ac}^\\text{opt} \\approx 0.06 \\times 2\\hbar\\omega_0$). Moreover, the noise added by tunneling of quasiparticles is of the order of $10^{-4}$ added photons, meaning that the amplifier operates near the quantum limit.\n\nWe now investigate the frequency dependence of phase-preserving gain and two-mode squeezing, and how they are controlled by dc- and ac-voltage amplitudes. First, we find that for the optimal voltages, $V^\\text{opt}$ and $V_\\text{ac}^\\text{opt}$, both gain and squeezing vanish at finite detuning ($\\omega\\neq 0$). This is explained by the fact that, at finite detuning, the absorption of single photons is the dominant process already for $eV = 2\\Delta - \\hbar(\\omega+\\omega_0) < V^\\text{opt}$, while parametric interaction is dominant for $eV \\lesssim 2\\Delta - \\hbar(\\omega+\\omega_0)$. The response of the junction at finite detuning $\\omega$ is similar to the one presented in Fig.~\\ref{fig1}(b). To achieve parametric amplification at finite detuning, we set dc-voltage to a value smaller than $V^\\text{opt}$. In this case, there is a frequency range [$e(V-V^\\text{opt})\/\\hbar,e(V^\\text{opt}-V)\/\\hbar$] where the parametric interaction is the dominant mechanism, leading to phase-preserving amplification and two-mode squeezing at finite detuning. This frequency range defines the operational bandwidth of the device.\n\nFigure~\\ref{fig3}(a) illustrates the phase-preserving gain while panel (b) shows two-mode squeezing as a function of the detuning $(\\omega-\\omega_0)\/\\omega_0$ for three different values of $V$ and for fixed $eV_\\text{ac} = 0.1 \\times 2\\hbar \\omega_0$. These results show striking features. The first is that the bandwidth is controlled by the dc-voltage and, as anticipated, equal to $2 e(V^\\text{opt}- V)\/\\hbar$ for $2(\\Delta- \\hbar\\omega_0) < eV < 2\\Delta - \\hbar\\omega_0$. As expected, the largest bandwidth occurs for $eV \\approx 2\\Delta - 2\\hbar\\omega_0$. Both gain and squeezing spectra are flat over a 4 GHz band, with more than 10 dB of phase-preserving gain and -14 dB of squeezing. Moreover, the broadband squeezing spectrum characterizes the generation of far-separated two-mode squeezed states.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig3.pdf}\n\\caption{Aluminum junction ($\\Delta\/\\hbar\\omega_0 \\approx 10.9$): phase-preserving gain (a) and two-mode squeezing (b) as a function of the frequency detuning from the pump frequency $\\omega_0\/2\\pi = 4$ GHz. We consider a fixed ac-voltage amplitude $V_\\text{ac} = 0.1 \\times 2\\hbar\\omega_0$ for three different values of dc-voltages, $V \\approx 2\\Delta-1.2 \\hbar \\omega_0$ (cyan line), $V \\approx 2\\Delta-1.5 \\hbar \\omega_0$ (dark-blue line), and $V \\approx 2\\Delta- 2 \\hbar \\omega_0$ (black line). At $\\omega=\\omega_0$, the amplifier is operated in the phase-sensitive mode. \\label{fig3}}\n\\end{center}\n\\end{figure}\n\nAs mentioned before, the tunneling of quasiparticles also generates noise preventing the amplifier from operating at the quantum-limit. To further characterize the SIS junction parametric amplifier, we fix $eV \\approx 2(\\Delta- \\hbar\\omega_0)$ and investigate the impact of the ac-voltage amplitude on the gain and added noise. Figs.~\\ref{fig4}(a) and (b) illustrate the gain and added noise for three different values of $V_\\text{ac}$, respectively. The gain is observed to raise with increasing ac-voltage amplitude, while adding noise and decreasing the bandwidth. More importantly, gain can be significantly increased while the added noise remains of the same order. For instance, increasing $V_\\text{ac}$ from $0.11 \\times 2\\hbar \\omega_0$ to $0.135 \\times 2\\hbar \\omega_0$, the gain jumps from 14 dB to approximately 30 dB, while the added noise remains practically the same. The bandwidth is reduced with the increased gain, nonetheless it is of the same order of the measured bandwidth of the Josephson traveling wave parametric amplifier \\cite{macklin-siddiqi-science2015}. The increase of both gain and added noise is due to the enhancement of photon-assisted tunneling of quasiparticles, which controls both parametric amplification and added noise [three last terms of Eq.~\\eqref{addednoise1}]. However, even with the increase of the added noise with $V_\\text{ac}$, the SIS amplifier operates near the high-gain quantum limit of an ideal phase-preserving amplifier [dashed line in Fig.~\\ref{fig4}(b)] \\cite{clerk-schoelkopf-rmp2010}.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig4.pdf}\n\\caption{ Aluminum junction ($\\Delta\/\\hbar\\omega_0 \\approx 10.9$): Gain spectrum (a) and added noise power spectral density (b) as a function of the frequency detuning from $\\omega_0\/2\\pi = 4$ GHz for $V\\approx 2\\Delta-2\\hbar\\omega_0$ and three different values of ac-voltage amplitude, $V_\\text{ac} \\approx 0.08 \\times 2\\hbar \\omega_0$ (cyan line), $V_\\text{ac} \\approx 0.11 \\times 2\\hbar \\omega_0$ (blue line), and $V_\\text{ac} \\approx 0.135 \\times 2\\hbar \\omega_0$ (black line). The dashed line marks the added noise for an ideal phase-preserving amplifier. In this case, half-photon is added during the amplification process. As the ac-voltage amplitude increases the gain increases and the bandwidth is reduced. However, the added noise still near the quantum limit for all values of $V_\\text{ac}$. \\label{fig4}}\n\\end{center}\n\\end{figure}\n\nFinally, a higher degree of amplification and squeezing can be obtained by fabricating a junction with normal state resistance smaller than 50 $\\Omega$. However, due to the impedance mismatch between the junction and the TL, the bandwidth goes to zero as $R_\\text{T}$ diminishes. This can be understood by the fact that internal reflections of the microwave signal, caused by the large impedance mismatch, can interfere destructively, thereby destroying the coherent coupling between the signal ($\\an{in}[\\omega+\\omega_0]$) and the idler ($\\ad{in}[\\omega_0 - \\omega]$) frequency modes. Also, as shown in Fig.~\\ref{fig2}, a SIS junction with large superconducting gap leads to higher gain and squeezing. For instance, a Niobium junction ($\\Delta\/h \\sim 750$ GHz) would, in principle, generate higher gain and squeezing than an Aluminum junction. \n\nFinally, a higher degree of amplification and squeezing can be obtained by fabricating a junction with normal state resistance smaller than 50 $\\Omega$. However, due to the impedance mismatch between the junction and the TL, the bandwidth and gain goes to zero as $R_\\text{T}$ diminishes. Also, as shown in Fig.~\\ref{fig2}, a SIS junction with large superconducting gap leads to higher gain and squeezing. For instance, a Niobium junction ($\\Delta\/h \\sim 750$ GHz) would, in principle, generate higher gain and squeezing than an Aluminum junction. Moreover, the operational bandwidth could be extended to the THz range with NbSi or NbN junctions.\n\n\\subsection{Effects of low-frequency noise \\label{resultsB}}\n\nThe results presented in the previous section were obtained considering an ideal SIS junction in the low-temperature limit $k_\\text{B} T \\ll 2\\Delta$ for which the transport response rises steeply for $eV=2\\Delta+n\\hbar\\omega_0$ \\cite{barone-paterno-book}. However, this is an idealized situation and, in practice, temperature or low-frequency noise can smoothen the transport response. Here, we consider the effects of low-frequency noise on gain and squeezing properties of the SIS amplifier. These effects are included by assuming that the junction interact with a low-frequency electromagnetic environment \\cite{hofheinz-esteve-prl2011} and that the transport properties are described by the $P(E)$-theory \\cite{ingold-nazarov-book1992}. Indeed, this approach has been shown to quantitatively explain the finite Dynes tunneling density of states, usually observed below the dc-transport gap in a normal-insulator-superconductor junctions and the corresponding smoothing of the BCS coherence peak \\cite{pekola-tsai-prl2010,barone-paterno-book}. In this approach, the low-frequency noise modifies the equilibrium noise current noise to \n\\begin{equation} \\label{noiseeq-pe}\nS_\\text{eq}^\\text{eff}(\\omega) = \\int_{-\\infty}^{\\infty} S_\\text{eq}(\\hbar\\omega - E) P(E) dE,\n\\end{equation}\nwhere $P(E)$ is the probability density of a tunneling electron emitting energy and takes the form for low-frequency noise \\cite{ingold-nazarov-book1992}\n\\begin{equation}\nP(E) = \\frac{1}{\\sqrt{4\\pi E_c k_\\text{B} T}}\\exp \\left( - \\frac{(E-E_c)^2}{4 E_c k_\\text{B} T} \\right) ,\n\\end{equation}\nwith $E_c = e^2\/2C_{bt}$ is the capacitor charging energy. This model is justified by the fact that the low noise biasing scheme consists of a resistive voltage divider followed by large capacitive filtering \\cite{hofheinz-esteve-prl2011,chen-rimberg-prb2014}. The low-frequency impedance is thus the parallel combination of a resistance with a large capacitance $C_\\text{bt}$. This filtering scheme reduces the bandwidth over which low-frequency voltage noise can develop to the kHz range \\cite{chen-rimberg-prb2014}, making the low-frequency voltage fluctuations fully classical. With this model, the theory developed in Sec.~\\ref{model} remains the same except for the replacement of $S_\\text{eq}(\\omega)$ by $S_\\text{eq}^\\text{eff}(\\omega)$ in Eq.~\\eqref{noise-PE}.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig5.pdf}\n\\caption{Aluminum junction ($\\Delta\/\\hbar\\omega_0 \\approx 10.9$): Gain (a) and squeezing (b) as a function of frequency detuning in the presence (solid lines) and in the absence (dashed line) of low-frequency electromagnetic environment. The two solid lines illustrate an efficient ($C_\\text{bt}\\sim 1$~nF - blue line) and inefficient ($C_\\text{bt}\\sim 10$~pF - cyan line) low-frequency noise filtering scheme. As expected, for an efficient filtering scheme, both gain and squeezing are quantitatively equal to the result obtained in absence of low-frequency electromagnetic environment. \\label{fig5}}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{fig5} illustrates the resulting effect on both (a) gain and (b) squeezing for $C_\\text{bt}\\sim 1$~nF (dark-blue line) and $C_\\text{bt}\\sim 10$~pF (light-blue line) corresponding to an efficient and inefficient low-frequency noise filtering scheme, respectively. These results are compared with the ideal case (dashed-line), where the junction does not interact with the low-frequency environment. For an Aluminum junction ($\\Delta\/\\hbar\\omega_0 \\approx 10.9$) and $T=15$ mK, $C_\\text{bt}\\sim 1$~nF corresponds to rms voltage fluctuations of $\\sqrt{k_\\text{B}T\/2C_\\text{bt}} \\sim 11$~nV (dark-blue line), the gain and squeezing are only weakly affected by the low-frequency environment which is efficiently filtered. The effect of low-frequency noise on $S_\\text{eq}^\\text{eff}(\\omega)$ is illustrated in the inset of Fig.~\\ref{fig5}. For $C_\\text{bt} \\sim 1$~nF, $S_\\text{eq}^\\text{eff}(\\omega)$ is almost indistinguishable from the ideal case (dashed line), a signature that the low-frequency noise is efficiently filtered. On the other hand, under less efficient filtering, $C_{bt} \\sim 10$~pF corresponding to $\\sim 0.1~\\mu$V rms voltage fluctuations (light-blue line), the equilibrium current noise rises smoothly and its behavior near $2\\Delta\/\\hbar$ deviates from the ideal case (see inset). In this case, low-frequency noise diminishes both gain and squeezing amplitudes (light-blue line). However, the bandwidth is only weakly modified.\n\n\\section{Conclusions \\label{conclusions}}\n\nWe have proposed a near-quantum-limited broadband amplifier and squeezer based on the photon-assisted tunneling of quasiparticles in a SIS junction. This device can function as a phase-sensitive or phase-preserving amplifier. The gain can be tuned by the dc- and ac-voltages amplitude to reach 30 dB over 2 GHz or 14 dB over 4 GHz. This device is also a source of two-mode squeezing with bandwidth over more than 4 GHz and $-14$ dB of squeezing. Moreover, gain and two-mode squeezing can be fine -tuned in-situ by changing the pumping tone frequency and dc-voltage bias. Therefore, the design and fabrication simplicity of this SIS amplifier, together with its operational mode flexibility, makes it a versatile near-quantum-limited microwave amplifier and squeezer, which can be easily integrated in many quantum microwaves experiments. \n\n\\section*{Acknowledgments}\n\nUCM thank S. Boutin, A. L. Grimsmo and M. Westig for fruitful discussion, and the Quantronics group for hospitality. UCM and AB were supported from Canada First Research Excellence Fund and NSERC. BR was supported by Canada Excellence Research Chairs, Government of Canada, Natural Sciences and Engineering Research Council of Canada, Qu\\'ebec MEIE, Qu\\'ebec FRQNT via INTRIQ, Universit\\'e de Sherbrooke via EPIQ, and Canada Foundation for Innovation. The research at CEA-Saclay has received funding from the European Research Council under the European Union's Programme for Research and Innovation (Horizon 2020)\/ERC Grant Agreement No. 639039, and support from the ANR AnPhoTeQ research contract.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWe introduce the rosette harmonic mappings, analogous to the $n$-cusped\nhypocycloid mappings. For each integer $n$ where $n\\geq3,$ we obtain a family\nof mappings, that in many instances have only cyclic rather than dihedral\nsymmetry. The mappings have $n$ cusps, or in some cases $n$ nodes rather than\ncusps. It is interesting to consider one particular mapping for each $n$ in\nwhich the boundary of the mapping is continuous, \\ but with arcs of constancy.\nOur main goal is to establish the univalence of the rosette harmonic mappings.\nAdditionally we describe the location and orientation of cusps and nodes. We\nalso define a fundamental set from which the full graph of a rosette mapping\ncan be reconstructed, which is useful for computational efficiency.\n\nWe begin by establishing some notation and standard terminology associated\nwith planar harmonic mappings. A \\textbf{harmonic mapping} $f$ is a complex\nvalued univalent harmonic function defined on a region in the complex plane $%\n\\mathbb{C}\n$. Harmonic mappings can be arrived at in a variety of ways, for example by\nadding different harmonic functions together, or by using the Poisson integral\nformula, and more recently by using the shear construction, first described in\n\\cite{ClunieSheilSmall}. Univalence is not guaranteed however, except for in\nthe latter approach. For any harmonic mapping $f,$ we write $f=h+\\bar{g}$\nwhere $h$ and $g$ are analytic, and call $h$ and $g$ the \\textbf{analytic} and\n\\textbf{co-analytic} parts of $f$, respectively. The decomposition is unique\nup to the constant terms of $h$ and $g,$ and $h+\\bar{g}$ is known as the\n\\textbf{canonical decomposition }of $f$\\textbf{. \\ }Our mappings are defined\non $U,$ the open unit disk in the complex plane. The Jacobian $J_{f}$ of $f$\nis given by $J_{f}\\left( z\\right) =\\left\\vert f_{z}\\left( z\\right)\n\\right\\vert ^{2}-\\left\\vert f_{\\bar{z}}\\left( z\\right) \\right\\vert\n^{2}=\\left\\vert h^{\\prime}\\left( z\\right) \\right\\vert ^{2}-\\left\\vert\ng^{\\prime}\\left( z\\right) \\right\\vert ^{2}$. We say that $f$ is\n\\textbf{sense-preserving} if $J_{f}\\left( z\\right) >0$ in $U.$ A theorem of\nL\\v{e}wy \\cite{Lewy} states that a harmonic function $f$ is locally one-to-one\nif $J_{f}$ is non-vanishing in $U$. Thus $f$ is locally one-to-one and\nsense-preserving if and only if $\\left\\vert g\\right\\vert <\\left\\vert\nh\\right\\vert $ and thus, there exists a meromorphic function known as the\n\\textbf{analytic dilatation} of $f,$ given by $\\omega_{f}\\left( z\\right)\n=g^{\\prime}\/h^{\\prime}.$ Note that the analytic dilatation is related to the\n\\textbf{complex dilatation }$\\mu_{f}=\\bar{g}^{\\prime}\/h^{\\prime}$ from the\ntheory of quasiconformal mappings. We refer here to $\\omega_{f}\\left(\nz\\right) $ simply as the \\textbf{dilatation} of $f $ - for more information\nsee \\cite{PeterHMBook} and \\cite{ECA}.\n\nThe rosette harmonic mappings of Definition \\ref{defnharmonic} are\nmodifications of a simple harmonic mapping known as the hypocycloid harmonic\nmapping, with image under the unit disk bounded by a $n$-cusped hypocycloid.\nThe symbol $\\partial A$ here denotes the topological boundary of the set $A$.\n\n\\begin{example}\n\\label{hypo}Let $n\\in%\n\\mathbb{N}\n,$ $n\\geq3.$ The \\textbf{hypocycloid harmonic mapping} is defined on $U$ by\n$f_{hyp}\\left( z\\right) =z+\\frac{1}{n-1}\\bar{z}^{n-1}$ with analytic and\nco-analytic parts $z$ and $\\frac{1}{n-1}z^{n-1}$. The dilatation is\n$\\omega_{f}\\left( z\\right) =z^{n-2}$ and $J_{f}\\left( z\\right)\n=1^{2}-\\left\\vert \\bar{z}^{n-2}\\right\\vert ^{2}=1-\\left\\vert z^{2n-4}%\n\\right\\vert .$ Clearly $J_{f}\\left( z\\right) >0$ in $U$ so $f$ is locally\none to one$.$ It is also univalent on $U$ (see for instance \\cite{PeterHMBook}\nor \\cite{ECA}). Upon extension to $\\bar{U},$ we can consider the boundary\ncurve $f\\left( e^{it}\\right) ,$ which for $n=4\\ $is the familiar astroid\ncurve from calculus. We consider the boundary extension $f_{hyp}\\left(\ne^{it}\\right) =e^{it}+\\frac{1}{n-1}e^{-i\\left( n-1\\right) t}$ which has\nsingular points (where the derivative is $0$ ) precisely when $t=2k\\pi\/n$ ,\n$k=1,2,...,n$. The left of Figure \\ref{rosette0} shows the image of $\\bar{U}$\nunder $f_{hyp} $ for $n=6$, where $f_{hyp}$ maps the unit circle $\\partial U$\nonto a $6$-cusped hypocycloid.\n\\end{example}\n\n\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[\nheight=2.3514in,\nwidth=5.028in\n]%\n{rosette0.eps}%\n\\caption{Images of a regular polar grid in $U$ under the 6-cusped hypocycloid\n(left) and a 6-cusped rosette mapping (right) defined by $f_{0}\\left(\nz\\right) =z\\,_{2}F_{1}\\left( \\frac{1}{2},\\frac{1}{12};\\frac{13}{12}%\n;z^{12}\\right) +\\frac{1}{5}\\bar{z}^{5}\\,\\overline{_{2}F_{1}\\left( \\frac\n{1}{2},\\frac{5}{12};\\frac{17}{12};z^{12}\\right) }$}%\n\\label{rosette0}%\n\\end{figure}\n\n\nThe rosette harmonic mappings introduced here can be viewed as modifications\nof the hypocycloid mappings, and are formulated by incorporating Gauss\nhypergeometric $_{2}F_{1}\\,\\ $factors into the analytic and co-analytic parts.\nThe rosette mappings $f_{\\beta}$ will be defined in Section 3, but an example\nof a 6-cusped rosette mapping appears on the right of Figure \\ref{rosette0}.\nIn comparison with cusps of the 6-cusped hypocycloid, the rosette has cusps\nthat are more \"pointy\". Figure \\ref{five} indicates further examples of\nrosette mappings for $n=6$ in which the images of the unit disk may have\nrotational but not reflectional symmetry. The process by which we obtain\nrosette mappings with essentially different features is by rotating the\nanalytic and co-analytic parts relative to one another.\n\nThis article is organized as follows: In Section 2, the properties of the\n$_{2}F_{1}$ \\linebreak hypergeometric functions utilized in the definition of\nthe rosette harmonic mappings are described. In Section 3, the rosette\nharmonic mappings are defined and their rotational and reflective symmetries\nare explored. In Section 4 we describe the boundary and arcs and the nodes and\ncusps seen there. In particular, we highlight a remarkable situation in which\nthe analytic and co-analytic parts \"cancel\" on $\\partial U,$ leading to arcs\nof constancy in the extension to the boundary. This has significant\nconsequences for the associated minimal surface that lifts from this map, as\ndescribed in \\cite{AbdullahMcDougall}. In Section 5 we use the argument\nprinciple for harmonic functions to show that the rosette harmonic mappings\nare in fact univalent. We also describe a computationally efficient way to\nconstruct the image of the unit disk by using rotations of a smaller\n\"fundamental set\".\n\n\\section{Hypergeometric functions\\label{hyper}}\n\nWe begin by defining two Gauss hypergeometric $_{2}F_{1}$ functions.\n\n\\begin{definition}\n\\label{hyperDef}Let $U$ denote the unit disk. For $n\\geq2$ and $z\\in\\bar{U},$\nconsider the Gauss $_{2}F_{1}$ hypergeometric functions\n\\begin{align*}\nH_{n}\\left( z\\right) & =\\,_{2}F_{1}\\left( \\frac{1}{2},\\frac{1}%\n{2n},1+\\frac{1}{2n},z\\right) \\\\\nG_{n}\\left( z\\right) & =\\,_{2}F_{1}\\left( \\frac{1}{2},\\frac{1}{2}%\n-\\frac{1}{2n},\\frac{3}{2}-\\frac{1}{2n},z\\right)\n\\end{align*}\nfor $n\\geq2.$ Note that when $n=2$ that $G_{2}=H_{2}$.\n\\end{definition}\n\nBy the Corollary in \\cite{MerkesScott}, $H_{n}$ and $G_{n}$ map the unit disk\nonto a convex region. Since $H_{n}\\left( 0\\right) =G_{n}\\left( 0\\right)\n=1,$ the convex regions $H_{n}\\left( U\\right) $ and $G_{n}\\left( U\\right)\n$ contain $1$. Thus $H_{n}$ and $G_{n}$ can be considered to be perturbations\nof the constant function of unit value$.$ In Figure \\ref{pert} where $n=6$,\none can see the distortion from unity in $H_{6}\\left( z\\right) $ and\n$G_{6}\\left( z\\right) .$ The reflectional symmetry in the real axis is also\napparent. These and further properties of the hypergeometric functions $H_{n}\n$ and $G_{n}$ are stated in Proposition \\ref{hyperProps}.%\n\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[\nheight=1.3699in,\nwidth=3.9297in\n]%\n{pert.eps}%\n\\caption{Images of a polar grid in $U$ under $H_{6}$ and $G_{6}.$}%\n\\label{pert}%\n\\end{figure}\n\n\n\\begin{proposition}\n\\label{hyperProps} Let $n\\geq3.$ \\newline(i) The Taylor coefficients $c_{m}$\nof $H_{n}\\left( z\\right) \\,$and $d_{m}$ of $G_{n}\\left( z\\right) $ are\ngiven by the formula\n\\[\nc_{m}=A_{m}\\frac{1}{2mn+1}\\text{ and }d_{m}=A_{m}\\frac{n-1}{n\\left(\n2m+1\\right) -1},\\text{ }m\\geq0,\n\\]\nwhere $A_{0}=1,$ and $A_{m}=\\left(\n\\begin{array}\n[c]{c}%\n2m-1\\\\\nm-1\n\\end{array}\n\\right) \/2^{2m-1},$ for $m\\geq1.$\\newline(ii) The hypergeometric functions\n$H_{n}$ and $G_{n}$ have reflectional symmetry\n\\[\nH_{n}\\left( \\bar{z}\\right) =\\overline{H_{n}\\left( z\\right) }\\text{ and\n}G_{n}\\left( \\bar{z}\\right) =\\overline{G_{n}\\left( z\\right) }.\n\\]\n(iii) Both $H_{n}\\ $and $G_{n}$ are univalent mappings of the open unit disk\nonto a bounded convex region in the right half plane. Moreover, both series\nconverge absolutely on the closed unit disk.\\newline(iv) Define $K_{n}%\n=H_{n}\\left( 1\\right) $. Both $H_{n}\\ $and $G_{n}$ are positive for\n$x\\in\\left[ -1,1\\right] ,$ with the interval $\\left[ H_{n}\\left(\n-1\\right) ,H_{n}\\left( 1\\right) \\right] $ strictly contained in $\\left[\nG_{n}\\left( -1\\right) ,G_{n}\\left( 1\\right) \\right] ,$ and the values at\n$1$ are as follows:\n\\begin{align}\nH_{n}\\left( 1\\right) & =K_{n}=\\sqrt{\\pi}\\frac{\\Gamma\\left( 1+\\frac{1}%\n{2n}\\right) }{\\Gamma\\left( \\frac{1}{2}+\\frac{1}{2n}\\right) },\\label{Kn}\\\\\nG_{n}\\left( 1\\right) & =\\left( n-1\\right) \\tan\\left( \\frac{\\pi}%\n{2n}\\right) K_{n}=\\sqrt{\\pi}\\frac{\\Gamma\\left( \\frac{3}{2}-\\frac{1}%\n{2n}\\right) }{\\Gamma\\left( 1-\\frac{1}{2n}\\right) }.\\nonumber\n\\end{align}\nMoreover,\n\\[\n5\/60$\nthe function values are positive, and increasing with $x$. Clearly\n$H_{n}\\left( 1\\right) >1>H_{n}\\left( -1\\right) ,$ because the $c_{m}$ are\npositive with $c_{0}=1,$ and for $x<0,$ we obtain an alternating series. For\n$G_{n},$ we see that $d_{0}-d_{1}d_{0}-d_{1}=1-\\frac{1}{2}\\frac{n-1}{3n-1}>1-\\frac\n{1}{6}=5\/6.\n\\]\nTo see that $G_{n}\\left( -1\\right) 2$. Thus\n$H_{n}\\left( 1\\right) 0$ (Section 29 of \\cite{Rainville}); here\n$\\operatorname{Re}\\left( c-a-b\\right) =1\/2\\,\\,$for both $G_{n}$ and $H_{n}$.\nBecause both $G_{n}$ and $H_{n}$ satisfy (ii) of the Corollary in\n\\cite{MerkesScott}, $_{2}F_{1}\\left( a,b,c,z\\right) $ maps the open disk\nunivalently onto a convex region. Because of positive Taylor coefficients, we\nhave $G_{n}\\left( -1\\right) <\\operatorname{Re}G_{n}\\left( z\\right)\n0$ in formula (\\ref{ga}).\n\\end{proof}\n\nWe now define the rosette harmonic mappings.\n\n\\begin{definition}\n\\label{defnharmonic}Let $n\\in%\n\\mathbb{N}\n$ with $n\\geq3,$ and let $h_{n}$ and $g_{n}$ be defined as in Definition\n\\ref{defnhg}. For each $\\beta\\in%\n\\mathbb{R}\n,$ define the \\textbf{rosette harmonic mapping}%\n\\[\nf_{\\beta}\\left( z\\right) =e^{i\\beta\/2}h_{n}\\left( z\\right) +e^{-i\\beta\n\/2}\\overline{g_{n}\\left( z\\right) },\\text{ }z\\in\\bar{U}.\n\\]\nThen $f_{\\beta}\\left( z\\right) $ is harmonic on $U,$ and continuous on\n$\\bar{U}.$ Denote the dilatation of $f_{\\beta}\\left( z\\right) $ by\n$\\omega\\left( z\\right) $ on $\\bar{U}\\backslash\\left\\{ \\zeta^{j}%\n:j=1,2,...,2n\\right\\} ,$ where $\\zeta$ is a primitive 2nth root of unity.\n\\end{definition}\n\n\\begin{remark}\nFor simplicity of the notation, we do not notate the value of $n,$ which is\napparent from context, and fixed as a constant in our discussions$.$ The\ndilatation is independent of $\\beta$ and we simplify our notation to $\\omega$\ninstead of using $\\omega_{f_{\\beta}}.$\n\\end{remark}\n\nThe derivatives of the analytic and co-analytic parts (for $\\beta=0$) in\nequation (\\ref{derivatives}) are the same as the corresponding derivatives for\nthe hypocycloid, except for the radical factor. This factor affects the\nargument of the summands in the canonical decomposition, but Corollary\n\\ref{rayshg}\\ shows that along rays $\\left\\{ re^{ij\\pi\/n}:r>0\\right\\} $\nwhere $j\\in%\n\\mathbb{Z}\n,$ that $h_{n}$ and $\\overline{g_{n}}$ are collinear. Moreover Corollary\n\\ref{rayshg} shows that on these same rays, that $h_{n}$ and $\\bar{g}_{n}$\nhave the same arguments as their analytic and anti-analytic counterparts in\nthe hypocycloid mapping. Thus $f_{0}\\left( z\\right) $ and $f_{hyp}\\left(\nz\\right) $ are collinear along these radial lines (as indicated in Figure\n\\ref{rosette0}). Figure \\ref{five} shows images $f_{\\beta}\\left( U\\right) $\nas $\\beta$ varies$,$ and suggests that different harmonic mappings $f_{\\beta}$\nare obtained for the four different values of $\\beta$ there. This contrasts\nwith the hypocycloid mapping, where rotating the analytic and anti-analytic\nparts $z$ and $\\frac{1}{n-1}\\bar{z}^{n-1}$ relative to one another does not\nyield a graph that is essentially different.\n\nOne may ask what other maps could be obtained through adding different\nrotations of $h_{n}$ and $\\overline{g_{n}}$. The next proposition shows that\nif we consider arbitrary rotations of the analytic and co-analytic parts, by\n$\\theta$ and $\\tilde{\\theta}$ say, then the result will in fact be a rotation\nof a rosette harmonic mapping.\n\n\\begin{proposition}\nLet $\\theta$ and $\\tilde{\\theta}$ be arbitrary real angles. Then $e^{i\\theta\n}h_{n}\\left( z\\right) +\\overline{e^{i\\tilde{\\theta}}g_{n}\\left( z\\right)\n}$ is a rotation $e^{i\\gamma}f_{\\beta}$ for some real angles $\\gamma$ and\n$\\beta.$\n\\end{proposition}\n\n\\begin{proof}\nNote that the harmonic function $e^{i\\theta}h_{n}\\left( z\\right)\n+\\overline{e^{i\\tilde{\\theta}}g_{n}\\left( z\\right) }$ can be rewritten%\n\\begin{align*}\n& e^{-i\\tilde{\\theta}}\\left( e^{i\\theta}e^{i\\tilde{\\theta}}h_{n}\\left(\nz\\right) +\\overline{g_{n}\\left( z\\right) }\\right) \\\\\n& =e^{-i\\tilde{\\theta}}e^{i\\left( \\theta+\\tilde{\\theta}\\right) \/2}\\left(\ne^{i\\left( \\theta+\\tilde{\\theta}\\right) \/2}h_{n}\\left( z\\right)\n+e^{-i\\left( \\theta+\\tilde{\\theta}\\right) \/2}\\overline{g_{n}\\left(\nz\\right) }\\right) \\\\\n& =e^{i\\left( \\theta-\\tilde{\\theta}\\right) \/2}f_{\\left( \\theta\n+\\tilde{\\theta}\\right) \/2}\\left( z\\right)\n\\end{align*}\nThus $e^{i\\theta}h_{n}\\left( z\\right) +\\overline{e^{i\\tilde{\\theta}}%\ng_{n}\\left( z\\right) }$ can be obtained by a rotation by $\\gamma=\\left(\n\\theta-\\tilde{\\theta}\\right) \/2$ of the map $f_{\\beta}$ where $\\beta=\\left(\n\\theta+\\tilde{\\theta}\\right) \/2.$\n\\end{proof}\n\nThus the family of mappings $\\left\\{ f_{\\beta}:\\beta\\in%\n\\mathbb{R}\n\\right\\} $ represents all of the different mappings, up to rotation, that\narise from arbitrary rotations of $h_{n}$ and $g_{n}$. We show in Proposition\n\\ref{parametrize} that $f_{\\beta+\\pi}$ is essentially the same mapping as\n$f_{\\beta}.$\n\n\\begin{proposition}\n\\label{parametrize}(i) For any \\ $\\beta\\in%\n\\mathbb{R}\n,$ the functions $f_{\\beta}$ and $f_{\\beta+\\pi}$ are related by\n\\begin{equation}\nf_{\\beta}\\left( z\\right) =e^{-i\\left( \\frac{\\pi}{2}+\\frac{\\pi}{n}\\right)\n}f_{\\beta+\\pi}\\left( e^{i\\frac{\\pi}{n}}z\\right) . \\label{rotbet}%\n\\end{equation}\nThus the image $f_{\\beta+\\pi}\\left( U\\right) $ is equal to a rotation of the\nimage $f_{\\beta}\\left( U\\right) $.\\newline(ii) If $\\beta<0,$ then the image\nof any point under $f_{\\beta}$ is a reflection in the real axis of the same\npoint under $f_{-\\beta},$ where $-\\beta>0.$ \\ Specifically, $f_{\\beta}\\left(\n\\bar{z}\\right) =\\overline{f_{-\\beta}\\left( z\\right) }.$\n\\end{proposition}\n\n\\begin{proof}\n(i) We compute, using $j=1$ in equation (\\ref{ga}),%\n\\[\nf_{\\pi+\\beta}\\left( e^{i\\frac{\\pi}{n}}z\\right) =e^{i\\left( \\pi\n\/2+\\beta\/2\\right) }e^{i\\pi\/n}h_{n}\\left( z\\right) +e^{-i\\left( \\pi\n\/2+\\beta\/2\\right) }\\overline{\\left( -1\\right) e^{-i\\pi\/n}g_{n}\\left(\nz\\right) }.\n\\]\nMultiplying by $e^{-i\\left( \\frac{\\pi}{2}+\\frac{\\pi}{n}\\right) },$ and\nnoting that $-e^{-i\\frac{\\pi}{2}}=e^{i\\frac{\\pi}{2}},$ $\\ $%\n\\[\ne^{-i\\left( \\frac{\\pi}{2}+\\frac{\\pi}{n}\\right) }f_{\\pi+\\beta}\\left(\ne^{i\\frac{\\pi}{n}}z\\right) =e^{i\\beta\/2}h_{n}\\left( z\\right) +e^{-i\\beta\n\/2}g_{n}\\left( z\\right) =f_{\\beta}\\left( z\\right) .\n\\]\n(ii) Once again, by computation:\n\\[\nf_{\\beta}\\left( \\bar{z}\\right) =e^{i\\frac{\\beta}{2}}h_{n}\\left( \\bar\n{z}\\right) +e^{-i\\frac{\\beta}{2}}\\overline{g_{n}\\left( \\bar{z}\\right)\n}=e^{i\\frac{\\beta}{2}}\\overline{h_{n}\\left( z\\right) }+e^{-i\\frac{\\beta}{2}%\n}g_{n}\\left( z\\right)\n\\]\nwhile\n\\[\nf_{-\\beta}\\left( z\\right) =e^{-i\\frac{\\beta}{2}}h_{n}\\left( z\\right)\n+e^{i\\frac{\\beta}{2}}\\overline{g_{n}\\left( z\\right) }.\n\\]\nThe last two expressions are conjugates of one another. $\\blacksquare$\n\\end{proof}\n\n\\begin{corollary}\n\\label{transit}Let $\\tilde{\\beta}\\in%\n\\mathbb{R}\n,$ and let $\\tilde{\\beta}=\\beta+l\\pi$ for some $l\\in%\n\\mathbb{Z}\n.$ Then%\n\\begin{equation}\nf_{\\beta+l\\pi}\\left( z\\right) =e^{il\\left( \\pi\/n+\\pi\/2\\right) }f_{\\beta\n}\\left( e^{-il\\pi\/n}z\\right) . \\label{trans}%\n\\end{equation}\n\n\\end{corollary}\n\n\\begin{proof}\nSolving equation (\\ref{rotbet}) we have $f_{\\beta+\\pi}\\left( z\\right)\n=e^{i\\left( \\pi\/2+\\pi\/n\\right) }f_{\\beta}\\left( e^{-i\\pi\/n}z\\right) .$\nEquation (\\ref{trans}) is obtained by repeated application of this equation.\n\\end{proof}\n\nProposition \\ref{parametrize} (i) shows that up to rotations (pre and post\ncomposed), all rosette mappings are represented in the set $\\left\\{ f_{\\beta\n}:\\beta\\in(-\\pi\/2,\\pi\/2]\\right\\} .$ We will see in Section 4 that these\nrosette mappings are all distinct from one another in that no rosette mapping\nin the set can be obtained by rotations from another. Proposition\n\\ref{parametrize} (ii) allows us to consider just $\\left\\{ f_{\\beta}:\\beta\n\\in\\left[ 0,\\pi\/2\\right] \\right\\} $ to obtain all rosette mappings, up to\nrotations and reflection$.$\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[\nheight=4.3811in,\nwidth=4.3811in\n]%\n{five.eps}%\n\\caption{Images of $U$ under $f_{\\beta}$ with $n=5.$ The darker \"radial curve\"\nindicates the image of $\\left[ 0,1\\right] .$ The graphs $f_{\\pi\/3}$ and\n$f_{-\\pi\/3}$ illustrate Proposition \\ref{parametrize} (ii). The images\n$f_{-\\pi\/3}\\left( U\\right) $ and $f_{\\pi\/3}\\left( U\\right) $ have cyclic\nsymmetry while $f_{0}\\left( U\\right) $ and $f_{\\pi\/2}\\left( U\\right) $\nexhibit dihedral symmetry, illustrating Theorem \\ref{symmetry}. For $f_{\\pi\n\/2}$ the line of symmetry $\\arg z=\\pi\/4-\\pi\/\\left( 2n\\right) $ is\nindicated.}%\n\\label{five}%\n\\end{figure}\n\n\nFigure \\ref{five} illustrates that $f_{-\\pi\/3}\\left( U\\right) $ and\n$f_{\\pi\/3}\\left( U\\right) $ are reflections of one another. The example\ngraphs in Figure \\ref{five} also demonstrate the rotational and reflectional\nsymmetries apparent within any particular graph $f_{\\beta}\\left( U\\right) ,$\nas stated in the following theorem$.$\n\n\\begin{theorem}\n\\label{symmetry}Let $n\\in%\n\\mathbb{N}\n$ and $n\\geq3.$ \\newline(i) The harmonic functions $f_{\\beta}\\left( z\\right)\n$, $\\beta\\in%\n\\mathbb{R}\n$ have dilatation $\\omega\\left( z\\right) =z^{n-2}$ for $z\\in U.$\\newline(ii)\nThe rosette mapping $f_{\\beta}\\left( z\\right) $, $\\beta\\in%\n\\mathbb{R}\n$ has n-fold rotational symmetry, that is\n\\begin{equation}\nf_{\\beta}\\left( e^{i2k\\pi\/n}z\\right) =e^{i2k\\pi\/n}f_{\\beta}\\left( z\\right)\n,\\text{ where }z\\in\\bar{U},\\text{ }k\\in%\n\\mathbb{Z}\n. \\label{RS}%\n\\end{equation}\n\\newline(iii) If $\\beta$ is an integer multiple of $\\pi\/2,$ then the image\n$f_{\\beta}\\left( U\\right) $ has reflectional symmetry$.$ For $\\beta=0$ and\n$\\beta=\\pi\/2$ the reflections are%\n\\begin{equation}\nf_{0}\\left( \\overline{z}\\right) =\\overline{f_{0}\\left( z\\right) }\\text{\nand }e^{i\\eta}f_{\\pi\/2}\\left( e^{i\\gamma}\\overline{z}\\right) =\\overline\n{e^{i\\eta}f_{\\pi\/2}\\left( e^{i\\gamma}z\\right) }, \\label{reflect}%\n\\end{equation}\nwhere $\\eta=\\pi\/\\left( 2n\\right) -\\pi\/4$ and $\\gamma=-\\pi\/\\left( 2n\\right)\n.$ Thus if $z$ and $z^{\\prime}$ are reflections in $\\arg z=\\gamma,$ then\n$f_{\\pi\/2}\\left( z\\right) $ and $f_{\\pi\/2}\\left( z^{\\prime}\\right) $ are\nreflections in $\\arg z=\\eta$. \\newline\n\\end{theorem}\n\n\\begin{proof}\n(i) We compute $\\omega\\left( z\\right) =$ $g_{n}^{\\prime}\\left( z\\right)\n\/h_{n}^{\\prime}\\left( z\\right) $ from the derivative expressions in\n(\\ref{derivatives}), noting that the constant $e^{i\\beta\/2},$ and the\nradicals, cancel leaving $z^{n-2}.$ \\newline(ii) From equation (\\ref{ga}) with\n$j=2k$ we have $h_{n}\\left( e^{i2k\\pi\/n}z\\right) =e^{i2k\\pi\/n}h_{n}\\left(\nz\\right) $ and $\\overline{g_{n}\\left( e^{i2k\\pi\/n}z\\right) }=e^{i2k\\pi\n\/n}\\overline{g_{n}\\left( z\\right) }$. Thus%\n\\[\nf_{\\beta}\\left( e^{i2k\\pi\/n}z\\right) =e^{i2k\\pi\/n}\\left( e^{i\\beta\/2}%\nh_{n}\\left( z\\right) +e^{-i\\beta\/2}\\overline{g_{n}\\left( z\\right)\n}\\right) =e^{i2k\\pi\/n}f_{\\beta}\\left( z\\right) \\text{.}%\n\\]\n\\allowbreak(iii) From Proposition \\ref{analco}, $f_{0}(\\overline{z}%\n)=h_{n}(\\bar{z})+\\overline{g_{n}\\left( \\bar{z}\\right) }=\\overline{h_{n}%\n(z)}+g_{n}\\left( z\\right) =\\overline{f_{0}\\left( z\\right) }.$ For\n$f_{\\pi\/2},$ consider the expressions $f_{\\pi\/2}\\left( e^{i\\gamma}%\n\\overline{z}\\right) $ and $f_{\\pi\/2}\\left( e^{i\\gamma}z\\right) $:\\newline%\n\\begin{align*}\nf_{\\pi\/2}\\left( e^{i\\gamma}\\overline{z}\\right) & =e^{i\\pi\/4}h_{n}\\left(\ne^{-i\\pi\/\\left( 2n\\right) }\\bar{z}\\right) +e^{-i\\pi\/4}\\overline\n{g_{n}\\left( e^{-i\\pi\/\\left( 2n\\right) }\\bar{z}\\right) }\\\\\n& =e^{i\\pi\/4}\\overline{h_{n}\\left( e^{i\\pi\/\\left( 2n\\right) }z\\right)\n}+e^{-i\\pi\/4}g_{n}\\left( e^{i\\pi\/\\left( 2n\\right) }z\\right) \\\\\n& =e^{i\\pi\/4}\\overline{e^{i\\pi\/\\left( 2n\\right) }zH\\left( -z^{2n}\\right)\n}+e^{-i\\pi\/4}ie^{-i\\pi\/\\left( 2n\\right) }\\frac{z^{n-1}G\\left(\n-z^{2n}\\right) }{n-1}\\\\\n& =e^{i\\left( \\pi\/4-\\pi\/\\left( 2n\\right) \\right) }\\left( \\overline\n{zH\\left( -z^{2n}\\right) }+\\frac{z^{n-1}G\\left( -z^{2n}\\right) }%\n{n-1}\\right) .\n\\end{align*}\nNote that $\\left( e^{i\\pi\/\\left( 2n\\right) }\\right) ^{n-1}=ie^{-i\\pi\n\/\\left( 2n\\right) }$ and $\\left( e^{-i\\pi\/\\left( 2n\\right) }z\\right)\n^{2n}=-z^{2n},$ as used in the calculation above$.$ We also use $\\left(\ne^{-i\\pi\/\\left( 2n\\right) }\\right) ^{n-1}=-ie^{i\\pi\/\\left( 2n\\right) }%\n\\ $below:\\newline%\n\\begin{align*}\nf_{\\pi\/2}\\left( e^{i\\gamma}z\\right) & =e^{i\\pi\/4}h_{n}\\left(\ne^{-i\\frac{\\pi}{2n}}z\\right) +e^{-i\\pi\/4}\\overline{g_{n}\\left(\ne^{-i\\frac{\\pi}{2n}}z\\right) }\\\\\n& =e^{i\\pi\/4}\\left( e^{-i\\frac{\\pi}{2n}}z\\right) H\\left( e^{-i\\pi}%\nz^{2n}\\right) +e^{-i\\pi\/4}\\overline{\\left( -i\\right) e^{i\\frac{\\pi}{2n}%\n}\\frac{z^{n-1}}{n-1}G\\left( -z^{2n}\\right) }\\\\\n& =e^{i\\left( \\pi\/4-\\pi\/\\left( 2n\\right) \\right) }\\left( zH\\left(\n-z^{2n}\\right) +\\overline{\\frac{z^{n-1}}{n-1}G\\left( -z^{2n}\\right)\n}\\right) .\n\\end{align*}\nMultiplying each of $f_{\\pi\/2}\\left( e^{i\\gamma}\\overline{z}\\right) $ and\n$f_{\\pi\/2}\\left( e^{i\\gamma}z\\right) $ by $e^{i\\eta}=e^{i\\left( \\pi\n\/4-\\pi\/\\left( 2n\\right) \\right) },$ we see that $e^{i\\eta}$ $f_{\\pi\n\/2}\\left( e^{i\\gamma}\\overline{z}\\right) $ and $e^{i\\eta}f_{\\pi\/2}\\left(\ne^{i\\gamma}z\\right) $ are conjugates of one another. $\\ $Thus the stated\nequations in (iii) hold.\n\\end{proof}\n\nIn Section 4 we use features of the boundary $\\partial f_{\\beta}\\left(\nU\\right) $ that allow us to demonstrate the precise symmetry group for each\ngraph $f_{\\beta}\\left( U\\right) $, $\\beta\\in%\n\\mathbb{R}\n$ (see Corollary \\ref{noreflect}). Theorem \\ref{symmetry} also shows that for\na given $n\\geq3$ the rosette mappings $f_{\\beta}$ and $n$-cusped hypocycloid\nall have the same dilatation. The equal dilatations result in similarities in\nthe tangents of the rosette and hypocycloid boundary curves, to be discussed\nfurther in Section 4.\n\nWe finish this section by laying out geometric features that are specific to\nrosette mappings for $\\beta$ in the interval $\\beta\\in(-\\pi\/2,\\pi\/2].$ These\nfacts in combination with equation (\\ref{trans}) will allow us to extend our\nconclusions for any $\\beta\\in%\n\\mathbb{R}\n.$\n\n\\begin{lemma}\n\\label{convexconcave} Let $n\\in%\n\\mathbb{N}\n,$ $n\\geq3,$ and recall $K_{n}=\\sqrt{\\pi}\\,\\Gamma\\left( 1+\\frac{1}%\n{2n}\\right) \/\\Gamma\\left( \\frac{1}{2}+\\frac{1}{2n}\\right) .$ \\newline(i)\nFor $\\beta\\in(-\\pi\/2,\\pi\/2],$ we have polar forms for $f_{\\beta}\\left(\n1\\right) $ and $f_{\\beta}\\left( e^{i\\pi\/n}\\right) $ given by magnitudes\n$\\left\\vert f_{\\beta}\\left( 1\\right) \\right\\vert $ and $\\left\\vert f_{\\beta\n}\\left( e^{i\\pi\/n}\\right) \\right\\vert $ which are, respectively,\n\\begin{equation}\nK_{n}\\sqrt{\\sec^{2}\\left( \\frac{\\pi}{2n}\\right) +2\\tan\\left( \\frac{\\pi}%\n{2n}\\right) \\cos\\beta}\\text{ and }K_{n}\\sqrt{\\sec^{2}\\left( \\frac{\\pi}%\n{2n}\\right) -2\\tan\\left( \\frac{\\pi}{2n}\\right) \\cos\\beta}, \\label{cnmag}%\n\\end{equation}\nand by the arguments $\\psi=\\arg\\left( f_{\\beta}\\left( 1\\right) \\right) $\nand $\\pi\/n+\\psi^{\\prime}=\\arg\\left( f_{\\beta}\\left( e^{i\\pi\/n}\\right)\n\\right) ,$ where%\n\\begin{equation}\n\\tan\\psi=\\frac{1-\\tan\\left( \\frac{\\pi}{2n}\\right) }{1+\\tan\\left( \\frac{\\pi\n}{2n}\\right) }\\tan\\left( \\frac{\\beta}{2}\\right) \\text{ and }\\tan\n\\psi^{\\prime}=\\frac{1+\\tan\\left( \\frac{\\pi}{2n}\\right) }{1-\\tan\\left(\n\\frac{\\pi}{2n}\\right) }\\tan\\left( \\frac{\\beta}{2}\\right) . \\label{cnarg}%\n\\end{equation}\nIf $\\beta=0$ then both $\\psi$ and $\\psi^{\\prime}$ are zero. If $\\beta=\\pi\/2,$\nthese angles reduce to \\linebreak$\\psi=\\pi\/4-\\pi\/\\left( 2n\\right) $ and\n$\\psi^{\\prime}=\\pi\/4+\\pi\/\\left( 2n\\right) .$\\newline(ii) If $\\beta\\in\n(0,\\pi\/2],$ then the curves $f_{\\beta}\\left( r\\right) $ and $f_{\\beta\n}\\left( re^{i\\pi\/n}\\right) $ have strictly increasing magnitude. Moreover,\n$\\arg\\frac{\\partial}{\\partial r}f_{\\beta}\\left( r\\right) $ decreases\nstrictly with $r$, and $\\arg\\frac{\\partial}{\\partial r}f_{\\beta}\\left(\nre^{i\\pi\/n}\\right) ,$ increases strictly with $r$. We also have the following\narguments for the the tangents of these curves at the origin\n\\[\n\\lim_{r\\rightarrow0^{+}}\\arg\\frac{\\partial}{\\partial r}f_{\\beta}\\left(\nr\\right) =\\beta\/2\\text{ and }\\lim_{r\\rightarrow0^{+}}\\arg\\frac{\\partial\n}{\\partial r}f_{\\beta}\\left( re^{i\\pi\/n}\\right) =\\beta\/2+\\pi\/n,\n\\]\nand at the boundary\n\\[\n\\lim_{r\\rightarrow1^{-}}\\arg\\frac{\\partial}{\\partial r}f_{\\beta}\\left(\nr\\right) =0\\text{ and }\\lim_{r\\rightarrow1^{-}}\\arg\\frac{\\partial}{\\partial\nr}f_{\\beta}\\left( re^{i\\pi\/n}\\right) =\\pi\/2+\\pi\/n.\n\\]\n\\newline(iii) For $\\beta=0$ the curves $f_{0}\\left( r\\right) $ and\n$f_{0}\\left( re^{i\\pi\/n}\\right) $ are straight lines - the images have\nconstant argument of $0$ and $\\pi\/n,$ respectively.\n\\end{lemma}\n\n\\begin{proof}\nTo prove (i), we compute\n\\begin{align*}\nf_{\\beta}\\left( r\\right) & =e^{i\\beta\/2}h_{n}\\left( r\\right)\n+e^{-i\\beta\/2}g_{n}\\left( r\\right) \\\\\n& =\\cos\\left( \\beta\/2\\right) \\left( h_{n}\\left( r\\right) +g_{n}\\left(\nr\\right) \\right) +i\\sin\\left( \\beta\/2\\right) \\left( h_{n}\\left(\nr\\right) -g_{n}\\left( r\\right) \\right) ,\n\\end{align*}\nand, upon recalling from Corollary \\ref{rayshg} that $\\overline{g_{n}\\left(\nre^{i\\pi\/n}\\right) }=e^{-i\\pi\/n}g_{n}\\left( r\\right) ,$\n\\begin{align*}\nf_{\\beta}\\left( re^{i\\pi\/n}\\right) & =e^{i\\beta\/2}h_{n}\\left( re^{i\\pi\n\/n}\\right) +e^{-i\\beta\/2}\\overline{e^{-i\\pi\/n}g_{n}\\left( r\\right) }\\\\\n& =e^{i\\pi\/n}\\left( \\cos\\left( \\beta\/2\\right) \\left( h_{n}\\left(\nr\\right) -g_{n}\\left( r\\right) \\right) +i\\sin\\left( \\beta\/2\\right)\n\\left( h_{n}\\left( r\\right) +g_{n}\\left( r\\right) \\right) \\right) .\n\\end{align*}\nWe recall the formulae $h_{n}\\left( 1\\right) =H_{n}\\left( 1\\right) =K_{n}$\nand $g_{n}\\left( 1\\right) =G_{n}\\left( 1\\right) \/\\left( n-1\\right)\n=\\tan\\left( \\pi\/\\left( 2n\\right) \\right) K_{n}$ from Proposition\n\\ref{hyperProps}. Using continuity of $f_{\\beta}$ on $\\bar{U}$, we take limits\nas $r\\rightarrow1^{-}$ to get%\n\\begin{align*}\nf_{\\beta}\\left( 1\\right) & =K_{n}\\left( \\cos\\left( \\beta\/2\\right)\n\\left( 1+\\tan\\left( \\frac{\\pi}{2n}\\right) \\right) +i\\sin\\left(\n\\beta\/2\\right) \\left( 1-\\tan\\left( \\frac{\\pi}{2n}\\right) \\right) \\right)\n\\text{ and}\\\\\nf_{\\beta}\\left( e^{i\\pi\/n}\\right) & =e^{i\\pi\/n}K_{n}\\left( \\cos\\left(\n\\beta\/2\\right) \\left( 1-\\tan\\left( \\frac{\\pi}{2n}\\right) \\right)\n+i\\sin\\left( \\beta\/2\\right) \\left( 1+\\tan\\left( \\frac{\\pi}{2n}\\right)\n\\right) \\right) .\n\\end{align*}\nMoreover, writing $\\psi=\\arg f_{\\beta}\\left( 1\\right) ,$ and $\\pi\n\/n+\\psi^{\\prime}=\\arg f_{\\beta}\\left( e^{i\\pi\/n}\\right) ,$ we take ratios of\nimaginary to real parts of $f_{\\beta}\\left( 1\\right) $ and $f_{\\beta}\\left(\ne^{i\\pi\/n}\\right) $ to easily obtain the stated formulae for $\\tan\\psi$ and\n$\\tan\\psi^{\\prime}.$ We also readily obtain\n\\begin{align*}\n\\left\\vert f_{\\beta}\\left( 1\\right) \\right\\vert ^{2} & =K_{n}^{2}\\left(\n1+2\\tan\\left( \\frac{\\pi}{2n}\\right) \\cos\\beta+\\tan^{2}\\left( \\frac{\\pi}%\n{2n}\\right) \\right) \\text{ and }\\\\\n\\left\\vert f_{\\beta}\\left( e^{i\\pi\/n}\\right) \\right\\vert ^{2} & =K_{n}%\n^{2}\\left( 1-2\\tan\\left( \\frac{\\pi}{2n}\\right) \\cos\\beta+\\tan^{2}\\left(\n\\frac{\\pi}{2n}\\right) \\right)\n\\end{align*}\nWe can rewrite $1+\\tan^{2}\\left( \\pi\/\\left( 2n\\right) \\right) =\\sec\n^{2}\\left( \\pi\/\\left( 2n\\right) \\right) $ to obtain the magnitudes stated\nin (i)$.$ For (ii), taking derivatives of the formulae first derived for\n$f_{\\beta}\\left( r\\right) $ and $f_{\\beta}\\left( re^{i\\pi\/n}\\right) ,$ we\nobtain\n\\begin{align*}\n\\frac{\\partial}{\\partial r}f_{\\beta}\\left( r\\right) & =\\cos\\left(\n\\beta\/2\\right) \\left( \\frac{1+r^{n-2}}{\\sqrt{1-r^{2n}}}\\right)\n+i\\sin\\left( \\beta\/2\\right) \\left( \\frac{1-r^{n-2}}{\\sqrt{1-r^{2n}}%\n}\\right) \\text{ and }\\\\\n\\frac{\\partial}{\\partial r}f_{\\beta}\\left( re^{i\\pi\/n}\\right) &\n=e^{i\\pi\/n}\\left( \\cos\\left( \\beta\/2\\right) \\left( \\frac{1-r^{n-2}}%\n{\\sqrt{1-r^{2n}}}\\right) +i\\sin\\left( \\beta\/2\\right) \\left( \\frac\n{1+r^{n-2}}{\\sqrt{1-r^{2n}}}\\right) \\right) .\n\\end{align*}\n\n\nThe arguments of these derivatives, namely $\\arg\\frac{\\partial}{\\partial\nr}f_{\\beta}\\left( r\\right) $ and $\\arg\\frac{\\partial}{\\partial r}f_{\\beta\n}\\left( re^{i\\pi\/n}\\right) $ are respectively\\\n\\[\n\\arctan\\left( \\tan\\left( \\beta\/2\\right) \\frac{1-r^{n-2}}{1+r^{n-2}}\\right)\n\\text{ and }\\pi\/n+\\arctan\\left( \\tan\\left( \\beta\/2\\right) \\frac{1+r^{n-2}%\n}{1-r^{n-2}}\\right) ,\n\\]\nwhence the stated monotonicity of $\\arg\\frac{\\partial}{\\partial r}f_{\\beta\n}\\left( r\\right) $ and $\\arg\\frac{\\partial}{\\partial r}f_{\\beta}\\left(\nre^{i\\pi\/n}\\right) $ in (ii).\\linebreak The limits in (ii) are now easily\ncomputed. Note that as $r\\rightarrow1^{-},$ the ratio $\\frac{1+r^{n-2}%\n}{1-r^{n-2}}$ becomes infinite, and $\\arctan\\left( \\tan\\left( \\beta\n\/2\\right) \\frac{1+r^{n-2}}{1-r^{n-2}}\\right) $ approaches $\\pi\/2.\\ $We can\nalso calculate the magnitudes $\\left\\vert \\frac{\\partial}{\\partial r}f_{\\beta\n}\\left( r\\right) \\right\\vert ^{2}$ and $\\left\\vert \\frac{\\partial}{\\partial\nr}f_{\\beta}\\left( re^{i\\pi\/n}\\right) \\right\\vert ^{2},$ respectively, as\n$\\frac{1\\pm2r^{n-2}\\cos\\beta+r^{2n-4}}{1-r^{2n}}>\\frac{\\left( 1-r^{n-2}%\n\\right) ^{2}}{1-r^{2n}}>0.$ Thus both curves $f_{\\beta}\\left( r\\right) $\nand $f_{\\beta}\\left( re^{i\\pi\/n}\\right) $ have increasing magnitude as $r$\nincreases. Finally to prove (iii), where $\\beta=0,$ we have \\newline%\n$f_{0}\\left( r\\right) =\\left( 1\/\\sqrt{2}\\right) \\left( h_{n}\\left(\nr\\right) +g_{n}\\left( r\\right) \\right) >0$. Moreover, $\\arg f_{0}\\left(\nre^{i\\pi\/n}\\right) =\\frac{\\pi}{n}+\\arctan\\left( 0\\right) =\\frac{\\pi}{n},$\nso $f_{0}\\left( re^{i\\frac{\\pi}{n}}\\right) $ maps to a ray emanating from\nthe origin with argument $\\pi\/n.$\n\\end{proof}\n\n\\section{Cusps and Nodes}\n\nWe now examine the boundary curves for the rosette harmonic mappings, which\nallows us to describe the cusps and other features that are apparent in the\ngraphs$.$ The boundary curve is also the key in our approach in Section 5 to\nproving univalence of the rosette harmonic mappings.\n\nFor a fixed $n\\geq3$, a rosette harmonic mapping $f_{\\beta}$ of Definition\n\\ref{defnharmonic} extends analytically to $\\partial U$, except at isolated\nvalues on $\\partial U$. Indeed recall that $h_{n}$ and $g_{n}$ of Definition\n\\ref{defnhg} are analytic except at the $2n$th roots of unity (Proposition\n\\ref{analco}). In contrast, the $n$-cusped hypocycloid $f_{hyp}$ of Example\n1.1 extends continuously to $\\bar{U}$ but with just $n$ values in $\\partial U\n$ where the boundary function is not regular.\n\nNevertheless, Figures \\ref{rosette0}\\ and \\ref{five} show rosette mappings\n$f_{\\beta},$ where exactly $n$ cusps are apparent. A striking similarity\nbetween the rosette and hypocycloid mappings is the common dilatation\n$\\omega\\left( z\\right) =z^{n-2}.$ We note the following consequence for a\nharmonic function $f$ on $U.$ Where $\\alpha\\left( t\\right) =f\\left(\ne^{it}\\right) $ exists with a continuous derivative $\\alpha^{\\prime}\\left(\nt\\right) $, Corollary 2.2b of \\cite{HenSchob} implies that $\\operatorname{Im}%\n\\left( \\sqrt{\\omega_{f}\\left( e^{it}\\right) }\\alpha^{\\prime}\\left(\nt\\right) \\right) =0$ (see also Section 7.4 of \\cite{PeterHMBook}). Thus on\nintervals where $\\alpha^{\\prime}\\left( t\\right) $ is continuous and\nnon-zero, and for dilatation $\\omega_{f}\\left( z\\right) =z^{n-2}$, we have\n\\begin{equation}\n\\arg\\alpha^{\\prime}\\left( t\\right) \\equiv-\\arg\\sqrt{e^{i\\left( n-2\\right)\nt}}\\equiv-\\left( n\/2-1\\right) t\\text{ }(\\operatorname{mod}\\pi).\n\\label{HSarg}%\n\\end{equation}\nWe will find for rosette mappings $f_{\\beta}$ of Definition \\ref{defnharmonic}\nthat have cusps, $n$ of the $2n$ singular points are \"removable\" in a sense to\nbe described. Furthermore, for $n\\geq3$ and consistent with (\\ref{HSarg}), the\nformula\n\\begin{equation}\n\\fbox{$\\arg\\alpha^\\prime\\left( t\\right) =k\\pi-\\left( \\frac{n}{2}-1\\right)\nt$} \\label{compass}%\n\\end{equation}\nholds (except at possibly one point) on each of the $n$ intervals $\\left(\n\\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) $, $\\ k=1,2,...,n,$ both when\n$\\alpha$ is the boundary function of an $n$-cusped hypocycloid \\textit{and\n}when $\\alpha$ is the boundary function of a rosette (provided it has cusps).\nFormula (\\ref{compass}) will not be valid for example for a rosette mapping\n$f_{\\pi\/2}$, where there are arcs for which the boundary function is\nconstant$.$ To proceed we first give a definition of cusp, node, and singular\npoint of a curve.\n\n\\begin{definition}\n\\label{cusp}An \\textbf{isolated singular point} on a curve $\\alpha\\left(\nt\\right) $ is a point $\\alpha\\left( t_{0}\\right) $ at which either\n$\\alpha^{\\prime}\\left( t_{0}\\right) =0$ or $\\alpha^{\\prime}\\left(\nt_{0}\\right) $ is not defined, but for which $\\alpha^{\\prime}\\left(\nt\\right) $ is defined and non-zero in a neighborhood of $t_{0}$. Define the\nquantities\n\\[\n\\arg\\alpha^{\\prime}\\left( t_{0}\\right) ^{-}=\\lim_{t\\nearrow t_{0}^{-}}%\n\\arg\\alpha^{\\prime}\\left( t\\right) \\text{ and }\\arg\\alpha^{\\prime}\\left(\nt_{0}\\right) ^{+}=\\lim_{t\\searrow t_{0}^{+}}\\arg\\alpha^{\\prime}\\left(\nt\\right)\n\\]\nwhere they exist. An isolated singular point for which $\\arg\\alpha^{\\prime\n}\\left( t_{0}\\right) ^{+}$and $\\arg\\alpha^{\\prime}\\left( t_{0}\\right) ^{-}\n$ differ by $\\pi$ is defined to be a \\textbf{cusp.} The line $L$ through the\ncusp $\\alpha\\left( t_{0}\\right) $ containing points with argument equal to\n$\\arg\\alpha^{\\prime}\\left( t_{0}\\right) ^{+}$ (or $\\arg\\alpha^{\\prime\n}\\left( t_{0}\\right) ^{-}$) is called the \\textbf{axis of the cusp}, or\nsimply the \\textbf{axis}$.$ Define a \\textbf{node} to be an isolated singular\npoint on the curve at which $\\arg\\alpha^{\\prime}\\left( t_{0}\\right)\n^{+}-\\arg\\alpha^{\\prime}\\left( t_{0}\\right) ^{-}=\\theta\\not \\equiv\n\\pi\\,\\left( \\operatorname{mod}2\\pi\\right) .$ In this case, $\\theta$ is the\n\\textbf{exterior angle} of the node. The interior angle at the node is then\n$\\pi-\\theta.$ If the exterior angle is $0,$ then we call the node a\n\\textbf{removable node}. A node is described as a \\textbf{corner} in some sources.\n\\end{definition}\n\nBoth $h_{n}$ and $g_{n}$ have nodes with exterior (and interior) angle\n$\\pi\/2,$ as seen in Figure \\ref{canon6}: since $\\overline{g_{n}}$ is a\nreflection of $g_{n}$, the nodes of $\\overline{g_{6}}$ in Figure \\ref{canon6}\nappear with exterior angle $-\\pi\/2$ rather than $\\pi\/2$. The lower right image\nin Figure \\ref{five} indicates a rosette mapping with nodes rather than cusps,\nbut the remaining images in Figure \\ref{five} show examples with cusps. The\nfollowing lemma provides a convenient way to invoke equation (\\ref{compass})\nand draw conclusions about cusps of the boundary extension.\n\n\\begin{lemma}\n\\label{cusps}Let $n\\geq3,$ and $f\\left( z\\right) $ be a harmonic mapping on\n$U$, with continuous extension to $\\bar{U},$ so that $\\alpha\\left( t\\right)\n=f\\left( e^{it}\\right) $ is defined on $\\partial U$. Let $k=1,2,...,n.$\n\\newline(i)\\ Suppose that $\\alpha^{\\prime}$ is defined and non-zero except at\n$t=2k\\pi\/n,$ and that $\\alpha$ satisfies (\\ref{compass}) on each interval\n$\\left( \\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) .$ Then for each $k,$\n$\\alpha\\left( 2k\\pi\/n\\right) $ is a cusp$,$ and the cusp axis has argument\n$2k\\pi\/n.$\\newline(ii)\\ Suppose that $f$ has $n$-fold rotational symmetry\n$f\\left( e^{i2k\\pi\/n}z\\right) =e^{i2k\\pi\/n}f\\left( z\\right) ,$ and\nthat\\ (\\ref{compass}) holds for $k=1$ on the interval $\\left( 0,2\\pi\n\/n\\right) .$ Then $\\alpha$ satisfies (\\ref{compass}) on each interval\n$\\left( \\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) $ and the conclusions of\n(i) hold.\n\\end{lemma}\n\n\\begin{proof}\nFor (i), we evaluate limits at $t=\\frac{2k\\pi}{n}$ as follows using\n(\\ref{compass}):\n\\begin{align}\n\\arg\\alpha^{\\prime}\\left( 2k\\pi\/n\\right) ^{-} & =\\lim_{t\\nearrow\n2k\\pi\/n^{-}}k\\pi-\\left( \\frac{n}{2}-1\\right) t=\\frac{2k\\pi}{n},\\text{\nand}\\label{cuspslopes}\\\\\n\\arg\\alpha^{\\prime}\\left( 2k\\pi\/n\\right) ^{+} & =\\lim_{t\\searrow\n2k\\pi\/n^{+}}\\left( k+1\\right) \\pi-\\left( \\frac{n}{2}-1\\right) t=\\pi\n+\\frac{2k\\pi}{n}.\\nonumber\n\\end{align}\nThus $\\arg\\alpha^{\\prime}\\left( 2\\pi\/n\\right) ^{+}$ and $\\arg\\alpha^{\\prime\n}\\left( 2\\pi\/n\\right) ^{-}$ differ by $\\pi,$ so $\\alpha\\left(\n2k\\pi\/n\\right) $ is a cusp. We also see the axis has argument $2k\\pi\/n$ (or\nequivalently $\\pi+2k\\pi\/n$). For (ii), we use the fact that $\\alpha\\left(\nt+2k\\pi\/n\\right) =e^{i2k\\pi\/n}\\alpha\\left( t\\right) $ to conclude\n\\begin{equation}\n\\arg\\alpha\\left( t+2k\\pi\/n\\right) =2k\\pi\/n+\\arg\\alpha\\left( t\\right) .\n\\label{badd}%\n\\end{equation}\nGiven that (\\ref{compass}) holds for $k=1,$ we can extend it to each interval\n$\\left( \\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) ,$ $k=2,3,...,n,$ using\n(\\ref{badd}), and so the conclusions of (i)\\ hold also.\n\\end{proof}\n\n\\begin{remark}\n\\label{extend}Lemma \\ref{cusps} remains valid (with appropriate adjustments to\nthe interval on which (\\ref{compass}) holds), even if for finitely many points\nin $\\left( \\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) ,$ $\\arg\\alpha^{\\prime\n}\\left( t\\right) $ does not exist. To compute the limits (\\ref{cuspslopes})\nwe only need equation (\\ref{compass}) to hold in a neighborhood of the\nendpoints $2k\\pi\/n$.\n\\end{remark}\n\nThe following proposition surely appears in the literature, but for\ncompleteness, we use Lemma \\ref{cusps} to demonstrate the properties of the\nhypocycloid cusps.\n\n\\begin{proposition}\n\\label{hypocycloid}Let $n\\geq3,$ and let $f_{hyp}\\left( z\\right) =z+\\frac\n{1}{n-1}\\bar{z}^{n-1}$ be the hypocycloid harmonic mapping, and let\n$\\alpha_{hyp}\\left( t\\right) =f_{hyp}\\left( e^{it}\\right) .$ For\n$k=1,2,...,n,$ formula (\\ref{compass}) holds with $\\alpha=\\alpha_{hyp}$ for\nall $t\\in\\left( \\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) .$ Moreover,\n$\\alpha_{hyp}$ has precisely $n$ cusps $\\alpha_{hyp}\\left( 2k\\pi\/n\\right)\n=\\frac{n}{n-1}e^{i2k\\pi\/n};$ the cusp axis has argument $2k\\pi\/n.$ In\ntraversing from one cusp to the next, $\\alpha_{hyp}$ has total curvature\n$\\pi-2\\pi\/n.$\n\\end{proposition}\n\n\\begin{proof}\nWe have $\\alpha_{hyp}\\left( t\\right) =h\\left( e^{it}\\right) +\\bar\n{g}\\left( e^{it}\\right) $ where $h\\left( e^{it}\\right) =e^{it}$ and\n$g\\left( e^{it}\\right) =\\frac{1}{n-1}e^{i\\left( n-1\\right) t}.$ We already\nnoted the singular points at $2k\\pi\/n$ in Example 1.1. With $z=e^{it}$ we\napply the chain rule, obtaining $\\frac{d}{dt}h\\left( e^{it}\\right) =ie^{it}$\nand $\\frac{d}{dt}g\\left( e^{it}\\right) =ie^{i\\left( n-1\\right) t}.$ The\nmagnitudes are both $1,$ so $\\alpha_{hyp}^{\\prime}\\left( t\\right) $ will\nhave argument equal to the mean of $\\arg\\frac{d}{dt}h\\left( e^{it}\\right) $\nand $\\arg\\frac{d}{dt}g\\left( e^{-it}\\right) ,$ when (for example) we choose\nbranches of the arguments that lie within $\\pi$ of one another. To this end,\nwe take\n\\begin{equation}\n\\arg\\frac{d}{dt}h\\left( e^{it}\\right) =\\pi\/2+t\\text{ and }\\arg\\frac{d}%\n{dt}g\\left( e^{-it}\\right) =3\\pi\/2-\\left( n-1\\right) t \\label{hypderivs}%\n\\end{equation}\non the interval $\\left( 0,2\\pi\/n\\right) .$ Thus the mean is $\\left(\n2\\pi-\\left( n\/2-1\\right) t\\right) \/2,$ which is equation (\\ref{compass})\nfor $k=1 $. We also have $f_{hyp}\\left( e^{i2k\\pi\/n}z\\right) =e^{i2k\\pi\n\/n}z+\\frac{1}{n-1}\\left( e^{-i2k\\pi\/n}\\bar{z}\\right) ^{n-1}.$ We factor out\nthe $e^{i2k\\pi\/n},$ using $\\left( e^{-i2k\\pi\/n}\\right) ^{n-1}=e^{-i2k\\pi\n\/n},$ and obtain $e^{i2k\\pi\/n}f_{hyp}\\left( z\\right) ,$ showing that\n$f_{hyp}$ has rotational symmetry$.$ By Lemma \\ref{cusps} (ii), $\\alpha\n_{hyp}\\left( 2k\\pi\/n\\right) $ is a cusp and the axis has argument $2k\\pi\/n$\nfor each $k=1,2,...,n.$ Using $z=1$ above, we also obtain $f_{hyp}\\left(\ne^{i2k\\pi\/n}\\right) =e^{i2k\\pi\/n}+\\frac{1}{n-1}\\left( e^{i2k\\pi\/n}\\right)\n=\\frac{n}{n-1}e^{i2k\\pi\/n}$. The total curvature of $\\alpha_{hyp}$ is measured\nwith the change in argument of the unit tangent, or equivalently the change in\n$\\arg\\alpha_{hyp}^{\\prime}.$ Since this is monotonic and linear in $t$\n(equation (\\ref{compass})), the total change in $\\arg\\alpha_{hyp}^{\\prime} $\nover any of the given intervals is equal to $\\left( n\/2-1\\right) $ times the\ninterval length $2\\pi\/n$, and so we obtain $\\pi-2\\pi\/n$.\n\\end{proof}\n\nWe compute formulae for the derivatives $\\left( d\/dt\\right) h_{n}\\left(\ne^{it}\\right) $ and $\\left( d\/dt\\right) g_{n}\\left( e^{it}\\right) .$ In\ncontrast with the hypocycloid, the arguments $\\left( d\/dt\\right)\nh_{n}\\left( e^{it}\\right) $ and $\\left( d\/dt\\right) \\overline{g_{n}\\left(\ne^{it}\\right) } $ differ by a constant angle of $\\pm\\pi\/2.$\n\n\\begin{proposition}\n\\label{linearangles}Let $n\\geq3,$ and let $\\zeta$ be a primitive 2nth root of\nunity, and consider $h_{n}$ and $g_{n}$ defined in Definition \\ref{defnhg},\nanalytic on $\\bar{U}\\backslash\\left\\{ \\zeta^{j}:j=1,2,...,2n\\right\\} .$ Let\n$j=1,2,...,2n.$\\newline i) On each interval $\\left( \\left( j-1\\right)\n\\pi\/n,j\\pi\/n\\right) ,\\ $derivatives $d\/dt$ of both $h_{n}\\left(\ne^{it}\\right) $ and $g_{n}\\left( e^{it}\\right) $ have magnitude\n$1\/\\left\\vert \\sqrt{1-e^{i2nt}}\\right\\vert ,$ and the arguments are linear\nmonotonic functions, expressible as%\n\\begin{align}\n\\arg\\left( \\frac{d}{dt}h_{n}\\left( e^{it}\\right) \\right) &\n=3\\pi\/4-\\left( n\/2-1\\right) t+\\left( j-1\\right) \\pi\/2,\\text{\nand}\\label{argghh}\\\\\n\\arg\\left( \\frac{d}{dt}g_{n}\\left( e^{it}\\right) \\right) &\n=3\\pi\/4+\\left( n\/2-1\\right) t+\\left( j-1\\right) \\pi\/2. \\label{argghg}%\n\\end{align}\n\\newline(ii) The functions $h_{n}\\left( e^{it}\\right) $ and $g_{n}\\left(\ne^{it}\\right) $ each have $2n$ singular points $h_{n}\\left( e^{ij\\pi\n\/n}\\right) $ and $g_{n}\\left( e^{ij\\pi\/n}\\right) ,$ which are each nodes\nwith exterior (and interior) angle $\\pi\/2.$\\newline(iii) The difference in the\narguments $\\arg\\frac{d}{dt}h_{n}\\left( e^{it}\\right) -\\arg\\frac{d}%\n{dt}\\overline{g_{n}}\\left( e^{it}\\right) $ is constant on $\\left( \\left(\nj-1\\right) \\pi\/n,j\\pi\/n\\right) ,$ and is alternately $+\\pi\/2$ when $j$ is\neven, and $-\\pi\/2$ when $j$ is odd$.$\n\\end{proposition}\n\n\\begin{proof}\n(i) Recall the derivatives $\\frac{dh}{dz}=\\frac{1}{\\sqrt{1-z^{2n}}}$ and\n$\\frac{dg}{dz}=\\frac{z^{n-2}}{\\sqrt{1-z^{2n}}}$ of Proposition \\ref{analco}\n(iii). \\ We have $\\frac{d}{dt}h_{n}\\left( e^{it}\\right) =\\frac{ie^{it}%\n}{\\sqrt{1-e^{i2nt}}}$ and $\\frac{d}{dt}g_{n}\\left( e^{it}\\right)\n=\\frac{ie^{i\\left( n-1\\right) t}}{\\sqrt{1-e^{i2nt}}}.$ Each derivative has\nmagnitude $1\/\\left\\vert \\sqrt{1-e^{i2nt}}\\right\\vert ,$ and singular points\noccur at the 2nth roots of unity, where $e^{i2nt}=1.$ We compute\n\\begin{align*}\n\\arg\\frac{d}{dt}h_{n}\\left( e^{it}\\right) & =\\arg ie^{it}-\\frac{1}{2}%\n\\arg\\left( 1-e^{i2nt}\\right) \\\\\n& =\\pi\/2+t+\\frac{1}{2}\\arctan\\frac{\\sin\\left( 2nt\\right) }{1-\\cos\\left(\n2nt\\right) }.\n\\end{align*}\nThe latter term reduces to $\\pi\/4-\\left( n\/2\\right) t,$ which can be seen\nfor example using the half angle formula for cotangent; $\\operatorname{arccot}%\n\\frac{\\sin\\left( 2nt\\right) }{1-\\cos\\left( 2nt\\right) }=nt$ and $\\arctan\nX=\\pi\/2-\\operatorname{arccot}X.$ Thus on $\\left( 0,\\pi\/n\\right) ,$%\n\\[\n\\arg\\frac{d}{dt}h_{n}\\left( e^{it}\\right) =3\\pi\/4-\\left( n\/2-1\\right) t.\n\\]\nAt $t=\\pi\/\\left( 2n\\right) ,$ $\\sqrt{1-e^{i2nt}}$ becomes real and\n$\\arg\\frac{d}{dt}h_{n}\\left( e^{i\\pi\/2n}\\right) =\\arg\\left( ie^{i\\pi\n\/2n}\\right) =\\pi\/2+\\pi\/\\left( 2n\\right) ,$ which is consistent with our\nformula on $\\left( 0,\\pi\/n\\right) $; this choice of branch of $\\arctan$\ngives $\\arg\\frac{d}{dt}h_{n}\\left( e^{it}\\right) $ an \"initial\" value\n$3\\pi\/4$ (the limit as $t\\searrow0^{+}$) and is evidently consistent with the\nargument at $t=e^{i\\pi\/2n}$ (see also Figure \\ref{canon6}). From the\nrotational symmetry equation (\\ref{ga}) for $h_{n}$%\n\\[\n\\arg\\frac{d}{dt}h_{n}\\left( e^{i\\left( t+j\\pi\/n\\right) }\\right) =\\arg\n\\frac{d}{dt}h_{n}\\left( e^{it}\\right) +j\\pi\/n.\n\\]\nAdding $\\left( j-1\\right) \\pi\/2$ extends our formula for $\\arg\\frac{d}%\n{dt}h_{n}\\left( e^{it}\\right) $ from $\\left( 0,\\pi\/n\\right) $ to the\ninterval $\\left( \\left( j-1\\right) \\pi\/n,j\\pi\/n\\right) ,$ giving\n(\\ref{argghh}). Similarly on $\\left( 0,\\pi\/n\\right) $ we obtain\n\\[\n\\arg\\frac{d}{dt}g_{n}\\left( e^{it}\\right) =\\pi\/2+\\left( n-1\\right)\nt+\\frac{1}{2}\\arctan\\frac{\\sin\\left( 2nt\\right) }{1-\\cos\\left( 2nt\\right)\n}=3\\pi\/4+\\left( n\/2-1\\right) t,\n\\]\na branch of the argument for which $\\arg\\frac{d}{dt}g_{n}\\left( e^{i\\pi\n\/2n}\\right) =\\pi-\\pi\/\\left( 2n\\right) $ as expected$,$ so again with\ninitial value $3\\pi\/4$ on $\\left( 0,\\pi\/n\\right) .$ From the rotational\nsymmetry equation (\\ref{ga}) for $g_{n}$ we have $g_{n}\\left( e^{ij\\pi\n\/n}z\\right) =e^{ij\\left( \\pi-\\pi\/n\\right) }g_{n}\\left( z\\right) ,$ so\n\\[\n\\arg\\frac{d}{dt}g_{n}\\left( e^{i\\left( t+j\\pi\/n\\right) }\\right) =\\arg\n\\frac{d}{dt}g_{n}\\left( e^{it}\\right) +j\\left( \\pi-\\pi\/n\\right) .\n\\]\nWe extend our formula for $\\arg\\frac{d}{dt}g_{n}\\left( e^{it}\\right) $ for\n$t\\in$ $\\left( \\left( j-1\\right) \\pi\/n,j\\pi\/n\\right) $ as before, adding\n$\\left( j-1\\right) \\pi\/2$, leading to equation (\\ref{argghg}). For (ii) we\nnote that as we pass from the interval $\\left( \\left( j-1\\right) \\pi\n\/n,j\\pi\/n\\right) $ to $\\left( j\\pi\/n,\\left( j+1\\right) \\pi\/n\\right) ,$\nboth $\\arg\\frac{d}{dt}h_{n}\\left( e^{it}\\right) $ and $\\arg\\frac{d}{dt}%\ng_{n}\\left( e^{it}\\right) $ increase by $\\pi\/2$ at $j\\pi\/n.$ Thus\n$h_{n}\\left( e^{ij\\pi\/n}\\right) $ and $g_{n}\\left( e^{ij\\pi\/n}\\right) $\neach are nodes with exterior angle $\\pi\/2.$ To prove (iii) we compute the\ndifference in $\\arg\\frac{d}{dt}h_{n}\\left( e^{it}\\right) $ and $\\arg\\frac\n{d}{dt}\\overline{g_{n}}\\left( e^{it}\\right) $ as $\\arg\\frac{d}{dt}%\nh_{n}\\left( e^{it}\\right) +\\arg\\frac{d}{dt}g_{n}\\left( e^{it}\\right) ,$ so\nfor \\newline$t\\in\\left( \\left( j-1\\right) \\pi\/n,j\\pi\/n\\right) ,$\n\\begin{equation}\n\\arg\\frac{d}{dt}h_{n}\\left( e^{it}\\right) -\\arg\\frac{d}{dt}\\overline{g_{n}%\n}\\left( e^{it}\\right) =3\\pi\/2+\\left( j-1\\right) \\pi=\\left( 2j+1\\right)\n\\pi\/2. \\label{hgdiff}%\n\\end{equation}\n\n\\end{proof}\n\n\\begin{remark}\nThe rosette mappings are distinguished from the hypocycloid in that $\\arg$\n$\\frac{d}{dt}h_{n}\\left( e^{it}\\right) $ and $\\arg\\frac{d}{dt}%\n\\overline{g_{n}\\left( e^{it}\\right) }$ are decreasing in lockstep. As a\nresult the curves $h_{n}\\left( e^{it}\\right) $ and $\\overline{g_{n}}\\left(\ne^{it}\\right) ,$ and ultimately $g_{n}\\left( e^{it}\\right) $ must be rigid\nmotions of one another, as illustrated in Figure \\ref{canon6}, and Figure\n\\ref{transmirror}, and proved in Corollary \\ref{rigid}. For the hypocycloid,\nthe arguments of the derivatives of the analytic and anti-analytic parts\n(\\ref{hypderivs}) are also linear, but with non-equal slopes with differing sign.\n\\end{remark}\n\n\\begin{corollary}\n\\label{rigid}The graph $h_{n}\\left( e^{it}\\right) $ on an interval $\\left(\n\\left( j-1\\right) \\pi\/n,j\\pi\/n\\right) $ and the graph of $g_{n}\\left(\ne^{it}\\right) $ on an interval $\\left( \\left( j^{\\prime}-1\\right)\n\\pi\/n,j^{\\prime}\\pi\/n\\right) $ are identical, up to a translation and\nrotation$,$ where $j,j^{\\prime}=1,2,...,2n.$ Moreover the two curves have\nopposite orientation.\n\\end{corollary}\n\n\\begin{proof}\nThe tangents $\\frac{d}{dt}h_{n}\\left( e^{it}\\right) $ and $\\frac{d}%\n{dt}\\overline{g_{n}}\\left( e^{it}\\right) $ have equal magnitudes on\n\\linebreak$\\left( \\left( j-1\\right) \\pi\/n,j\\pi\/n\\right) $, where their\narguments differ by a constant. Thus $h_{n}\\left( e^{it}\\right) $ and\n$\\overline{g_{n}\\left( e^{it}\\right) }$ have equal arclength and curvature,\nand so are equal up to a translation and rotation by the fundamental theorem\nof plane curves. Both $h_{n}\\left( e^{it}\\right) $ and $g_{n}\\left(\ne^{it}\\right) $ have rotational symmetry (Proposition \\ref{analco}) so the\nprevious statement is true even when $h_{n}\\left( e^{it}\\right) $ and\n$g_{n}\\left( e^{it}\\right) $ are defined on different arcs. Proposition\n\\ref{analco} also shows that the curve $h_{n}\\left( e^{int}\\right) $ also\nhas reflectional symmetry, so the graph of $h_{n}$ has symmetry group\n$D_{2n}.$ Thus $\\overline{h_{n}\\left( e^{it}\\right) }$ is also a rotation of\n$h_{n}\\left( e^{it}\\right) $ on any interval $\\left( \\left( j-1\\right)\n\\pi\/n,j\\pi\/n\\right) ,$ where the pair has opposite orientation. We conclude\n$\\overline{h_{n}\\left( e^{it}\\right) }$ and $\\overline{g_{n}\\left(\ne^{it}\\right) }$ are rigid motions of one another with opposite orientation,\nand the Corollary follows upon conjugation.\n\\end{proof}\n\nProposition \\ref{linearangles} allows us to compute the derivative of the\nboundary function of a rosette harmonic mapping. The cosine rule for triangles\nis useful for adding numbers of the same magnitude, and we recall its\napplication in the following remark.\n\n\\begin{remark}\n\\label{cosine} For $X>0,$ the cosine rule yields\n\\[\n\\left\\vert Xe^{i\\theta_{1}}+Xe^{i\\theta_{2}}\\right\\vert =X\\left\\vert\ne^{i\\theta_{1}}+e^{i\\theta_{2}}\\right\\vert =\\sqrt{2}X\n\\sqrt{ \n1+\\cos \\left\\vert \\theta_{1}-\\theta_{2}\\right\\vert \n} .\n\\]\n\n\\end{remark}\n\n\\begin{theorem}\n\\label{fbdry}For $n\\geq3$ and $\\beta\\in(-\\pi\/2,\\pi\/2],$ and let $f_{\\beta}$ be\na rosette harmonic mapping defined in Definition \\ref{defnharmonic}. Consider\nthe boundary curve $\\alpha_{\\beta}\\left( t\\right) =f_{\\beta}\\left(\ne^{it}\\right) ,$ $t\\in\\partial U.$ The derivative $\\alpha_{\\beta}^{\\prime\n}\\left( t\\right) $ exists and is continuous on $\\partial U, $ except at the\n$2n$ multiples of $\\pi\/n.$ Let $k=1,2,...,n.$ \\newline(i) For $\\left\\vert\n\\beta\\right\\vert <\\pi\/2,$ $\\alpha_{\\beta}$ satisfies (\\ref{compass}) on\n$\\left( \\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) ,$ except at $t=\\left(\n2k-1\\right) \\pi\/n$ where $\\alpha_{\\beta}^{\\prime}\\left( t\\right) $ is\nundefined. Moreover the magnitude of $\\alpha_{\\beta}^{\\prime}\\left( t\\right)\n$, which is strictly non-zero, is%\n\\begin{equation}\n\\left\\vert \\alpha_{\\beta}^{\\prime}\\left( t\\right) \\right\\vert =\\sqrt{2}%\n\\sqrt{1\\pm\\sin\\left( \\beta\\right) }\/\\left\\vert \\sqrt{1-e^{i2nt}}\\right\\vert\n. \\label{mag}%\n\\end{equation}\nHere the $\\sin$ term is subtracted on the first half of the interval $\\left(\n\\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) ,$ and added on the second\nhalf.\\newline(ii) When $\\beta=\\pi\/2,$ on the first half of the interval\n$\\left( \\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) ,$ $\\alpha_{\\pi\/2}$\nsatisfies (\\ref{compass})$,$ while $\\alpha_{\\pi\/2}^{\\prime}\\left( t\\right) $\nis strictly non-zero there with $\\left\\vert \\alpha_{\\pi\/2}^{\\prime}\\left(\nt\\right) \\right\\vert =\\sqrt{2}\/\\left\\vert \\sqrt{1-e^{2int}}\\right\\vert $. On\nthe second half of the interval $\\left( \\left( 2k-2\\right) \\pi\n\/n,2k\\pi\/n\\right) ,$ $\\alpha_{\\pi\/2}^{\\prime}\\left( t\\right) =0$ and\n$\\alpha_{\\pi\/2}\\left( t\\right) $ is constant.\n\\end{theorem}\n\n\\begin{proof}\nNote that the summands $\\frac{d}{dt}e^{i\\beta\/2}h_{n}\\left( e^{it}\\right) $\nand $\\frac{d}{dt}e^{-i\\beta\/2}g_{n}\\left( e^{-it}\\right) $ of $\\alpha\n_{\\beta}^{\\prime}\\left( t\\right) =\\frac{d}{dt}\\arg f_{\\beta}\\left(\ne^{it}\\right) $ have the same magnitude, namely $1\/\\left\\vert \\sqrt\n{1-e^{2int}}\\right\\vert $. \\ From equation (\\ref{hgdiff}), the angle between\n$\\frac{d}{dt}h_{n}\\left( e^{it}\\right) $ and $\\frac{d}{dt}\\overline{g_{n}%\n}\\left( e^{it}\\right) $ is the constant $\\left( -1\\right) ^{j}\\pi\/2$ on\neach interval $\\left( \\left( j-1\\right) \\pi\/n,j\\pi\/n\\right) ,$ and the\npresence of $\\beta$ changes this difference to $\\beta+\\left( -1\\right)\n^{j}\\pi\/2.$ Thus from the cosine rule (see Remark \\ref{cosine}) we obtain the\nmagnitude $\\left\\vert \\alpha_{\\beta}^{\\prime}\\left( t\\right) \\right\\vert\n=\\frac{\\sqrt{2}\\sqrt{1+\\cos\\left( \\beta+\\left( -1\\right) ^{j}\\pi\/2\\right)\n}}{\\left\\vert \\sqrt{1-e^{i2nt}}\\right\\vert }=\\frac{\\sqrt{2}\\sqrt{1+\\left(\n-1\\right) ^{j}\\sin\\left( \\beta\\right) }}{\\left\\vert \\sqrt{1-e^{i2nt}%\n}\\right\\vert },$ for $t\\neq j\\pi\/n.$ This proves equation (\\ref{mag}). Note\nthat since $\\left\\vert \\beta\\right\\vert <\\pi\/2$, $\\alpha_{\\beta}^{\\prime\n}\\left( t\\right) \\neq0.$ We turn to the argument of $\\alpha_{\\beta}^{\\prime\n}.$ We can utilize arithmetic means involving (\\ref{argghh}) and\n(\\ref{argghg}), \\ or simply make use of (\\ref{HSarg}). For either approach,\nthe initial argument $\\arg\\alpha_{\\beta}^{\\prime}\\left( 0\\right) ^{+}$ must\nbe determined as either $\\pi$ or $0.$ The initial values of $\\arg\\frac{d}%\n{dt}h_{n}\\left( e^{it}\\right) $ and $\\arg\\frac{d}{dt}\\overline{g_{n}}\\left(\ne^{it}\\right) $ as $t\\searrow0^{+}$ are $3\\pi\/4$ and $-3\\pi\/4$\nrespectively$.$ Therefore the initial angle $\\arg\\left( \\alpha_{0}^{\\prime\n}\\left( 0\\right) ^{+}\\right) $ is $\\pi$ rather than $0,$ and $\\arg\n\\alpha_{0}^{\\prime}\\left( t\\right) =\\pi-\\left( n\/2-1\\right) t.$ This\nformula for $\\arg\\alpha_{0}^{\\prime}\\left( t\\right) $ holds throughout\n$\\left( 0,\\pi\/n\\right) ,$ since $\\alpha_{\\beta}\\left( t\\right) $ is\ncontinuous$.$ The initial angles $\\frac{d}{dt}h_{n}\\left( e^{it}\\right) $\nand $\\frac{d}{dt}\\overline{g_{n}}\\left( e^{it}\\right) $ as $t\\searrow0^{+}$\non $\\left( \\pi\/n,2\\pi\/n\\right) $ become $3\\pi\/4+\\pi\/n$ and $\\pi\/4+\\pi\/n$\nrespectively (using Proposition \\ref{linearangles}, or rotational symmetry).\nThe initial value of $\\arg\\alpha_{0}^{\\prime}\\left( t\\right) $ on $\\left(\n\\pi\/n,2\\pi\/n\\right) $ is therefore $\\pi\/2+\\pi\/n,$ consistent with\n(\\ref{compass}). Thus the formula $\\arg\\alpha_{0}^{\\prime}\\left( t\\right)\n=\\pi-\\left( n\/2-1\\right) t$ holds throughout $\\left( 0,2\\pi\/n\\right) $,\nwhere defined. For \\linebreak$\\left\\vert \\beta\\right\\vert <\\pi\/2,$ the means\nof $\\arg\\frac{d}{dt}e^{i\\beta\/2}h_{n}\\left( e^{it}\\right) $ and $\\arg\n\\frac{d}{dt}e^{-i\\beta\/2}\\overline{g_{n}}\\left( e^{it}\\right) $ are the same\nas for $\\beta=0$: the initial angles $\\arg\\frac{d}{dt}e^{i\\beta\/2}h_{n}\\left(\ne^{it}\\right) $ and $\\arg\\frac{d}{dt}e^{-i\\beta\/2}\\overline{g_{n}}\\left(\ne^{it}\\right) $ as $t\\searrow0^{+}$ on $\\left( 0,\\pi\/n\\right) $ are\nrespectively $3\\pi\/4+\\beta\/2$ (third quadrant) and $-3\\pi\/4-\\beta\/2$ (fourth\nquadrant)$.$ Thus the initial angle $\\arg\\left( \\alpha_{\\beta}^{\\prime\n}\\left( 0\\right) ^{+}\\right) $ maintains the value $\\pi$ (rather than\n$0$)$.$ Similarly the initial angles $\\arg\\frac{d}{dt}e^{i\\beta\/2}h_{n}\\left(\ne^{it}\\right) $ and $\\arg\\frac{d}{dt}e^{-i\\beta\/2}\\overline{g_{n}}\\left(\ne^{it}\\right) \/2$ as $t\\searrow0^{+}$ on $\\left( \\pi\/n,2\\pi\/n\\right) $ are\nrespectively $3\\pi\/4+\\pi\/n+\\beta\/2$ and $\\pi\/4+\\pi\/n-\\beta\/2.$ Thus the\ninitial angle $\\arg\\left( \\alpha_{\\beta}^{\\prime}\\left( \\pi\/n\\right)\n^{+}\\right) $ also remains fixed as $\\pi\/2+\\pi\/n.$ We conclude that the\nequation for $\\arg\\alpha_{0}^{\\prime}\\left( t\\right) $ is valid for\n$\\arg\\alpha_{\\beta}^{\\prime}\\left( t\\right) $ on $\\left( 0,2\\pi\/n\\right) $\n(note this would \\textit{not} be the case for $\\left\\vert \\beta\\right\\vert\n\\in\\left( \\pi\/2,\\pi\\right) $). Thus equation (\\ref{compass}) holds for\n$k=1,$ with $\\alpha_{\\beta}$ in place of $\\alpha,$ except at $t=\\pi\/n$ where\n$\\arg\\alpha_{\\beta}^{\\prime}\\left( t\\right) $ is not defined$.$ By Theorem\n\\ref{symmetry} (ii), $f_{\\beta}\\ $has $n$-fold rotational symmetry needed to\ninvoke Lemma \\ref{cusps} (ii), and in view of Remark \\ref{extend}, formula\n(\\ref{compass}) holds on each interval $\\left( \\left( 2k-2\\right)\n\\pi\/n,2k\\pi\/n\\right) ,$ for $t\\neq\\left( 2k-1\\right) \\pi\/n.$ This proves (i).\n\nWe now consider (ii), with $\\beta=\\pi\/2$. From Proposition \\ref{linearangles}\n(iii), the angle between $\\frac{d}{dt}e^{i\\pi\/4}h_{n}\\left( e^{it}\\right) $\nand $\\arg\\frac{d}{dt}e^{-i\\pi\/4}\\overline{g_{n}}\\left( e^{it}\\right) $ is\n$\\beta+\\left( -1\\right) ^{j}\\pi\/2$ (mod $2\\pi$), which is either $0$ or\n$\\pi.$ For even $j,$ we see that $\\frac{d}{dt}e^{i\\pi\/4}h_{n}\\left(\ne^{it}\\right) $ and $\\frac{d}{dt}e^{-i\\pi\/4}\\overline{g_{n}}\\left(\ne^{it}\\right) $ cancel, since their arguments differ by $\\pi.$ Then\n$\\alpha_{\\beta}^{\\prime}\\left( t\\right) $ $=0,$ and $\\alpha_{\\beta}$ is\nconstant on $\\left( \\left( j-1\\right) \\pi\/n,j\\pi\/n\\right) .$ For odd $j,$\nthe two summands have the same argument, so $\\arg\\alpha_{\\pi\/2}^{\\prime\n}\\left( t\\right) =\\arg\\frac{d}{dt}e^{i\\pi\/4}h_{n}\\left( e^{it}\\right) .$\nAdding $\\pi\/4$ to formula (\\ref{argghh}), we obtain equation $\\arg\\alpha\n_{\\pi\/2}^{\\prime}\\left( t\\right) =\\pi-\\left( \\frac{n}{2}-1\\right)\nt+\\left( j-1\\right) \\pi\/2$ on $\\left( \\left( j-1\\right) \\pi\n\/n,j\\pi\/n\\right) .$ But this subinterval is the \"first half\" of the interval\n$\\left( \\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) $ where $j=2k-1,$ so\nreplacing $j$ in our formula for $\\arg\\alpha_{\\pi\/2}^{\\prime}\\left( t\\right)\n$ we obtain $\\pi-\\left( \\frac{n}{2}-1\\right) t+\\left( 2k-2\\right) \\pi\/2$,\nwhich is (\\ref{compass}). Moreover the non-zero magnitude of $\\alpha_{\\pi\n\/2}^{\\prime}\\left( t\\right) $ is $\\left\\vert \\alpha_{\\pi\/2}^{\\prime}\\left(\nt\\right) \\right\\vert =\\sqrt{2}\/\\left\\vert \\sqrt{1-e^{2int}}\\right\\vert .$\n\\end{proof}\n\n\\begin{corollary}\n\\label{cuspdir}For $\\beta\\in\\left( -\\pi\/2,\\pi\/2\\right) $, let $\\alpha\n_{\\beta}\\left( t\\right) =f_{\\beta}\\left( e^{it}\\right) ,$ and\n$k=1,2,...,n. $ Then the singular points\\ of $\\alpha_{\\beta}$ occurring at\nmultiples of $\\pi\/n\\ $are alternately cusps, and removable nodes. At\n$t=\\left( 2k-1\\right) \\pi\/n$, the discontinuity in $\\arg\\alpha_{\\beta\n}^{\\prime}$ is removable, and $\\alpha_{\\beta}\\left( \\left( 2k-1\\right)\n\\pi\/n\\right) $ is a removable node. In traversing from the cusp\n$\\alpha_{\\beta}\\left( \\left( 2k-2\\right) \\pi\/n\\right) $ to the cusp\n$\\alpha_{\\beta}\\left( 2k\\pi\/n\\right) ,$ $\\alpha_{\\beta}$ has total curvature\n$\\pi-2\\pi\/n.$ The total curvature over the first half of the interval, is\nequal to the total curvature over the second half of the interval $\\left(\n\\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) ,$ and is $\\pi\/2-\\pi\/n$.\n\\end{corollary}\n\n\\begin{proof}\nAs noted in the proof of (i) above, Lemma \\ref{cusps} still applies with\n(\\ref{compass}) holding on $\\left( \\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right)\n$ except at the center $t=\\left( 2k-1\\right) \\pi\/n$ of the interval$.$ We\ntherefore use (\\ref{compass}) on the punctured interval to evaluate the\nlimits\n\\begin{equation}\n\\arg\\alpha_{\\beta}^{\\prime}\\left( \\left( 2k-1\\right) \\pi\/n\\right)\n^{-}=\\arg\\alpha_{\\beta}^{\\prime}\\left( \\left( 2k-1\\right) \\pi\/n\\right)\n^{+}=\\pi\/2+\\left( 2k-1\\right) \\pi\/n, \\label{nodedir}%\n\\end{equation}\nso the exterior angle is $0$ and $\\alpha_{\\beta}\\left( \\left( 2k-1\\right)\n\\pi\/n\\right) $ is a removable node. We note that the discontinuity in\n$\\arg\\alpha_{\\beta}^{\\prime}$ at $\\left( 2k-1\\right) \\pi\/n$ is removable.\nAgain by Lemma \\ref{cusps}, $\\alpha_{\\beta}\\left( 2k\\pi\/n\\right) $ is a cusp\nand the axis has argument $2k\\pi\/n.$ The equation (\\ref{compass}) is monotonic\nin $t$, so the total change in $\\arg\\alpha_{\\beta}^{\\prime}$ is equal to\n$\\pi-2\\pi\/n$ on the interval $\\left( \\left( 2k-2\\right) \\pi\/n,2k\\pi\n\/n\\right) .$ Since (\\ref{compass}) is linear, half of this change, namely\n$\\pi\/2-\\pi\/n,$ occurs on each half of the interval $\\left( \\left(\n2k-2\\right) \\pi\/n,2k\\pi\/n\\right) .$\n\\end{proof}\n\n\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[\nheight=2.226in,\nwidth=4.3742in\n]%\n{fiveL2.eps}%\n\\caption{For $n=5$ with $\\beta=\\pi\/4$ (left) and $\\beta=2\\pi\/5$ (right), the\nnodes and cusps are indicated by a dots, and interlace with one another by\nargument. The argument of each node and cusp increases with $\\beta$ by\nequation (\\ref{cnarg}). Node and cusp locations are as described $\\left\\vert\n\\beta\\right\\vert <\\pi\/2$ in Theorem \\ref{cuspsnodes}. }%\n\\label{cuspnodepic5}%\n\\end{figure}\nBy Corollary \\ref{cuspdir}, the rosette harmonic mappings $f_{\\beta}$ for\n$\\left\\vert \\beta\\right\\vert <\\pi\/2$ have $n$-cusps, just as for the\n$n$-cusped hypocycloid mappings. Moreover with $n$ fixed, corresponding cusps\nfor different mappings have cusp axes that are parallel. This can be seen in\nFigure \\ref{cuspnodepic5} and in Figure \\ref{five} where cusps $\\alpha_{\\beta\n}\\left( 0\\right) $ have axes parallel to the real axis$.$ The parallelism of\naxes follows from the identical unit tangent values of the boundary\nextensions, which also explains the total curvature of $\\pi-2\\pi\/n$ from one\ncusp to the next described both in Proposition \\ref{hypocycloid} and Corollary\n\\ref{cuspdir}.\n\nThe last graph $f_{\\pi\/2}\\left( U\\right) $ in Figure \\ref{five} does not\nhave cusps, but nodes with an acute interior angle, which we now examine.\n\n\\begin{corollary}\n\\label{nodes}Let $\\alpha_{\\pi\/2}\\left( t\\right) =f_{\\pi\/2}\\left(\ne^{it}\\right) ,$ and let $k=1,2,...,n.$ On the first half of the interval\n$\\left( \\left( 2k-2\\right) \\pi\/n,2k\\pi\/n\\right) ,\\ $the total curvature of\n$\\alpha_{\\pi\/2}$ is $\\pi\/2-\\pi\/n,$ while $\\alpha_{\\pi\/2}$ is constant\notherwise$.$ There is a piecewise smooth parametrization $\\tilde{\\alpha}%\n_{\\pi\/2},$ with the same graph as $\\alpha_{\\pi\/2}$ over $\\partial U$, and just\n$n$ singular points at $\\tilde{\\alpha}_{\\pi\/2}\\left( 2k\\pi\/n\\right)\n=\\alpha_{\\pi\/2}\\left( 2k\\pi\/n\\right) $ which are nodes of $\\tilde{\\alpha\n}_{\\pi\/2}$ with interior angle $\\pi\/2-\\pi\/n.$\n\\end{corollary}\n\n\\begin{proof}\nSince $\\alpha_{\\pi\/2}^{\\prime}\\left( t\\right) =0$ on $\\left( \\left(\n2k-1\\right) \\pi\/n,2k\\pi\/n\\right) ,$ the singularities are not isolated, and\nmoreover these intervals are arcs of constancy for the boundary function of\n$f_{\\pi\/2}$. We define $\\tilde{\\alpha}_{\\pi\/2},$ defined piecewise on\n$[0,2\\pi)$ by%\n\\begin{equation}\n\\tilde{\\alpha}_{\\pi\/2}\\left( t\\right) =\\alpha_{\\pi\/2}\\left( \\left(\nk-1\\right) \\pi\/n+t\/2\\right) ,\\text{ }t\\in\\lbrack\\left( 2k-2\\right)\n\\pi\/n,2k\\pi\/n). \\label{halfspeed}%\n\\end{equation}\n\n\nThen on each interval in (\\ref{halfspeed}) the curve $\\tilde{\\alpha}_{\\pi\/2} $\nhas the values\\newline$\\alpha_{\\pi\/2}\\left( \\left[ \\left( 2k-2\\right)\n\\pi\/n,\\left( 2k-1\\right) \\pi\/n\\right] \\right) $. Moreover, $\\tilde{\\alpha\n}_{\\pi\/2}\\left( 2k\\pi\/n\\right) =\\alpha_{\\pi\/2}\\left( 2k\\pi\/n\\right) .$\nThus $\\tilde{\\alpha}_{\\pi\/2}$ is also continuous, with the same image as\n$\\alpha_{\\pi\/2}$ on $\\left[ 0,2\\pi\\right] .$ We compute\n\\begin{align*}\n\\lim_{t\\rightarrow2k\\pi\/n^{-}}\\arg\\tilde{\\alpha}_{\\pi\/2}^{\\prime}\\left(\nt\\right) & =\\lim_{t\\rightarrow2k\\pi\/n^{-}}\\arg\\alpha_{\\pi\/2}^{\\prime\n}\\left( \\left( k-1\\right) \\pi\/n+t\/2\\right) \\\\\n& =\\lim_{t^{\\prime}\\rightarrow\\left( 2k-1\\right) \\pi\/n}2k\\pi\/2-\\left(\n\\frac{n}{2}-1\\right) t^{\\prime}=k\\pi-\\left( \\frac{n}{2}-1\\right) \\left(\n2k-1\\right) \\frac{\\pi}{n}.\n\\end{align*}\nand using the formula for $\\arg\\tilde{\\alpha}_{\\pi\/2}^{\\prime}$ on $\\left(\n2k\\pi\/n,2\\left( k+1\\right) \\pi\/n\\right) $ obtain\n\\begin{align*}\n\\lim_{t\\rightarrow2k\\pi\/n^{+}}\\arg\\tilde{\\alpha}_{\\pi\/2}^{\\prime}\\left(\nt\\right) & =\\lim_{t\\rightarrow2k\\pi\/n^{+}}\\arg\\alpha_{\\pi\/2}^{\\prime\n}\\left( k\\pi\/n+t\/2\\right) \\\\\n& =\\lim_{t^{\\prime}\\rightarrow2k\\pi\/n}\\left( 2k+2\\right) \\pi\/2-\\left(\n\\frac{n}{2}-1\\right) t^{\\prime}\\\\\n& =\\left( k+1\\right) \\pi-\\left( \\frac{n}{2}-1\\right) 2k\\frac{\\pi}{n}.\n\\end{align*}\nThe difference $\\arg\\tilde{\\alpha}_{\\pi\/2}^{\\prime}\\left( 2k\\pi\/n\\right)\n^{+}-\\arg\\tilde{\\alpha}_{\\pi\/2}^{\\prime}\\left( 2k\\pi\/n\\right) ^{-}$ is thus\n$\\pi-\\left( \\frac{n}{2}-1\\right) \\pi\/n=\\pi\/2+\\pi\/n,$ so we have a node with\ninterior angle $\\pi\/2-\\pi\/n.$\n\\end{proof}\n\n\\begin{remark}\n\\label{node}The node $\\tilde{\\alpha}_{\\pi\/2}\\left( 2k\\pi\/n\\right) $ can be\nwritten $e^{i2k\\pi\/n}\\alpha_{\\pi\/2}\\left( t\\right) $ for any \\newline%\n$t\\in\\left[ \\left( 2k-1\\right) \\pi\/n,2k\\pi\/n\\right] ,$ where $\\alpha\n_{\\pi\/2}$ is constant$.$ We write the nodes of $\\tilde{\\alpha}_{\\pi\/2}$ as\n\\newline$e^{i2k\\pi\/n}f_{\\pi\/2}\\left( 1\\right) $ when convenient$,$ and refer\nto nodes of $\\tilde{\\alpha}_{\\pi\/2}$ as the nodes of $f_{\\pi\/2}.$\n\\end{remark}\n\n\\begin{example}\nWhen $\\beta=\\pi\/2$ and with $n=5,$ then on intervals $\\left( 0,\\pi\/5\\right)\n,$ $\\left( 2\\pi\/5,3\\pi\/5\\right) ,$ ... the boundary arcs $e^{i\\pi\/4}%\nh_{5}\\left( e^{it}\\right) $ and $e^{-i\\pi\/4}\\overline{g_{5}}\\left(\ne^{-it}\\right) $ are translates of one another, and on intervals $\\left(\n\\pi\/5,2\\pi\/5\\right) ,$ $\\left( 3\\pi\/5,5\\pi\/5\\right) ,$ ..., the boundary\narcs are mirrors of one another. Figure \\ref{transmirror} (right) shows the\narc of constancy $\\left( \\pi\/5,2\\pi\/5\\right) $ on which $\\ e^{i\\pi\/4}%\nh_{5}\\left( e^{it}\\right) $ and $e^{-i\\pi\/4}\\overline{g_{5}}\\left(\ne^{-it}\\right) $ are mirror images, and where $f_{\\pi\/2}\\left(\ne^{it}\\right) $ is equal to the node $e^{i2\\pi\/5}f_{\\pi\/2}\\left( 1\\right) $\n(indicated with a larger dot in the second quadrant).\n\\begin{figure}[ptb]%\n\\centering\n\\includegraphics[\nheight=2.4059in,\nwidth=5.0548in\n]%\n{transmir5.eps}%\n\\caption{Images of sectors in $U$ of $e^{i\\pi\/4}h_{5}$ and $e^{-i\\pi\n\/4}\\overline{g_{5}}$ with $\\arg z\\in\\left( 0,\\pi\/5\\right) $ shaded (left)\nand $\\arg z\\in\\left( \\pi\/5,2\\pi\/5\\right) $ shaded (right), for $n=5.$ The\nbounding curves (thickened)$,$ are translates (left) and reflections (right).\nA tangent is indicated for $h_{5}\\left( e^{it}\\right) $ and $\\overline\n{g_{5}\\left( e^{it}\\right) }$ in each case, along with a portion of\n$f_{\\pi\/2}\\left( e^{it}\\right) $, which is constant (right) on $\\left(\n\\pi\/5,2\\pi\/5\\right) .$}%\n\\label{transmirror}%\n\\end{figure}\n\n\\end{example}\n\nWe now complete our description of the symmetries within the graphs of the\nrosette mappings.\n\n\\begin{corollary}\n\\label{noreflect}Let $\\beta\\in%\n\\mathbb{R}\n$ and $n\\geq3.$ If $\\beta$ is not a multiple of $\\pi\/2,$ then the image set\n$f_{\\beta}\\left( U\\right) $ does not have reflectional symmetry. In this\ncase, $f_{\\beta}\\left( U\\right) $ has symmetry group $C_{n}.$ Otherwise,\n$\\beta$ is a multiple of $\\pi\/2,$ and $f_{\\beta}\\left( U\\right) $ has\nsymmetry group $D_{n}.$\n\\end{corollary}\n\n\\begin{proof}\nAll rosette mappings have at least $n$ fold rotational symmetry, by\nProposition \\ref{symmetry} (ii). Since $f_{\\beta}$ has either exactly $n$\ncusps, or exactly $n$ non-removable nodes, $f_{\\beta}\\left( U\\right) $\ncannot have a higher order of symmetry. For $\\left\\vert \\beta\\right\\vert\n\\in\\left( 0,\\pi\/2\\right) ,$ the axis of the cusp through $f_{\\beta}\\left(\n1\\right) $ is parallel to the real axis, while the radial ray through $0$ and\n$f_{\\beta}\\left( 1\\right) $ has argument $\\psi=\\arg f_{\\beta}\\left(\n1\\right) ,$ distinct for each $\\beta$ in $\\left( -\\pi\/2,\\pi\/2\\right) $ by\nequation (\\ref{cnarg})$.$ Moreover as noted in Lemma \\ref{convexconcave},\n$\\psi$ is acute$,$ and has the same sign as $\\beta.$ Thus if $\\left\\vert\n\\beta\\right\\vert \\in\\left( 0,\\pi\/2\\right) ,$ then any reflection of\n$f_{\\beta}\\left( U\\right) $ results changing the sign of the angle between\nthe reflected cusp axis and the reflected radial ray, resulting in a distinct\nreflected image set. Thus the symmetry group of $f_{\\beta}\\left( U\\right) $\nis $C_{n}$ for $\\left\\vert \\beta\\right\\vert \\in\\left( 0,\\pi\/2\\right) .$ This\nfact extends by formula (\\ref{trans}) to any real $\\beta$ that is not a\nmultiple of $\\pi\/2.$ If $\\beta=0$ or $\\beta=\\pi\/2$ we already established that\n$f_{\\beta}\\left( U\\right) $ has reflectional symmetry. We conclude that the\nsets $f_{0}\\left( U\\right) $ and $f_{\\pi\/2}\\left( U\\right) $ have dihedral\nsymmetry group $D_{n}$. If $\\beta=l\\pi\/2 $ for some $l\\in%\n\\mathbb{Z}\n$ then by formula (\\ref{trans}), $f_{\\beta}\\left( U\\right) $ is a rotation\nof either $f_{\\pi\/2}\\left( U\\right) $ or $f_{0}\\left( U\\right) ,$ and thus\nhas rotational symmetry also.\n\\end{proof}\n\n\\begin{corollary}\n\\label{differ}Let $n\\geq3.$ For distinct $\\beta$ and $\\beta^{\\prime}$ in the\ninterval $(-\\pi\/2,\\pi\/2],$ the image sets $f_{\\beta}\\left( U\\right) $ and\n$f_{\\beta^{\\prime}}\\left( U\\right) $ are not scalings or rotations of one\nanother. Moreover with the parameter $\\beta$ within the set $\\left[\n0,\\pi\/2\\right] ,$ all images of the unit disk under a rosette harmonic\nmapping are obtained, up to rotation and reflection.\n\\end{corollary}\n\n\\begin{proof}\nThe proof of Corollary \\ref{noreflect} shows that if $\\beta\\in\\left(\n-\\pi\/2,\\pi\/2\\right) ,$ the angle between the cusp axis through $f_{\\beta\n}\\left( 1\\right) $ and the radial line from $0$ to $f_{\\beta}\\left(\n1\\right) $ intersect at an angle that is distinct for each choice of $\\beta.\n$ Thus $f_{\\beta}\\left( U\\right) $ is different from any rotation or scaling\nof $f_{\\beta^{\\prime}}\\left( U\\right) $ for any other $\\beta^{\\prime}%\n\\in\\left( -\\pi\/2,\\pi\/2\\right) .$ Moreover $f_{\\pi\/2}$ is the only $f_{\\beta\n}$ without cusps for $\\beta\\in(-\\pi\/2,\\pi\/2].$ To prove the second statement,\nlet $\\tilde{\\beta}\\in%\n\\mathbb{R}\n\\backslash(-\\pi\/2,\\pi\/2].$ Upon reducing $\\tilde{\\beta}$ modulo $\\pi,$ we\nobtain an equivalent $\\beta\\equiv\\tilde{\\beta}$ $(\\operatorname{mod}$ $\\pi)$\nwhere $\\beta\\in(-\\pi\/2,\\pi\/2].$ By possibly repeated application of\n(\\ref{rotbet}) in Proposition \\ref{parametrize} (i), $f_{\\tilde{\\beta}}\\left(\nz\\right) $ is obtained as a pre- and post- composition of a rotation with\n$f_{\\beta}\\left( z\\right) .$ Thus it is sufficient to consider $\\beta\n\\in(-\\pi\/2,\\pi\/2].$ If $\\beta<0,$ then by Proposition \\ref{parametrize} (ii),\n$f_{\\beta}\\left( z\\right) $ is a reflection in the real axis of $f_{-\\beta\n}\\left( z\\right) $ where $-\\beta\\in\\left[ 0,\\pi\/2\\right] .$\n\\end{proof}\n\nWe turn to describing features of the graphs $f_{\\beta}\\left( U\\right) $ for\n$\\beta$ restricted to $(-\\pi\/2,\\pi\/2].$ We may use equation (\\ref{trans}) to\nidentify cusps and nodes for $\\beta$ outside this interval, but we make the\ncautionary observation that while for $\\beta\\in\\left( -\\pi\/2,\\pi\/2\\right) $\n$f_{\\beta}\\left( 1\\right) $ is a cusp and $f_{\\beta}\\left( e^{i\\pi\n\/n}\\right) $ is a node, adding $\\pi$ to $\\beta$ results in $f_{\\beta+\\pi\n}\\left( 1\\right) $ being a node and $f_{\\beta+\\pi}\\left( e^{i\\pi\/n}\\right)\n$ being a cusp. Quantities such as the magnitude of a cusp, the magnitude of a\nnode, and the distribution of angle between cusps neighboring nodes however$,$\nare independent of the interval to which $\\beta$ is restricted$.$ These\nquantities are derived in Theorem \\ref{cuspsnodes}, with the help of Lemma\n\\ref{convexconcave}.\n\nTheorem \\ref{cuspsnodes} shows that for fixed $n,$ the image $f_{0}\\left(\nU\\right) $ has maximal diameter, and $diam\\left( f_{\\left\\vert\n\\beta\\right\\vert }\\left( U\\right) \\right) $ decreases with $\\left\\vert\n\\beta\\right\\vert \\in\\left( 0,\\pi\/2\\right) .\\,$This is illustrated in Figure\n\\ref{five}, where all four graphs are plotted on the same scale. We also see\nthat for $\\beta=0,$ the cusps and nodes are equally spaced with angular\nseparation $\\pi\/n,$ but this becomes increasingly unbalanced as $\\left\\vert\n\\beta\\right\\vert $ increases to $\\pi\/2.$ As $\\beta\\nearrow\\pi\/2^{-},$ each\nnode approaches the subsequent\\footnote{Subsequent cusp (or node) here\nindicates the cusp (or node) with the next largest argument.} cusp, as shown\nin Theorem \\ref{cuspsnodes} (ii).\n\n\\begin{theorem}\n\\label{cuspsnodes}Let $n\\geq3,$ $\\beta\\in(-\\pi\/2,\\pi\/2],$ and let\n$\\alpha_{\\beta}\\left( t\\right) =f_{\\beta}\\left( e^{it}\\right) $ where\n$f_{\\beta}$ is a rosette harmonic mapping$.$ Recall the constant $K_{n}$\ndefined in Lemma \\ref{convexconcave}$.$ \\newline(i)$\\ $If $\\beta\\in\\left(\n-\\pi\/2,\\pi\/2\\right) $ then the magnitude of the cusps decreases strictly if\n$\\left\\vert \\beta\\right\\vert \\in\\lbrack0,\\pi\/2),$ from a maximum $K_{n}%\n\\sec\\left( \\frac{\\pi}{2n}\\right) ,$ and with infimum $K_{n}\\left(\n1+\\tan\\left( \\frac{\\pi}{2n}\\right) \\right) .$ The removable nodes have\nmagnitudes that increase strictly if $\\left\\vert \\beta\\right\\vert \\in\n\\lbrack0,\\pi\/2),$ from the minimum $K_{n}\\left( 1+\\tan\\left( \\frac{\\pi}%\n{2n}\\right) \\right) $ and with supremum $K_{n}\\sec\\left( \\frac{\\pi}%\n{2n}\\right) .$\\newline(ii) If $\\beta\\in\\left( -\\pi\/2,\\pi\/2\\right) $ then\nthe angle between a cusp neighboring node is\n\\begin{equation}\n\\pi\/n\\pm\\arctan\\left( \\frac{2\\tan\\left( \\frac{\\pi}{2n}\\right) \\sin\\left(\n\\beta\\right) }{1-\\tan^{2}\\left( \\frac{\\pi}{2n}\\right) }\\right) ,\n\\label{sep}%\n\\end{equation}\nwhere we use the positive sign when the node has the more positive argument.\nFor $\\beta=0,$ the cusps and neighboring nodes are separated by equal angles\nof $\\pi\/n$. The difference in arguments of a cusp and subsequent node\nincreases from $0$ to $2\\pi\/n$ with $\\beta\\in\\left( -\\pi\/2,\\pi\/2\\right) $.\n\\newline(iii) For $\\beta=\\pi\/2,$ the $n$ nodes of the rosette mapping\n$f_{\\pi\/2}$ have magnitude \\linebreak$K_{n}\\sec\\frac{\\pi}{2n},$ and for the\nnode $f_{\\pi\/2}\\left( 1\\right) $ we have $\\arg\\left( f_{\\pi\/2}\\left(\n1\\right) \\right) =\\pi\/4-\\pi\/\\left( 2n\\right) $.\n\\end{theorem}\n\n\\begin{proof}\nWe begin with (i). Due to the $\\cos\\beta$ term in (\\ref{cnmag}), the cusp\n$\\left\\vert f_{\\beta}\\left( 1\\right) \\right\\vert $ is decreasing with\n$\\left\\vert \\beta\\right\\vert $ and the node $\\left\\vert f_{\\beta}\\left(\ne^{i\\pi\/n}\\right) \\right\\vert $ is increasing with $\\left\\vert \\beta\n\\right\\vert .$ Note that\n\\[\n\\sec^{2}\\left( \\frac{\\pi}{2n}\\right) \\pm2\\tan\\left( \\frac{\\pi}{2n}\\right)\n\\cos\\beta=1+\\tan^{2}\\left( \\frac{\\pi}{2n}\\right) \\pm2\\tan\\left( \\frac{\\pi\n}{2n}\\right) =\\left( 1\\pm\\tan\\left( \\frac{\\pi}{2n}\\right) \\right) ^{2}.\n\\]\nIn view of this equation, the maximum of $\\left\\vert f_{\\beta}\\left(\n1\\right) \\right\\vert $ is $K_{n}\\left( 1+\\tan\\left( \\frac{\\pi}{2n}\\right)\n\\right) $ when $\\beta=0,$ and $\\left\\vert f_{\\beta}\\left( 1\\right)\n\\right\\vert $ approaches $K_{n}\\sec\\left( \\frac{\\pi}{2n}\\right) $ as $\\beta$\napproaches $\\pi\/2.$ Similarly, the minimum of $\\left\\vert f_{\\beta}\\left(\ne^{i\\pi\/n}\\right) \\right\\vert $ is $K_{n}\\left( 1-\\tan\\left( \\frac{\\pi}%\n{2n}\\right) \\right) $ when $\\beta=0,$ and approaches $K_{n}\\left(\n\\sec\\left( \\frac{\\pi}{2n}\\right) \\right) $ as an upper bound as $\\left\\vert\n\\beta\\right\\vert $ approaches $\\pi\/2,$ proving (i)$.$ For (ii) formula\n(\\ref{cnarg}) shows us $\\arg f_{\\beta}\\left( 1\\right) $ and $\\arg f_{\\beta\n}\\left( e^{i\\pi\/n}\\right) $ are increasing with $\\beta$ on $\\left(\n-\\pi\/2,\\pi\/2\\right) .$ Clearly $\\arg\\left( f_{0}\\left( 1\\right) \\right)\n=0$ and $\\arg f_{0}\\left( e^{i\\pi\/n}\\right) =\\pi\/n.$ Additionally we have\n$\\arg\\left( f_{\\beta}\\left( 1\\right) \\right) <\\pi\/n<\\arg f_{\\beta}\\left(\ne^{i\\pi\/n}\\right) \\,\\ $for $\\beta\\in\\left( 0,\\pi\/2\\right) .$ We compute the\nangle between this cusp and node to be $\\arg f_{\\beta}\\left( e^{i\\pi\n\/n}\\right) -\\arg f_{\\beta}\\left( 1\\right) =\\pi\/n+\\psi^{\\prime}-\\psi,$ and\nwe use the arctangent formula $\\arctan\\psi^{\\prime}-\\arctan\\psi=\\arctan\\left(\n\\frac{\\psi^{\\prime}-\\psi}{1+\\psi^{\\prime}\\psi}\\right) $ where $\\psi^{\\prime}$\nand $\\psi$ are as defined in equation (\\ref{cnarg}). Thus after a short\ncalculation we obtain formula (\\ref{sep}). This difference increases from\n$\\pi\/n$ and approaches $2\\pi\/n$ as $\\beta$ increases through the interval\n$[0,\\pi\/2)$. Now suppose that $\\beta\\in\\left( -\\pi\/2,0\\right) .$ The\nreflection of the node $f_{\\beta}\\left( e^{i\\pi\/n}\\right) $ in the real axis\nis the node $f_{-\\beta}\\left( e^{-i\\pi\/n}\\right) $ of $f_{-\\beta}$ and the\nreflection of $f_{\\beta}\\left( 1\\right) $ is the cusp $f_{-\\beta}\\left(\n1\\right) .$ From rotational symmetry, the angular difference between $\\arg\nf_{-\\beta}\\left( 1\\right) $ and $\\arg f_{-\\beta}\\left( e^{-i\\pi\/n}\\right)\n$ is the same as the that of $\\arg f_{-\\beta}\\left( e^{i2\\pi\/n}\\right) $ and\n$\\arg f_{-\\beta}\\left( e^{i\\pi\/n}\\right) .$ Again using rotational symmetry,\nthis is $2\\pi\/n-\\left( \\psi-\\psi^{\\prime}\\right) =\\pi\/n-\\arctan\\left(\n\\frac{\\psi^{\\prime}-\\psi}{1+\\psi^{\\prime}\\psi}\\right) =\\pi\/n-\\arctan\\left(\n\\frac{2\\tan\\left( \\frac{\\pi}{2n}\\right) \\sin\\left( -\\beta\\right) }%\n{1-\\tan^{2}\\left( \\frac{\\pi}{2n}\\right) }\\right) =\\pi\/n+\\arctan\\left(\n\\frac{2\\tan\\left( \\frac{\\pi}{2n}\\right) \\sin\\left( \\beta\\right) }%\n{1-\\tan^{2}\\left( \\frac{\\pi}{2n}\\right) }\\right) $. For (iii), with\n$\\beta=\\pi\/2, $ we noted in Remark \\ref{node} after Corollary \\ref{nodes} that\nthe nodes can be expressed as rotations of $f_{\\pi\/2}\\left( 1\\right) ,$ for\nwhich we observed the stated quantities in Lemma \\ref{convexconcave} (i)$.$\n\\end{proof}\n\nThe next example illustrates the separation of cusps and nodes as it varies\nwith $\\beta,$ as described in Theorem \\ref{cuspsnodes} (see also Figure\n\\ref{cuspnodepic5})$.$ The relative proximity of a node and neighboring cusp\ncoupled with equal total curvature of the boundary curve between any node and\ncusp (Corollary \\ref{cuspdir}) gives rise to the appearance of a cresting wave\nat the cusp.\n\n\\begin{example}\n\\label{cuspnody}For $n=5,$ the arguments of the nodes and cusps of $f_{0}$ are\nequally spaced by $\\pi\/n=\\pi\/5.$ By Lemma \\ref{convexconcave}, as $\\beta\n\\in\\lbrack0,\\pi\/2)$ increases, the separation of a cusp and subsequent node\ngrows from $\\pi\/5=36^{\\circ}$ towards $2\\pi\/5=72^{\\circ}. $ For $\\beta=\\pi\/4$\nthis separation is $\\pi\/5+\\arctan\\sqrt{5\/2-\\sqrt{5}}\\approx63^{\\circ},$ while\nfor $\\beta=2\\pi\/5$ it grows to $\\pi\/5+\\arctan\\sqrt{5\/4-\\sqrt{5}\/4}%\n\\approx71^{\\circ}$ (see Figure \\ref{cuspnodepic5}). For a specific cusp or\nnode $f_{\\beta}\\left( e^{ij_{0}\\pi\/n}\\right) ,$ $\\arg f_{\\beta}\\left(\ne^{ij_{0}\\pi\/n}\\right) $ increases with $\\beta$ (see proof of Theorem\n\\ref{cuspsnodes} (ii))$,\\ $which is also illustrated in Figure\n\\ref{cuspnodepic5} as $\\beta$ increases from $\\pi\/4$ to $2\\pi\/5.$ By Corollary\n\\ref{cuspdir}, the total curvature of the boundary of $f_{\\beta}$ between\nneighboring cusps is $\\pi-2\\pi\/n=3\\pi\/5=108^{\\circ}.$ The total curvature of\nthe boundary from a cusp to a neighboring node is half of this, namely\n$54^{\\circ}$. Figure \\ref{cuspnodepic5} also indicates the phenomenon\ndescribed in Theorem \\ref{cuspsnodes}, that while both nodes and cusps \"rotate\ncounterclockwise\" as $\\beta\\in\\left( 0,\\pi\/2\\right) $ increases, the nodes\ndo so at a greater rate..\n\\end{example}\n\nFinally we point out that the argument of the boundary curve fails to be\nstrictly increasing on the whole of $\\partial U,$ for $\\beta$ with $\\left\\vert\n\\beta\\right\\vert \\in\\left( 0,\\pi\/2\\right) .$\n\n\\begin{corollary}\n\\label{parallel}Let $n\\geq3$, $\\beta\\in\\left( -\\pi\/2,\\pi\/2\\right) $ and\n$\\alpha_{\\beta}$ be the boundary function of the rosette harmonic mapping\n$f_{\\beta}.$ Then for each $k=1,2,...,n,$ $\\arg\\alpha_{\\beta}\\left( \\left(\n2k-2\\right) \\pi\/n\\right) ,\\ $\\newline$\\arg\\alpha_{\\beta}\\left( \\left(\n2k-1\\right) \\pi\/n\\right) ,$ and $\\arg\\alpha_{\\beta}\\left( 2k\\pi\/n\\right) $\noccur in increasing order$,$ but there exists an interval on which\n$\\arg\\left( \\alpha_{\\beta}\\left( t\\right) \\right) $ is decreasing.\n\\end{corollary}\n\n\\begin{proof}\nTheorem \\ref{cuspsnodes} (ii) shows that the angle $\\gamma$ between\\ a cusp\nand subsequent node is given by formula (\\ref{sep}) which belongs to $\\left(\n0,2\\pi\/n\\right) $. Because the cusps are separated by angle $2\\pi\/n,$ the\nangle from a cusp to a subsequent node is then $2\\pi\/n-\\gamma\\in\\left(\n0,2\\pi\/n\\right) .$ Finally with $\\beta\\in\\left( 0,\\pi\/2\\right) ,$ suppose\nthat $\\arg\\alpha_{\\beta}\\left( -\\epsilon\\right) \\leq\\psi=\\arg\\alpha_{\\beta\n}\\left( 0\\right) $ for some $-\\epsilon\\in\\left( -\\pi\/n,0\\right) ,$ but for\nwhich $\\arg\\alpha_{\\beta}^{\\prime}\\left( -\\epsilon\\right) <\\tan\\psi.$ Such\nan $\\epsilon$ exists because $\\alpha_{\\beta}$ satisfies (\\ref{cuspslopes})\nwith $k=0,$ so $\\arg\\alpha_{\\beta}^{\\prime}\\left( 0\\right) ^{-}=0.$ Since\n$\\arg\\alpha_{\\beta}^{\\prime}\\left( t\\right) $ is decreasing, we have\n$\\arg\\alpha_{\\beta}\\left( t\\right) <\\psi$ for all $t\\in\\left(\n-\\epsilon\/2,0\\right) ,$ and so $\\alpha_{\\beta}\\left( t\\right) $ lies in a\nhalf-plane formed by a line parallel to to $\\arg z=\\psi,$ but with smaller\nimaginary part, and this half plane does not contain $\\alpha_{\\beta}\\left(\n0\\right) .$ This contradicts continuity of $\\alpha_{\\beta}\\left( t\\right)\n\\rightarrow\\alpha_{\\beta}\\left( 0\\right) $ as $t\\nearrow0^{-}.$ Thus there\nis an interval $\\left( -\\epsilon,0\\right) $ on which $\\arg\\alpha_{\\beta\n}\\left( t\\right) >\\psi.$ On this interval, $\\arg\\alpha_{\\beta}\\left(\nt\\right) $ decreases to $\\psi,$ and so rotates negatively relative to the\norigin. A similar argument holds when $\\beta\\in\\left( -\\pi\/2,0\\right) .$\n\\end{proof}\n\n\\section{Univalence and Fundamental Sets}\n\nOur approach to proving the univalence of $f_{\\beta}$ is to use the argument\nprinciple for harmonic functions. We note that various proofs of univalence\nfor rosette mappings $f_{\\beta}$ are possible. The following theorem describes\nthe winding number of the boundary curve $\\alpha\\left( t\\right) =f_{\\beta\n}\\left( e^{it}\\right) ,$ so that we can apply the argument principle.\n\n\\begin{lemma}\n\\label{unibdry}For fixed $\\beta\\in\\lbrack0,\\pi\/2),$ the boundary\n$\\alpha_{\\beta}\\left( t\\right) =f_{\\beta}\\left( e^{it}\\right) ,$\n$t\\in\\partial U$ is a simple, positively oriented closed curve. While\n$\\alpha_{\\pi\/2}$ has arcs of constancy, the parametrization $\\tilde{\\alpha\n}_{\\pi\/2}$ of equation (\\ref{halfspeed}) is a simple, positively oriented\nclosed curve on $\\partial U.$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\beta\\in\\left( 0,\\pi\/2\\right) .$ We give a separate argument for\n$\\beta=0$ and for $\\beta=\\pi\/2.$ We first show that $\\alpha_{\\beta}$ is one to\none when restricted to $\\left( 0,2\\pi\/n\\right) ,$ and that the boundary\ncurve portion $\\alpha_{\\beta}\\left( \\left( 0,2\\pi\/n\\right) \\right) $ lies\nin a sector $S.$ We then show that the portion of the graph of $\\alpha_{\\beta\n}$ restricted to the interval $\\left( 2k\\pi\/n,2\\left( k+1\\right)\n\\pi\/n\\right) $ is contained within the set $e^{i2k\\pi\/n}S.$ The sets\n$\\left\\{ e^{i2k\\pi\/n}S:k=0,1,...,n-1\\right\\} $ are then seen to be pairwise\ndisjoint, so that the curve $\\alpha_{\\beta}$ has no self intersections on\n$[0,2\\pi),$ and we conclude $\\alpha_{\\beta}$ is a simple closed curve on\n$\\partial U.$\n\nDefine $L_{k}$ to be the axis of the cusp $\\alpha_{\\beta}\\left( 2k\\pi\\right)\n,$ so with argument equal to $2k\\pi\/n,$ $k=0,1,..,n-1.$ Recall from Theorem\n(\\ref{fbdry}) that $\\alpha_{\\beta}$ satisfies (\\ref{compass}) (which holds\nexcept at $t=\\left( 2k+1\\right) \\pi\/n$). We begin with $\\alpha_{\\beta}$ on\nthe interval $\\left( 0,2\\pi\/n\\right) .$ Let $p$ be the intersection of\n$L_{0}$ (parallel to the real axis) and $L_{1}.$ These non-collinear lines\nform the boundary of four unbounded open connected sectors, and we define $S$\nto be the sector with vertex $p$ \\ for which the (distinct) cusps\n$\\alpha_{\\beta}\\left( 0\\right) \\in L_{0}$ and $\\alpha_{\\beta}\\left(\n2\\pi\/n\\right) \\in L_{1}$ belong to $\\partial S.$ By equation (\\ref{compass}%\n)$,$ $\\arg\\alpha_{\\beta}^{\\prime}\\left( t\\right) $ decreasing and\n$\\arg\\alpha_{\\beta}^{\\prime}\\left( 0\\right) ^{+}=\\pi.$ Thus the curve\n$\\alpha_{\\beta}$ lies \"above\" $L_{0}$. If $\\alpha_{\\beta}$ were to cross\n$L_{1}$ at some $t_{1}\\in\\left( 0,2\\pi\/n\\right) ,$ then $\\arg\\alpha_{\\beta\n}^{\\prime}\\left( t_{1}^{\\prime}\\right) <2\\pi\/n$ for some $t_{1}^{\\prime}%\n\\in\\left( t_{1},2\\pi\/n\\right) $ in order for $\\arg\\alpha_{\\beta}^{\\prime\n}\\left( 2\\pi\/n\\right) ^{-}=2\\pi\/n$ in equation (\\ref{cuspslopes}) to hold.\nHowever $\\arg\\alpha_{\\beta}^{\\prime}\\left( \\left( 0,2\\pi\/n\\right) \\right)\n\\subset\\left( 2\\pi\/n,\\pi\\right) $ so no such intersection can occur, and we\nconclude that $\\alpha_{\\beta}$ restricted to $\\left( 0,2\\pi\/n\\right) $ lies\nwithin the region $S.$ Moreover, because $\\arg\\alpha_{\\beta}^{\\prime}\\left(\nt\\right) $ decreases strictly, with total change $\\pi-2\\pi\/n<\\pi,$\n$\\alpha_{\\beta}$ must be one to one on $\\left( 0,2\\pi\/n\\right) .$ Now let\n$k=1,2,..,n,$ and consider the curve $\\alpha_{\\beta}$ restricted to $\\left(\n2k\\pi\/n,2\\left( k+1\\right) \\pi\/n\\right) .$ By rotational symmetry of\n$f_{\\beta}$, $\\alpha_{\\beta}\\left( t+2k\\pi\/n\\right) =e^{i2k\\pi\/n}%\n\\alpha_{\\beta}\\left( t\\right) $, $\\alpha_{\\beta}$ is also one to one on\n$\\left( 2k\\pi\/n,2\\left( k+1\\right) \\pi\/n\\right) $, and the graph\n$\\alpha_{\\beta}\\left( \\left( 2k\\pi\/n,2\\left( k+1\\right) \\pi\/n\\right)\n\\right) \\subseteq e^{i2k\\pi\/n}S.$ We complete the proof for $\\beta\\in\\left(\n0,\\pi\/2\\right) $ by showing that the sets $\\left\\{ e^{i2k\\pi\/n}%\nS:k=0,1,...,n-1\\right\\} $ are pairwise disjoint. Because of rotational\nsymmetry, the line $e^{i2k\\pi\/n}L_{0}$ passes through $e^{i2k\\pi\/n}%\n\\alpha_{\\beta}\\left( 0\\right) =\\alpha_{\\beta}\\left( 2k\\pi\/n\\right) ,$ a\ncusp, and $e^{i2k\\pi\/n}L_{0}$ has argument $2k\\pi\/n,$ so\\linebreak%\n\\ $e^{i2k\\pi\/n}L_{0}=L_{k}.$ Similarly $e^{i2k\\pi\/n}L_{1}=L_{k+1}.$ Thus\n$e^{i2k\\pi\/n}S$ is a sector with sides $L_{k}$ and $L_{k+1},$ and vertex at\ntheir point of intersection $e^{i2k\\pi\/n}p.$ The points $\\left\\{ e^{i2k\\pi\n\/n}p:k=0,..,n-1\\right\\} $ form the vertices of a regular $n$-gon, that we\ndenote by $P.$ Then $P$ is centered on the origin, and we have\n$\\operatorname{Im}p>0,$ since the cusp $\\alpha_{\\beta}\\left( 0\\right) $ has\nargument $\\psi=\\arg\\alpha_{\\beta}\\left( 0\\right) \\in\\left( 0,\\pi\/4\\right)\n$ by Lemma \\ref{convexconcave} (ii). Note also that $L_{n}=L_{0},$ and\n$L_{n-1}=e^{-i2\\pi\/n}L_{0}.$ Thus $L_{0}$ and $L_{n-1}$ intersect at\n$e^{-i2\\pi\/n}p,$ with $\\operatorname{Im}\\left( e^{-i2\\pi\/n}p\\right)\n=\\operatorname{Im}\\left( p\\right) .$ Therefore $p$ is in the second\nquadrant, and $e^{-i2\\pi\/n}p$ is in the first quadrant. We conclude that one\nbounding side of the sector $S$ is $\\left\\{ z\\in L_{0}:\\arg z\\leq\\arg\np\\right\\} $. By rotational symmetry, the second side of $S$ is\\linebreak%\n\\ $\\left\\{ z\\in L_{1}:\\arg z\\leq2\\pi\/n+\\arg p\\right\\} $. Thus the rays\n$\\left\\{ z\\in L_{k}:\\arg z\\leq2k\\pi\/n+\\arg p\\right\\} $ that extend the sides\nof $P,$ each originating at $e^{i2k\\pi\/n}p$ and extending in the direction of\ndecreasing argument, form the boundaries of the sectors $e^{i2k\\pi\/n}S,$ where\n$k=0,1,...,n-1$ (see Figure 7 (left) where $S$ is shaded).%\n\n{\\includegraphics[\nheight=2.2364in,\nwidth=2.4561in\n]%\n{sectorsUni.eps}%\n}\n\\ \\\n{\\includegraphics[\nheight=2.1724in,\nwidth=2.1136in\n]%\n{sectorsPrime.eps}%\n}\n\\ \\ \\newline\\textsc{Figure 7.} {\\small The sector }$S${\\small \\ and regular\n}$n${\\small -gon }$P${\\small \\ (left). The set }$S^{\\prime}$ {\\small lies on\nthe non-zero side of the line }$L_{0},${\\small \\ and is bounded by }%\n$L_{0}^{\\prime}${\\small \\ and }$L_{1}^{\\prime}${\\small \\ (right).\\medskip}\n\nVarious approaches are possible to demonstrate that the sectors $e^{i2k\\pi\n\/n}S$ are disjoint. For instance, note that $\\arg p=\\pi\/2+\\pi\/n,$ and consider\nthe related sector\n\\[\nS^{\\prime}=\\left\\{ z\\in\\text{ext}\\left( P\\right) :\\arg z\\in\\left(\n\\pi\/2-\\pi\/n,\\pi\/2+\\pi\/n\\right) \\right\\} .\n\\]\nThe sets $e^{i2k\\pi\/n}S^{\\prime}$ are clearly disjoint for $k=0,1,...,n-1,$\nsince the arguments of points in the sets $e^{i2k\\pi\/n}S^{\\prime}$ for\ndistinct $k\\in\\left\\{ 0,1,...,n-1\\right\\} $ are in disjoint intervals (see\nthe right of Figure 7, where $S^{\\prime}$ is region that is shaded). Let\n$L_{k}^{\\prime}$ be the line through the origin and with argument\n$\\pi\/2+\\left( 2k-1\\right) \\pi\/n,$ and let $E$ be the set $E=\\left\\{ z\\in\next\\left( P\\right) :\\arg z\\leq\\pi\/2-\\pi\/n,\\text{ }\\operatorname{Im}%\nz>\\operatorname{Im}p\\right\\} ,$ bounded by $L_{0}$ and $L_{0}^{\\prime}$ and\n$P.$ We obtain $S$ from $S^{\\prime}$ by including the set $E,$ and excluding\nthe set $e^{i2k\\pi\/n}E$ (darker shaded region in the right of Figure 7) from\n$S^{\\prime}.$ Then $S=S^{\\prime}\\cup E\\backslash e^{i2\\pi\/n}E,$ and by\nrotational symmetry,\n\\[\ne^{i2k\\pi\/n}S=e^{i2k\\pi\/n}S^{\\prime}\\cup e^{i2k\\pi\/n}E\\backslash e^{i2\\left(\nk+1\\right) \\pi\/n}E.\n\\]\nThus to obtain $e^{i2k\\pi\/n}S$ we have simply removed a sector with vertex\n$e^{i2\\left( k-1\\right) \\pi\/n},$ namely $e^{i2k\\pi\/n}E,$ from the set\n$e^{i2k\\pi\/n}S^{\\prime}$ and included its rotation $e^{i2\\left( k+1\\right)\n\\pi\/n}E$ to obtain $e^{i2\\left( k+1\\right) \\pi\/n}S^{\\prime}.$ Because the\nsets $e^{i2k\\pi\/n}S^{\\prime}$ are disjoint, so are the sets $\\left\\{\ne^{i2k\\pi\/n}S:k=0,1,..,n-1\\right\\} .$\n\nIf $\\beta=0$ then $p=0$ and the argument above does not apply, but a simple\nproof follows if we adapt the proof for $\\beta\\in\\left( 0,\\pi\/2\\right) $ and\ndefine the sector $S$ to be $S=\\left\\{ z\\in%\n\\mathbb{C}\n:\\arg z\\in\\left( 0,2\\pi\/n\\right) \\right\\} ,$ in which case the sectors\n$\\left\\{ e^{2k\\pi i\/n}S:k=0,1,...,n-1\\right\\} $ are clearly disjoint$.$\n\nIf $\\beta=\\pi\/2,$ then we adapt the proof so that $L_{k}$ is the line passing\nthrough node $\\alpha_{\\pi\/2}\\left( 2k\\pi\/n\\right) $ with argument\n$\\alpha_{\\pi\/2}^{\\prime}\\left( 2k\\pi\/n\\right) ^{+}$. The sectors then are\nbounded by the lines $L_{k},$ and the same same argument applies to show\n$\\tilde{\\alpha}_{\\pi\/2}$ is one to one on $S$ (note that $\\arg\\tilde{\\alpha\n}_{\\pi\/2}\\left( \\left( 0,2\\pi\/n\\right) \\right) \\subset\\left( \\pi\n\/2+\\pi\/\\left( 2n\\right) ,\\pi\\right) $ rather than $\\arg\\alpha_{\\beta\n}^{\\prime}\\left( \\left( 0,2\\pi\/n\\right) \\right) \\subset\\left( 2\\pi\n\/n,\\pi\\right) $).\n\\end{proof}\n\n\\begin{theorem}\n\\label{univalent}For any $n\\geq3$ in $%\n\\mathbb{N}\n,$ the harmonic functions $f_{\\beta}\\left( z\\right) $, $\\beta\\in%\n\\mathbb{R}\n$ defined in Definition \\ref{defnharmonic} are univalent, and so they are\nharmonic mappings.\n\\end{theorem}\n\n\\begin{proof}\nWe apply the argument principle for harmonic functions of\n\\cite{DurenHenLaugesen} to obtain univalence of $f_{\\beta}.$ We proved in\nLemma \\ref{unibdry} that for $\\beta\\in\\lbrack0,\\pi\/2),$ the boundary curve\n$\\alpha_{\\beta}\\left( t\\right) =f_{\\beta}\\left( e^{it}\\right) $ on\n$\\partial U$ is a simple closed curve about the origin$.$ Although\n$\\alpha_{\\pi\/2}$ has arcs of constancy, the winding number about each point\nenclosed by the curve $\\alpha_{\\beta}$ is still $1,$ so the argument principle\napplies for $\\beta=\\pi\/2$. Choose an arbitrary point $w_{0}$ enclosed by the\ncurve $\\alpha_{\\beta}.$ Then define $f=f_{\\beta}-w_{0},$ a harmonic function\ncontinuous in $\\bar{U}$, which does not have a zero on $\\partial U$. Moreover,\nsince $\\left\\vert \\omega_{f}\\right\\vert =\\left\\vert \\omega\\right\\vert <1$ in\n$U$, $f$ does not have any singular zeros in $U$. We see that $f\\left(\ne^{it}\\right) =\\alpha_{\\beta}\\left( t\\right) -w_{0}$ has index 1 about the\norigin for $t\\in\\partial U$, so it follows from the argument principle that\n$f$ has exactly one zero in $U$. Thus $f_{\\beta}\\left( z_{0}\\right) =w_{0}$\nfor a unique $z_{0}\\in U.$ Since the choice of $z_{0}\\in U$ was arbitrary, we\nsee that $f_{\\beta}$ is onto the region enclosed by $\\alpha_{\\beta}.$ If\n$w_{1}$ is in the exterior of the region enclosed by the curve $\\alpha_{\\beta\n},$ then consider the function $\\tilde{f}=$ $f_{\\beta}-w_{1}.$ The harmonic\nfunction $\\tilde{f}$ satisfies the same hypotheses as did $f,$ but the winding\nnumber of $\\tilde{f}\\left( e^{it}\\right) $ about the origin is zero. Thus\nthere is no $z_{1}\\in U$ for which $f_{\\beta}\\left( z_{1}\\right) =w_{1}.$\nTherefore $f_{\\beta}\\left( U\\right) $ is contained in the interior of the\nregion bounded by $\\alpha_{\\beta}.$ If $\\beta\\in(-\\pi\/2,0),$ then $f_{\\beta}$\nis univalent since it is a reflection of $f_{-\\beta}$ where $-\\beta>0.$\nFinally if $\\beta\\not \\in (-\\pi\/2,\\pi\/2]$ then $f_{\\beta}$ $f_{\\beta}\\left(\nz\\right) $ is a rotation of $f_{\\beta}\\left( e^{-il\\pi\/n}z\\right) $ for\nsome $l\\in%\n\\mathbb{Z}\n$ by Corollary \\ref{transit}, so is also univalent.\n\\end{proof}\n\n\\begin{remark}\nOn sectors $S$ of the closed unit disk for which $g_{n}\\left( S\\right) $ is\nconvex, we can show that $g_{n}$ is relatively more contractive than $h_{n},$\nin that for any two points $z_{0}$ and $z_{1}$ in the sector,\n\\[\n\\left\\vert h_{n}\\left( z_{0}\\right) -h_{n}\\left( z_{1}\\right) \\right\\vert\n\\geq\\left\\vert g_{n}\\left( z_{0}\\right) -g_{n}\\left( z_{1}\\right)\n\\right\\vert\n\\]\nMoreover the inequality is strict when at least one point is not on $\\partial\nU.$ This fact can be used to show that $f_{\\beta}$ is one to one in the\nsectors $S$ of $\\bar{U}$ with argument in the range $[0,\\pi\/n)$, or with\nargument in the range $[\\pi\/n,2\\pi\/n).$ This leads to a direct proof of\nunivalence that does not rely on the argument principle.\n\\end{remark}\n\nWe now define a fundamental set, rotations of which make up the graph of the\nrosette harmonic mapping $f_{\\beta}$.\\ This set has an interesting\ndecomposition into two curvilinear triangles when $\\beta=0,$ or into a\ncurvilinear triangle and curvilinear bigon for $\\left\\vert \\beta\\right\\vert\n\\in(0,\\pi\/2]$. Moreover, the triangle is nowhere convex for $\\left\\vert\n\\beta\\right\\vert \\in(0,\\pi\/2],$ and when $\\beta=\\pi\/2$ the bigon has\nreflectional symmetry.\n\n\\setcounter{figure}{7}\n\n\\begin{definition}\n\\label{fundamental}For an interval $I$ with $\\left\\vert I\\right\\vert <2\\pi,$\ndefine the sector $S_{I}$ of the closed unit disk $\\bar{U}$ to be\n$S_{I}=\\left\\{ z\\in\\bar{U}:\\arg z\\in I,\\text{ }0\\leq r\\leq1\\right\\} $. For\n$n\\in%\n\\mathbb{N}\n,$ $n\\geq3,$ $\\beta\\in(-\\pi\/2,\\pi\/2],$ let $f_{\\beta}$ be the rosette mapping\ndefined in Definition \\ref{defnharmonic}, and define the\\textbf{\\ fundamental\nset of the rosette mapping }$f_{\\beta}$ to be\n\\[\n\\mathcal{A}_{\\beta,n}=f_{\\beta}\\left( S_{[0,2\\pi\/n)}\\right) =\\left\\{\nf_{\\beta}\\left( z\\right) :0\\leq\\left\\vert z\\right\\vert \\leq1,\\text{ }%\n0\\leq\\arg z<2\\pi\/n\\right\\} .\n\\]%\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[\nheight=2.1793in,\nwidth=4.3517in\n]%\n{fund.eps}%\n\\caption{Fundamental sets $\\mathcal{A}_{\\pi\/5,5}$ (left) and $\\mathcal{A}%\n_{\\pi\/2,5}$ (right)}%\n\\label{fund}%\n\\end{figure}\n\n\\end{definition}\n\nThe univalence of $f_{\\beta}$ implies that the set $f_{\\beta}\\left(\nS_{[a,b)}\\right) $ is bounded by the image of $f_{\\beta}\\left( \\partial\nS_{[a,b)}\\right) ,$ for $\\left\\vert b-a\\right\\vert <2\\pi.$ Thus the\nfundamental set $\\mathcal{A}_{\\beta,n}$ is bounded by $f_{\\beta}\\left(\n\\partial S_{[0,2\\pi\/n)}\\right) .$ Definition \\ref{fundamental} and the\nfollowing proposition apply only to $\\beta\\in(-\\pi\/2,\\pi\/2],$ but we will use\nthe fundamental sets $\\mathcal{A}_{\\beta,n}$ with $\\beta$ from this restricted\nrange to express the graph of the rosette mapping $f_{\\tilde{\\beta}}\\left(\n\\bar{U}\\right) ,$ for any $\\tilde{\\beta}\\in%\n\\mathbb{R}\n.$\n\nFor $\\beta\\in(-\\pi\/2,\\pi\/2]$ we note a further interesting decomposition in\nProposition \\ref{fundunion} of the set $\\mathcal{A}_{\\beta,n}$ for $\\beta\n\\neq0$ into a curvilinear triangle with sides that are nowhere convex, and a\ncurvilinear bigon, the latter having reflectional symmetry when $\\beta=\\pi\/2$\n(see Figure \\ref{fund}).\n\n\\begin{proposition}\n\\label{fundunion} Let $n\\in%\n\\mathbb{N}\n,$ $n\\geq3$ and $\\beta\\in(-\\pi\/2,\\pi\/2].$ Let $\\mathcal{A}_{\\beta,n}$ be the\nfundamental set defined in \\ref{fundamental} for the rosette mapping\n$f_{\\beta}$ of Definition \\ref{defnhg}. The set $\\mathcal{A}_{\\beta,n}$ is a\ncurvilinear triangle subtending an angle of $2\\pi\/n$ at the origin. Moreover,\n$\\mathcal{A}_{\\beta,n}$ is the disjoint union%\n\\[\n\\mathcal{A}_{\\beta,n}=f_{\\beta}\\left( S_{[0,\\pi\/n)}\\right) \\cup f_{\\beta\n}\\left( S_{[\\pi\/n,2\\pi\/n)}\\right) .\n\\]\n(i) For $\\beta\\in\\left( 0,\\pi\/2\\right) $, the set $f_{\\beta}\\left( \\partial\nS_{[0,\\pi\/n)}\\right) $ is a curvilinear triangle with sides that are nowhere\nconvex$.$ The set $f_{\\beta}\\left( \\partial S_{[\\pi\/n,2\\pi\/n)}\\right) $ is a\ncurvilinear bigon, where one side is strictly convex, and the other side has a\nsingle inflection point. The angle subtended at the origin, both by the bigon\nand the triangle, is $\\pi\/n.$ The remaining angles in the bigon and triangle\nare $0.$ \\newline(ii)\\ If $\\beta=\\pi\/2$ then the statements in (i) hold,\nexcept the angle subtended by the bigon at the node of $f_{\\pi\/2}$ is\n$\\pi\/2-\\pi\/n.$ Additionally, the bigon $f_{\\pi\/2}\\left( \\partial\nS_{[\\pi\/n,2\\pi\/n)}\\right) $ is symmetric about the line through the vertices\nof the bigon$. $\\newline(iii) When $\\beta\\in\\left( -\\pi\/2,0\\right) $, the\nconclusions of (i) except $f_{\\beta}\\left( \\partial S_{[\\pi\/n,2\\pi\n\/n)}\\right) $ is the curvilinear triangle and $f_{\\beta}\\left( \\partial\nS_{[0,\\pi\/n)}\\right) $ is the curvilinear bigon.\\newline(iv)\\ When $\\beta=0,$\nthe sides of $f_{\\beta}\\left( \\partial S_{[0,\\pi\/n)}\\right) $ and $f_{\\beta\n}\\left( \\partial S_{[\\pi\/n,2\\pi\/n)}\\right) $ incident with the origin are\nline segments, and the curvilinear triangles $f_{\\beta}\\left( \\partial\nS_{[0,\\pi\/n)}\\right) $ and $f_{\\beta}\\left( \\partial S_{[\\pi\/n,2\\pi\n\/n)}\\right) $ are reflections of one another in the line containing their\ncommon side.\n\\end{proposition}\n\n\\begin{proof}\nWe let $\\alpha_{\\beta}\\left( t\\right) =f_{\\beta}\\left( e^{it}\\right) $ on\n$\\partial U.$ The proofs that follow combine facts about the image of\n$f_{\\beta}$ along radial lines of $U$ in Lemma \\ref{convexconcave}, and limits\nof $\\arg\\alpha_{\\beta}^{\\prime}$ at cusps and nodes. The image $\\mathcal{A}%\n_{\\beta,n}=f_{\\beta}\\left( S_{[0,2\\pi\/n)}\\right) $ is bounded by $f_{\\beta\n}\\left( \\partial S_{[0,2\\pi\/n)}\\right) .$ Since $f_{\\beta}\\left( 1\\right)\n$ and $f_{\\beta}\\left( e^{i2\\pi\/n}\\right) $ are distinct, $f_{\\beta}\\left(\n\\partial S_{[0,2\\pi\/n)}\\right) $ is a curvilinear triangle (even when\n$\\beta=\\pi\/2$). At the origin, Lemma \\ref{convexconcave} (ii) states that\n$f_{\\beta}\\left( r\\right) $ and $f_{\\beta}\\left( re^{i\\pi\/n}\\right) $ have\ntangents with arguments $\\beta\/2$ and $\\beta\/2+\\pi\/n.$ The side $f_{\\beta\n}\\left( re^{i2\\pi\/n}\\right) $ is a rotation by $2\\pi\/n$ of $f_{\\beta}\\left(\nr\\right) ,$ and so the argument of its tangent at $0$ is $\\beta\/2+2\\pi\/n.$\nThus the angle subtended by the vertex of $\\mathcal{A}_{\\beta,n}$ at the\norigin is $2\\pi\/n$. We now prove (i). Our observations about $\\frac{\\partial\n}{\\partial r}\\arg f_{\\beta}\\left( r\\right) ,$ $\\frac{\\partial}{\\partial\nr}\\arg f_{\\beta}\\left( re^{i\\pi\/n}\\right) ,$ and $\\frac{\\partial}{\\partial\nr}\\arg f_{\\beta}\\left( re^{i2\\pi\/n}\\right) $ at the origin show that the\nangle subtended by $f_{\\beta}\\left( \\partial S_{[0,\\pi\/n)}\\right) $ and by\n$f_{\\beta}\\left( \\partial S_{[\\pi\/n,2\\pi\/n)}\\right) $ at the origin is\n$\\pi\/n.$ At the boundary, we have by equation (\\ref{nodedir}) with $k=1$ in\nCorollary \\ref{cuspdir} that $\\arg\\alpha_{\\beta}^{\\prime}\\left( t\\right)\n\\rightarrow\\pi\/2+\\pi\/n$ as $t\\rightarrow\\pi\/n^{+}.$ This matches\n$\\frac{\\partial}{\\partial r}\\arg f_{\\beta}\\left( re^{i\\pi\/n}\\right) $ at\n$f_{\\beta}\\left( e^{i\\pi\/n}\\right) =\\alpha_{\\beta}\\left( \\pi\/n\\right) $,\nby Lemma \\ref{convexconcave} (ii)$.$ Thus $f_{\\beta}\\left( re^{i\\pi\n\/n}\\right) $ and $\\alpha_{\\beta}\\left( t\\right) $ join to form a single\nsmooth curve, together forming one side of $f_{\\beta}\\left( \\partial\nS_{[0,2\\pi\/n)}\\right) ,$ whence the bigon. From Lemma \\ref{convexconcave}\n(ii), $\\frac{\\partial}{\\partial r}\\arg f_{\\beta}\\left( re^{i\\pi\/n}\\right) $\nis strictly increasing, yet $\\arg\\alpha_{\\beta}^{\\prime}\\left( e^{it}\\right)\n$ is strictly decreasing by (\\ref{compass}). Thus $f_{\\beta}\\left( e^{i\\pi\n\/n}\\right) $ is an inflection point where curvature changes sign on the\nbigon. We also have $\\frac{\\partial}{\\partial r}\\arg f_{\\beta}\\left(\nr\\right) $ is strictly decreasing, while $\\frac{\\partial}{\\partial r}\\arg\nf_{\\beta}\\left( re^{i\\pi\/n}\\right) $ is strictly increasing. Thus the sides\nof the triangle $f_{\\beta}\\left( \\partial S_{[0,\\pi\/n)}\\right) $ are nowhere\nconvex. Returning to the bigon, we now see that its second side $f_{\\beta\n}\\left( re^{i2\\pi\/n}\\right) $ is strictly convex, since $f_{\\beta}\\left(\nre^{i2\\pi\/n}\\right) $ is a rotation about the origin of $f_{\\beta}\\left(\nr\\right) $. We complete the proof of (i) by observing that when $\\beta\n\\in\\lbrack0,\\pi\/2),$ the vertices $f_{\\beta}\\left( e^{i2\\pi\/n}\\right) $ and\n$f_{\\beta}\\left( 1\\right) $ are cusps of $\\alpha_{\\beta}, $ and the angle\nsubtended at $f_{\\beta}\\left( e^{i2\\pi\/n}\\right) $ and $f_{\\beta}\\left(\n1\\right) $ is $0.$\n\nNow we turn to (ii), with $\\beta=\\pi\/2.$ We observed reflectional symmetry in\nTheorem \\ref{symmetry} (iii) of $f_{\\pi\/2}\\left( re^{-i\\pi\/n}\\right) $ and\n$f_{\\pi\/2}\\left( r\\right) $ about the axis through $0$ and the node\n$f_{\\pi\/2}\\left( 1\\right) .$ Rotating by $2\\pi\/n$, the ray through $0$ and\nthe node $f_{\\pi\/2}\\left( e^{i2\\pi\/n}\\right) $ is also an axis of\nreflectional symmetry, in which the sides $f\\left( re^{i\\pi\/n}\\right) $ and\n$f\\left( re^{i2\\pi\/n}\\right) $ of the bigon are reflected. To see the\nsubtended angle at the boundary $\\alpha_{\\pi\/2}$, we use Lemma\n\\ref{convexconcave} (ii) as above to see the tangent of $f_{\\pi\/2}\\left(\nre^{i\\pi\/n}\\right) $ approaches $\\pi\/2+\\pi\/n$ as $r\\rightarrow1^{-}.$ By the\nformula in the proof of Corollary \\ref{nodes} with $k=1$, $\\arg\\tilde{\\alpha\n}_{\\pi\/2}^{\\prime}\\left( t\\right) $ also approaches $\\pi\/2+\\pi\/n$ as\n$t\\rightarrow2\\pi\/n^{-}.$ We conclude that the side $f_{\\pi\/2}\\left(\nre^{i\\pi\/n}\\right) $ of the bigon becomes tangent to the boundary\n$\\tilde{\\alpha}_{\\pi\/2}$ at the node $f_{\\pi\/2}\\left( e^{i\\pi\/n}\\right)\n=f_{\\pi\/2}\\left( e^{i2\\pi\/n}\\right) $ as $t\\rightarrow2\\pi\/n^{-}.$ By\nreflectional symmetry, the second side of the bigon $f_{\\pi\/2}\\left(\nre^{i2\\pi\/n}\\right) $ becomes tangent to the boundary $\\tilde{\\alpha}_{\\pi\n\/2}$ at as $t\\rightarrow2\\pi\/n^{+}.$ Thus the angle subtended in the bigon at\n$f_{\\pi\/2}\\left( e^{i2\\pi\/n}\\right) $ is the same as the interior angle of\nthe nodes of $\\tilde{\\alpha}_{\\pi\/2},$ namely $\\pi\/2-\\pi\/n$ by Corollary\n\\ref{nodes}$.$\n\nThe statements in (iii) follow readily using reflectional symmetry. For\n(iv)\\ we already noted in Lemma \\ref{convexconcave} (iii) that the images of\n$f_{0}\\left( r\\right) ,$ $f_{0}\\left( re^{i\\pi\/n}\\right) $ and\n$f_{0}\\left( re^{i2\\pi\/n}\\right) $ are linear. Reflectional symmetry was\nestablished in Corollary \\ref{noreflect}.\n\\end{proof}\n\nWe finish by decomposing the graph of an arbitrary rosette mapping$\\ $into a\ndisjoint union of $n$ rotations of a fundamental set of Definition\n\\ref{fundamental}.\n\n\\begin{corollary}\nLet $n\\geq3$ and $\\tilde{\\beta}\\in%\n\\mathbb{R}\n.$ The set $f_{\\tilde{\\beta}}\\left( \\bar{U}\\right) $ can be written as a\ndisjoint union of rotations of $\\mathcal{A}_{\\beta,n}.$ Specifically if\n$\\tilde{\\beta}=\\beta+l\\pi$ for $\\beta\\in(-\\pi\/2,\\pi\/2]$ then\n\\[\nf_{\\tilde{\\beta}}\\left( \\bar{U}\\right) =i^{l}%\n{\\displaystyle\\bigcup_{k=1}^{n}}\ne^{i\\left( 2k+l\\right) \\pi\/n}\\mathcal{A}_{\\beta,n}.\n\\]\n\n\\end{corollary}\n\n\\begin{proof}\nBy Corollary \\ref{transit} we have $f_{\\tilde{\\beta}}\\left( z\\right)\n=e^{il\\left( \\pi\/n+\\pi\/2\\right) }f_{\\beta}\\left( e^{-il\\pi\/n}z\\right) .$\nThus\n\\[\nf_{\\tilde{\\beta}}\\left( \\bar{U}\\right) =f_{\\tilde{\\beta}}\\left( e^{il}%\n\\bar{U}\\right) =e^{il\\left( \\pi\/n+\\pi\/2\\right) }f_{\\beta}\\left( \\bar\n{U}\\right) =i^{l}e^{il\\pi\/n}%\n{\\displaystyle\\bigcup_{k=1}^{n}}\ne^{i2k\\pi\/n}\\mathcal{A}_{\\beta,n}.\n\\]\n\n\\end{proof}\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\partial{\\partial}\n\\def\\partial_i{\\partial_i}\n\\def\\frac{1}{2}{\\frac{1}{2}}\n\\defA^{+}_0{A^{+}_0}\n\\def\\psi_+{\\psi_+}\n\\def\\psi_-{\\psi_-}\n\\def\\psi_2{\\psi_2}\n\\def\\psi_1{\\psi_1}\n\\def\\psi^{\\dagger}_+{\\psi^{\\dagger}_+}\n\\def\\psi^{\\dagger}_-{\\psi^{\\dagger}_-}\n\\def\\overline{\\psi}{\\overline{\\psi}}\n\\def\\psi^{\\dag}{\\psi^{\\dag}}\n\\def\\chi^{\\dag}{\\chi^{\\dag}}\n\\def\\sla#1{#1\\!\\!\\!\/}\n\\defx^{+}{x^{+}}\n\\defx^{-}{x^{-}}\n\\defy^{-}{y^{-}}\n\\defx_\\perp{x_\\perp}\n\\defy_\\perp{y_\\perp}\n\\def\\underline{x}{\\underline{x}}\n\\def\\underline{y}{\\underline{y}}\n\\def\\underline{p}{\\underline{p}}\n\\def\\underline{n}{\\underline{n}}\n\\newcommand{\\newcommand}{\\newcommand}\n\\defp_{\\ulin}{p_{\\underline{n}}}\n\\newcommand{\\intl}{\\int\\limits_{-L}^{+L}\\!dx^-}\n\\newcommand{\\intv}{\\int\\limits_{V}^{}\\!d^3\\underline{x}}\n\\newcommand{\\intvy}{\\int\\limits_{V}^{}\\!d^3\\underline{y}}\n\\newcommand{\\intly}{\\int\\limits_{-L}^{+L}\\!{{dy^-}\\over\\!2}}\n\\newcommand{\\zmint}{\\int\\limits_{-L}^{+L}\\!{{dx^-}\\over{\\!2L}}}\n\\newcommand{\\intp}{\\int\\limits_{0}^{+\\infty}\\!{{dp^+}\\over\\!4\\pi}}\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\begin{document}\n\\title{Higgs mechanism in a light front formulation}\n\\medskip\n\\author{ {\\sl L$\\!\\!$'ubom\\'{\\i}r Martinovi\\v c}\n\\footnote{E-mail: fyziluma@savba.sk}\\\\\nInstitute of Physics, Slovak Academy of Sciences\\\\ \nD\\'ubravsk\\'a cesta 9, 845 11 Bratislava, Slovakia\\\\\n\\phantom{.}\\\\\nand\\\\\n\\phantom{.}\\\\\n{\\sl Pierre Grang\\'e}\n\\footnote{E-mail: Pierre.GRANGE@LPTA.univ-montp2.fr}\\\\\nLaboratoire de Physique Th\\'eorique et Astroparticules\\\\ \nUniversit\\'e Montpellier II, Pl. E. Bataillon\\\\ \nMontpellier, F-34095 France}\n\\medskip\n\\maketitle\n\n\\begin{abstract}\nWe give a simple derivation of the Higgs phenomenon \nin an abelian light front gauge theory. It is based on a finite-volume \nquantization with antiperiodic scalar fields and \nperiodic gauge field. An infinite set of degenerate vacua \nin the form of coherent states of the scalar field, that minimize \nthe light front energy, is constructed. The corresponding effective \nHamiltonian describes a massive vector field whose third component is \ngenerated by the would-be Goldstone boson. This mechanism, \nunderstood here quantum mechanically in the form analogous to the spacelike \nquantization, is derived without gauge fixing as well as in the unitary \nand the light-cone gauge. \n\\end{abstract}\n\n\n\\vspace{5mm}\nSpontaneous symmetry breaking (SSB) of global as well as gauge symmetries \nhas not been fully understood in the light-front (LF) field theory, which \nis the formulation (most often hamiltonian) of relativistic dynamics \nthat uses the LF \nvariables $x^\\mu= (x^{+}, x^{-},x^1,x^2), x^\\pm=x^0 \\pm x^3$ and is quantized \non a surface of constant light-front time $x^{+}$ \\cite{Dir,LKS,Rohr}. \nConsequently, one deals with the LF field variables which satisfy field \nequations with a different structure than the equations of the conventional \nfield theory which parametrizes the spacetime by \n$x^\\mu = (t,x^1,x^2,x^3)$ and is quantized at $t=0$. \nThe main reason for difficulties in obtaining a clear picture of SSB \nin the LF theory is the positivity of the spectrum of the LF momentum operator \n$P^+$. Quite generally, the interacting-theory vacuum state coincides with \nthe free Fock vacuum if Fourier modes carrying $p^+=0$ (LF zero modes - ZM) can \nbe neglected. This simplifying feature is very useful in perturbative and \nbound-state calculations. However, it complicates the understanding of other \nnon-perturbative properties because it seems to prohibit any vacuum structure \nin LF theories and hence also the well established SSB pattern. Alternative \nschemes of the physics of broken phase have been given in the LF literature \n\\cite{lfssb,Yam,RT}. They are typically based on the operator scalar ZM \nwhich is present for periodic boundary conditions (BC) and which satisfies \nan equation of a constraint. The role of this \nvariable in the phase transition of the $\\lambda \\phi^4(1+1)$ model was \nanalyzed by means of the Haag expansion in \\cite{SGW}. \n\nThe Higgs mechanism in the LF formalism was studied on the tree level in the \ncontinuum formulation \\cite{Prem}. It was assumed that a scalar field contains \na c-number piece which gave a justification for performing a usual shift \nin the Lagrangian leading to the generation of the mass term for the gauge \nfield. A support for the above assumption comes from the fact that the solution \nof the zero-mode constraint of the real scalar field in the DLCQ analysis \ncontains such a constant non-operator part \\cite{Dave, LMSSB}. \n\nIn the present work, we study the SSB of an abelian symmetry in the Higgs \nmodel. Our approach is based on the discrete light-cone quantization method \n(DLCQ) considered as a hamiltonian analytical framework with large but finite \nnumber of Fourier modes to approximate quantum field \ntheory with its infinite number of degrees of freedom. A (regularized) unitary \noperator that shifts the scalar field by a constant will be used to transform \nthe Fock space. The motivation for this step is a natural physical \nrequirement to find ground states in the broken phase which would correspond \nto a lower LF energy than the usual Fock vacuum. This is suggested already \nby considering minima of the classical LF potential energy. A procedure, \nequivalent to transforming the states, is to work with a transformed \nHamiltonian and calculate its matrix elements between the usual Fock states. \nIn this way one naturally arrives at the effective type of the Hamiltonian \nthat incorporates the usual pattern of the Higgs mechanism. \n\n \nThe Lagrangian density of the abelian Higgs model that we wish to \nanalyze has the form \n\\begin{equation}\n{\\cal L}=-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu} + \\frac{1}{2} (D_\\mu\\phi)^\\dagger D^\\mu \\phi + \n\\frac{1}{2} \\mu^2 \\phi^\\dagger\\phi - \\frac{\\lambda}{4} (\\phi^\\dagger \\phi)^2, \n\\label{lagr}\n\\end{equation}\nwhere $F_{\\mu\\nu}=\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu,~~D^\\mu\\phi=\\partial^\\mu\\phi \n+ieA^\\mu\\phi, \\mu^2 > 0$. The Lagrangian is invariant under two groups of \ntransformations: the global rotations of the complex scalar field $\\phi(x) \n\\rightarrow \\exp{\\big(-i\\beta\\big)}\\phi(x)$ and the local gauge transformations \n\\begin{equation}\n\\phi(x) \\rightarrow \\exp{\\big(-i\\omega(x)\\big)}\\phi(x),~~A^\\mu (x) \\rightarrow \nA^\\mu (x) + \\partial^\\mu \\omega(x)\/e.\n\\label{GT}\n\\end{equation}\nIn terms of the LF variables, the Lagrangian (\\ref{lagr}) is \n\n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\\cal L}_{lf} = \n\\frac{1}{2} \\big(\\partial_+ A^+ - \\partial_- A^-\\big)^2 \n+ \\big(\\partial_+ A^i + \\frac{1}{2} \\partial_i A^-\\big)\\big(2\\partial_- A^i +\\partial_i \nA^+\\big) - \n\\nonumber \\\\ \n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! - \\frac{1}{2} \\big(\\partial_1 A_2 - \\partial_2 A_1)^2 + \n \\partial_+ \\phi^\\dagger\\partial_-\\phi + \\partial_-\\phi^\\dagger \\partial_+\\phi - \n\\frac{1}{2} \\partial_i\\phi^\\dagger\\partial_i\\phi - \n\\frac{ie}{2}\\phi^\\dagger\\stackrel {\\leftrightarrow} {\\partial_+}\\phi A^+ - \\nonumber \\\\ \n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! - \\frac{ie}{2}\\phi^\\dagger\\delrlm\n\\phi A^- - \\frac{ie}{2}\\phi^\\dagger\\delrli\\phi A^i + \n\\frac{e^2}{2}\\big(A^+A^- - A^iA^i\\big)\\phi^\\dagger\\phi + \\frac{\\mu^2}{2}\n\\phi^\\dagger\\phi - \\frac{\\lambda}{4}\\big(\\phi^\\dagger\\phi\\big)^2.\n\\label{lflagr}\n\\end{eqnarray} \nWriting $\\phi=\\sigma+i\\pi$, the conserved current corresponding to \nthe global symmetry is $J^\\mu (x)=-i\\phi^\\dagger (x) \\stackrel {\\leftrightarrow} {\\partial^\\gm} \\phi(x)=\n2\\sigma (x)\\stackrel {\\leftrightarrow} {\\partial^\\gm} \\pi (x)$. \n \nWe will work in a finite volume $V=8L^3$ with space coordinates restricted to \n$-L \\le x^{-},x^1,x^2 \\le L$. Our notation is $x^\\mu=(x^{+},\\underline{x}), \\underline{x}=\n(x^{-},x^1,x^2),~ p.x=\\frac{1}{2} p^-x^+ + \\frac{1}{2} p^+x^- - p^1x^1-p^2x^2$. The gauge \nfield will be chosen periodic in all three directions, while the scalar field \nwill be antiperiodic: $A^\\mu(x^{+},x^{-}=-L, x,y)=A^\\mu(x^{+},x^{-}=L,x,y), \n\\sigma(x^{+},x^{-}=-L,x,y)=-\\sigma(x^{+},x^{-}=L, x,y)$, and similarly in the \nperpendicular directions $x_\\perp \\equiv(x^1,x^2)$ \\cite{remark}.\nThe boundary conditions imply the discrete values \nof the three momentum labeled by a (half)integer and also lead to the \npresence of global and proper zero modes of the gauge field \\cite{Alex}. The \nproper ZM are constrained variables that can modify the LF Hamiltonian. For \nsmall coupling the corrections may be evaluated perturbatively \\cite{AD}. \nWe shall however neglect the gauge-field ZM in the present discussion \nbecause they are not crucial for the phenomenon under study. The \nfields below refer then to the sector of normal Fourier modes.\n\nThe LF Hamiltonian, obtained in the canonical way from the \nLagrangian (\\ref{lflagr}), reads\n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!P^-=\\intv \\Big\\{F_{12}^2 + \\Pi^2_{A^+}+2\\Pi_{A^+}\\partial_- A^--\\Pi_{A^i}\n\\partial_i A^-- 2e\\sigma\\delrlm\\pi A^- \n- 2e\\sigma\\delrli \\pi A^i - \\nonumber \\\\ \n&&\\!\\!\\!\\!\\! - e^2 A^2\\big(\\sigma^2+\\pi^2\\big)\n+ (\\partial_i\\sigma)^2 + (\\partial_i \\pi)^2 \n- \\mu^2\\big(\\sigma^2+\\pi^2\\big) + \n\\frac{\\lambda}{2}\\big(\\sigma^2+\\pi^2\\big)^2 \\Big\\}. \n\\label{Ham}\n\\end{eqnarray}\nHere $A^2=A^+A^- - A^iA^i, i=1,2$ and the canonical momenta are \n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!\\!\\Pi_{A^+}=\\partial_+ A^+-\\partial_- A^-,~\\Pi_{A^i} = \n2\\partial_- A^i + \\partial_i A^+ \n\\nonumber \\\\\n&&\\!\\!\\!\\!\\!\\!\\Pi_{A^-}=0,~\\Pi_\\sigma=2\\partial_- \\sigma-e\\pi A^+,\\Pi_\\pi=2\\partial_-\\pi+\ne\\sigma A^+.\n\\label{moms}\n\\end{eqnarray}\n\nAt $x^{+}=0$, we assume the usual LF commutation rules\n\\begin{eqnarray}\n&&\\big[\\sigma(x^{+},\\underline{x}),\\sigma(x^{+},\\underline{y})\\big] = \n\\frac{-i}{8}\\epsilon(x^{-}-y^{-})\\delta^2(x_\\perp-y_\\perp), \\nonumber \\\\\n&&\\big[\\pi(x^{+},\\underline{x}),\\pi(x^{+},\\underline{y})\\big] = \n\\frac{-i}{8}\\epsilon(x^{-}-y^{-})\\delta^2(x_\\perp-y_\\perp), \\nonumber \\\\\n&&\\big[A^+(x^{+},\\underline{x}),\\Pi_{A^+}(x^{+},\\underline{y})\\big]=\n\\frac{i}{2}\\delta^3(\\underline{x}-\\underline{y}) \\nonumber\\\\\n&&\\big[A^i(x^{+},\\underline{x}),\\Pi_{A^j}(x^{+},\\underline{y})\\big]= \n\\frac{i}{2}\\delta^{ij}\\delta^3(\\underline{x}-\\underline{y}).\n\\label{cr}\n\\end{eqnarray}\nThe mode expansions of the scalar fields are \n\\begin{eqnarray}\n\\sigma(0,\\underline{x})=\\frac{1}{\\sqrt{V}}\\sum\\limits_{\\underline{n}}^{}\\frac{1}{\\sqrt{p^+_n}}\n\\big[a(p_{\\underline{n}})e^{-i p_{\\underline{n}} .\\underline{x}} + a^\\dagger(p_{\\underline{n}})e^{i p_{\\underline{n}} .\\underline{x}} \\big], \n\\nonumber \\\\\n\\pi(0,\\underline{x})=\\frac{1}{\\sqrt{V}}\\sum\\limits_{\\underline{n}}^{}\\frac{1}{\\sqrt{p^+_n}}\n\\big[c(p_{\\underline{n}})e^{-i p_{\\underline{n}}. \\underline{x}} + c^\\dagger(p_{\\underline{n}})e^{i p_{\\underline{n}}.\\underline{x}} \\big],\n\\label{fexp}\n\\end{eqnarray}\nwhere $p_{\\underline{n}}=(p_n^+,p_{n_1},p_{n_2}), p_n^+=\\frac{2\\pi}\n{L}n, n=1\/2,3\/2,\\dots$, and similarly for the perpendicular components. \nThe global rotations are implemented by the unitary operators $V(\\beta)$ in terms of the charge $Q=\\intv J^+(x)$:\n\\begin{eqnarray}\n\\sigma(x) \\rightarrow V(\\beta)\\sigma(x)V^\\dagger(\\beta) = \\sigma(x)\\cos \\beta-\\pi(x)\\sin \\beta, \\nonumber \\\\\n\\pi(x) \\rightarrow V(\\beta)\\pi(x)V^\\dagger(\\beta) = \\sigma(x)\\sin \\beta+\\pi(x)\\cos \\beta.\n\\label{ccr}\n\\end{eqnarray}\nHere\n\\begin{equation}\nV(\\beta)=e^{i\\beta Q}\n\\label{uniq}\n\\end{equation}\nwith \n\\begin{equation}\nV(\\beta) = \\exp\\big\\{\\sum\\limits_{\\underline{n}}^{\\Lambda}\\Big(a^\\dagger(p_{\\underline{n}})\nc(p_{\\underline{n}}) - c^\\dagger(p_{\\underline{n}})a(p_{\\underline{n}})\\Big)\\big\\}. \n\\label{glim}\n\\end{equation}\nThe Hamiltonian (\\ref{Ham}) is invariant under $x^{+}$-independent gauge \ntransformations. They are implemented by the unitary operator\n\\begin{equation}\nU[\\omega(\\underline{x})]=\\exp \\Big\\{ i \\intv\\big[2\\Pi_{A^+}\\partial_- - \\Pi_{A^i}\\partial_i + \neJ^+\\big]\\omega(\\underline{x})\\Big\\} \\label{loim}\n\\end{equation}\nIndeed, we easily find \n\\begin{eqnarray}\n&&U[\\omega(\\underline{x})]\\phi(x)U^\\dagger[\\omega(\\underline{x})]=\\exp{\\big(-i\\omega(\\underline{x})\\big)}\n\\phi(x), \\nonumber \\\\\n&&U[\\omega(\\underline{x})]A^\\mu(x)U^\\dagger[\\omega(\\underline{x})]=\n\t\t\tA^\\mu(x) + e^{-1}\\partial^\\mu\\omega(\\underline{x}).\n\\label{gtimpl}\n\\end{eqnarray}\nConsider now the unitary operators \n\\begin{eqnarray}\nU_\\sigma(b)=\\exp\\Big\\{-2ib\\intv \\Pi_\\sigma(x)\\Big\\} \\nonumber \\\\\nU_\\pi(b)=\\exp\\Big\\{-2ib\\intv \\Pi_\\pi(x)\\Big\\}.\n\\label{shifto}\n\\end{eqnarray}\nThey shift the corresponding scalar field by a constant. To follow the usual \ntreatment, we will perform only shifts in the $\\sigma$-direction:\n\\begin{eqnarray}\nU_\\sigma(b) \\sigma(x) U^{-1}_\\sigma(b) = \\sigma(x) - 2ib \\intvy \\big[\\Pi_\\sigma(y),\\sigma(x)\\big] \n\\nonumber \\\\\n= \\sigma(x) - b \\epsilon_\\Lambda(L-x^{-}) \\epsilon_\\Lambda(L-x^1) \\epsilon_\\Lambda(L-x^2).\n\\label{shift}\n\\end{eqnarray}\nThe subscript $\\Lambda$ attached to the sign function $\\epsilon(\\underline{x})$ indicates that their \nFourier series is truncated at $\\Lambda$:\n\\begin{eqnarray}\n\\epsilon_\\Lambda(x^{-}) = \\frac{4i}{L} \\sum_{n=\\frac{1}{2}}^{\\Lambda} \\frac{1}{p_n^+} \\Big(e^{-ip_n^+ x^{-}} \n- e^{ip^+_n x^{-}}\\Big) \n\\label{sign}\n\\end{eqnarray}\nand analogously for the perpendicular components. The point is that one has to \ntake a large but finite number of field modes in all three space directions in order \nto have a well-defined operator $U_\\sigma(b)$. In practice, for $\\Lambda \\approx 10^3$ the \nsign functions are equal to unity to a very good approximation everywhere on the finite \ninterval $-L < x^{-},x^1,x^2 < L$ except for a very small neighborhood of the \nend-points. Therefore we \nwill not write these sign functions explicitly henceforth. \n\nBy means of the shift operator $U_\\sigma(b)$, we can define a set of states \n$\\vert b \\rangle = U_\\sigma(b)\\vert 0 \\rangle$ ($\\vert 0 \\rangle$ is the Fock \nvacuum). Minimizing the expectation value of the energy density $V^{-1}\\langle \nb \\vert P^- \\vert b \\rangle$, we easily find that the minimum of the LF energy, \nequal to $-\\frac{\\mu^4}{2\\lambda}$ is achieved for $b=\\frac{\\mu}{\\sqrt{\\lambda}} \\equiv \nv$. It is lower than the usual (vanishing) value of the LF energy in the \n``trivial'' vacuum $\\vert 0 \\rangle$. From Eq.(\\ref{shift}) we also have the \nproperty that the vacuum expectation value of the $\\sigma$-field is non-zero \nwhich is the indication of broken symmetry: \n\\begin{equation}\n\\langle v \\vert \\sigma(x) \\vert v \\rangle = \\langle 0 \\vert U^{-1}_\\sigma(v) \n\\sigma(x)U_\\sigma(v) \\vert 0 \\rangle = v.\n\\label{ssb}\n\\end{equation} \nHere, the sign functions multiplying the value $v$ are understood as in \nEq.(\\ref{shift}). The accompanying vacuum degeneracy is easily obtained by \nrotating our ``trial'' vacuum (chosen in the $\\sigma$-direction):\n\\begin{equation}\nV(\\beta)\\vert v \\rangle = V(\\beta)U_\\sigma(v)\\vert 0 \\rangle \\equiv \\vert v;\\beta \n\\rangle.\n\\label{fulv}\n\\end{equation}\nThus we have an infinite set of vacuum states corresponding to the above \nminimum of the LF energy and labeled by the angle $\\beta$.\n\nThe next step in the Hamiltonian formalism is to construct the space of states. \nA natural possibility would be to apply a string of creation operators \nof all fields to the new vacuum, chosen to be $\\vert v;0 \\rangle$, and \ncalculate the corresponding matrix elements of $P^-$. A simpler option is \nto build a usual set of Fock states from the Fock vacuum $\\vert 0 \\rangle$ \nand transform all of them by $U_\\sigma(v)$. This type of states is known as \ndisplaced number states in quantum optics \\cite{qo}. In either case one can \neasily see that instead of the original Hamiltonian (\\ref{Ham}) one actually \nworks with the new ``effective'' LF Hamiltonian\n\\begin{equation}\n\\tilde{P}^- = U^{-1}_\\sigma(v) P^- U_\\sigma(v)\n\\label{Hef}\n\\end{equation}\nin which the $\\sigma$-field is shifted by the value $v$. This of course leads to \nthe structure known from the lagrangian formalism in the conventional field \ntheory \\cite{IZ}: the mass term of the gauge field of the form $e^2v^2A^2$ is \ngenerated, \nthe pion field becomes massless and the $\\sigma$-field acquires mass equal to \n$\\sqrt{2}\\mu$. The change in the Hamiltonian density shows this explicitely: \n\\begin{eqnarray}\n&&\\delta P^- = -\\frac{\\mu^4}{2\\lambda} + 3\\mu^2 \\sigma^2 + \\mu^2 \\pi^2 - e^2v^2A^2 \n- 2e^2v\\sigma A^2 + \n\\nonumber\\\\\n&& + 2\\sqrt{\\lambda}\\mu \\sigma \\Big(\\sigma^2 + \\pi^2 \\Big)\n- 2ev\\Big(\\partial_- \\pi A^- + \\partial_i\\pi A^i\n\\Big). \n\\end{eqnarray}\nThe latter non-diagonal term and the kinetic term $(\\partial_i \\pi)^2$ can be \nremoved by introducing the new field $B$:\n\\begin{equation}\nB^{+}(x)=A^{+}+\\frac{2}{ev}\\partial_-\\pi,~ \nB^i(x) = A^i(x) -\\frac{1}{ev}\\partial_i \\pi(x), \n\\label{subst}\n\\end{equation}\nwhile $B^- = A^-$. In this way, the $\\pi$ field disappeared from the quadratic \npart of the Hamiltonian but it is still present in the interacting part. One \nmay suspect that it is actually a redundant degree of freedom because \nthe gauge freedom has not been removed.\n \nIn a full analogy with the space-like treatment, a clearer \nphysical picture is obtained in the unitary gauge. Introducing the radial \nand angular field variables: \n\\begin{equation}\n\\phi(x)=\\rho(x)e^{i\\Theta(x)\/v},\n\\label{radial}\n\\end{equation}\nthe LF Hamiltonian will take the form\n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!P^-_r=\\intv \\Big\\{\\Pi^2_{A^+} + 2\\Pi_{A^+}\\partial_- A^- - \n\\Pi_{A^i}\\partial_i A^- \n+ F_{12}^2 +(\\partial_i \\rho)^2 + \\nonumber \\\\\n&&\\!\\!\\!\\!\\! + \\rho^2(\\partial_i \\Theta\/v)^2 - 2e\\rho^2\\partial_- \nA^-\\Theta\/v - 2e\\rho^2A^i \\partial_i \\Theta \/v -e^2\\rho^2 A^2 -\\mu^2\\rho^2 + \n\\frac{\\lambda}{2}\\rho^4 \\Big\\}. \n\\label{hamr}\n\\end{eqnarray}\nTo fix the gauge at the classical Lagrangian level, one observes that the gauge \ntransformations simply shift the angular field variable $\\Theta(x)$ by the \ngauge function $\\omega(x)$. Choosing $\\omega(x)=-\\Theta(x)\/v$, one has \n\\begin{equation}\n\\! \\phi(x) \\rightarrow \\rho(x),~A^\\mu (x) \\rightarrow B^\\mu (x) = A^\\mu (x) - \n\\frac{1}{ev}\\partial^\\mu \\Theta (x)\n\\label{newa}\n\\end{equation}\nwith the corresponding Lagrangian\n\\begin{equation}\n{\\cal L}_u = -\\frac{1}{4}G_{\\mu \\nu}G^{\\mu \\nu} + \\frac{1}{2} |\\partial_\\gm \\rho \n-ieB_\\mu \\rho|^2 + \\frac{1}{2}\\mu^2 \\rho^2 \n-\\frac{\\lambda}{4}\\rho^4.\n\\label{ulag}\n\\end{equation} \nTaking this gauge fixing over to the quantum theory, defined by the \ncommutation relation at $x^{+}=0$\n\\begin{equation}\n\\Big[\\rho(x^{+},\\underline{x}),\\rho(x^{+},\\underline{y})\\Big] = -\\frac{i}{8}\n\\epsilon(x^{-}-y^{-})\\delta^2(x_\\perp-y_\\perp),\n\\label{crr}\n\\end{equation}\nwe find the quantum LF Hamiltonian $P^-_u$ in the unitary gauge. It coincides \nwith the Hamiltonian (\\ref{hamr}) except for the missing $\\Theta$-terms and \nthe $B^\\mu$ replacing the $A^\\mu$ field. The equal-LF time algebra (\\ref{crr}) \nenables us to introduce the shift operator ($\\Pi_\\rho = 2\\partial_- \\rho)$\n\\begin{equation}\nU_\\rho(v)=\\exp\\Big\\{-2i v\\intv \\Pi_\\rho(x) \\Big\\}\n\\label{shifr}\n\\end{equation}\nwhich defines the ``effective'' LF Hamiltonian \n$\\tilde{P}^-_u = U^{-1}_\\rho (v) P^-_u U_\\rho(v)$ \ncorresponding to the unitary gauge:\n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!\\!\\tilde{P}^-_u = \\intv \\Big\\{\\Pi_{B^+}^2 + 2\\Pi_{B^+}\\partial_- B^- \n- \\Pi_{B^i}\\partial_i B^- + G_{12}^2 + ~~~~ \n\\nonumber \\\\ \n&&\\!\\!\\!\\!\\!\\!+ (\\partial_i\\rho)^2 \n-e^2(\\rho + v)^2B^2 - \\mu^2(\\rho + v)^2 + \\frac{\\lambda}{2}(\\rho + v)^4\\Big\\}. \n\\label{Hunit}\n\\end{eqnarray}\nFrom its form it is easy to find that it describes one massive scalar field \n$\\rho$ and a vector field with the mass $e^2v^2$. The massive vector field \nemerged as a combination of the the massless gauge field $A^\\mu$ and the \nscalar $\\Theta$ field. \n\nAnother possibility is to analyze the symmetry breaking in the light-cone \ngauge. This means that we set $A^+ = 0$ in the normal-mode sector. The \nstarting Hamiltonian and conjugate momenta are then the expressions \n(\\ref{Ham}),(\\ref{moms}) without the terms containing $A^+$. One proceeds \nas in the case without the gauge fixing, namely defines the \nshift operator $U_{\\sigma}(v)$ and constructs the infinite set of degenerate \n(approximative) vacuum states by applying the unitary operator $V(\\beta)$ \n(Eq.(\\ref{glim})) to the coherent-state vacuum $\\vert v \\rangle$. The \ncorresponding effective LF Hamiltonian is obtained by the transformation \n(\\ref{Hef}). One observes an important difference as compared \nwith the unitary-gauge treatment. It is related to the fact that the choice \n$A^+ = 0$ eliminates the $A^+ A^-$ part of the vector field mass term \ngenerated by shifting the $\\sigma$ field in the $-e^2 A^2 (\\sigma^2 + \\pi^2)$ term \nin the Hamiltonian (\\ref{Ham}). Thus the massive vector field seems to have \nonly two components and this is not correct. The resolution of this difficulty \ncomes from the observation \\cite{Prem} that in the light-cone gauge the \nGauss' law becomes a constrained equation for the $A^-$ component of the \ngauge field: \n\\begin{equation}\n\\partial_-^2 A^-(x) + \\partial_- \\partial_i A^i(x) = e\\sigma(x) \\delrli \\pi(x). \n\\label{amcon}\n\\end{equation}\nThe shift of the $\\sigma$ field by means of the operator $U_\\sigma(v)$ generates \nan additional term of the form $ev\\partial_- \\pi(x)$ on the righ-hand side of this \nequation. Upon inserting the shifted constraint to the Hamiltonian, the latter \npiece leads to the new term $e^2v^2 \\pi^2$ ($i=1,2$): \n\\begin{equation}\n\\tilde{P}^-_{lc}= \\intv \\Big[F_{12}^2 + (\\partial_i A^i)^2 + (\\partial_i \\pi)^2 \n+ e^2v^2 \\big(\\pi^2 + A_i^2\\big) + \\dots \\Big].\n\\label{frep}\n\\end{equation}\nTo see that this Hamiltonian corresponds to a free massive vector meson field, \nit is useful to consider the gauge-invariant form of the scalar electrodynamics with a massive vector field \\cite{Sop}. It differs from the Lagrangian \nof the massless scalar QED by the term $\\frac{1}{2}(mA^\\mu-\\partial^\\mu B)^2$ which \nmakes the vector field massive. $B$ is a scalar field and $m$ a mass parameter. The usual formulation with the condition $\\partial_\\mu A^\\mu = 0$ corresponds \nto the gauge $B=0$. In the $A^+=0$ gauge we obtain \n\\begin{equation}\nP^- = \\intv \\Big[F_{12}^2 + \\big(\\partial_i A^i)^2 + (\\partial_i B)^2 + m^2 A_i^2 \n+m^2B^2\\Big], \n\\label{lfmvbh}\n\\end{equation} \nplus the interaction terms. Comparing the two Hamiltonians, one can see that \nalso in the light-cone gauge picture of the LF Higgs mechanism the gauge field \nbecame massive possessing three components $(\\pi,A^1,A^2)$ with the mass \n$m=ev$. The mass term of the $\\sigma$ field is generated as in the previous case. \n \nIn summary, we gave three versions of the Higgs phenomenon in the light front \nabelian Higgs model for different gauge choices. Our light front formulation \nwas based on the finite-volume quantization with antiperiodic boundary \nconditions for the scalar fields. Minimization of the LF energy led \nto the semiquantum description of the degenerate vacuum states. In this way, \nthe concept of the trivial LF vacuum containing no quanta was generalized to \na more complex vacuum state with the non-trivial structure. The overall picture of the spontaneous breaking of the (abelian) gauge symmetry was thus found to \nbe quite analogous to the conventional theory quantized on the space-like \nhypersurface, namely one scalar field and the gauge field become massive \n(the tree-level masses $e^2v^2$ and $\\sqrt{2}\\mu$, respectively) and there \nis no massless Goldstone boson in the particle spectrum.\n\nThis work was supported by the grant No. APVT 51-005704 of the Slovak Research \nand Development Agency, by the French NATO fellowship and by \nIN2P3-CNRS. Hospitality of the LPTA Laboratory at the Montpellier University \nis also gratefully acknowledged by L.M..\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdnbb b/data_all_eng_slimpj/shuffled/split2/finalzzdnbb new file mode 100644 index 0000000000000000000000000000000000000000..e9b32123231e37593ceb8ad18ddfc944dfca3358 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzdnbb @@ -0,0 +1,5 @@ +{"text":"\\section{INTRODUCTION}\n\nMany complex systems might be represented as network structures, for example, human interactions or mobile phone telecommunications, food webs or gene interactions. In recent works, a lot of attention has been focused on the extraction of meaningful clusters to characterize networks at different levels \\cite{Arenas2008}. This clustering is essential to comprehend large networks and extract relevant statistical properties. Many researchers have proposed appropriate measures and algorithms to unfold community structures, i.e. groups of densely connected nodes \\cite{Porter2009, Fortunato2010}. However, this structural distribution of nodes in networks is not always representative and lack generalization in practical contexts. For instance, bipartite networks or cycle graphs do not contain communities although they may be heavily structured. Less attention has been paid to uncover more general structures which is known as roles extraction or block modeling \\cite{Wasserman1994,Cason2012}. In previous work \\cite{reichardt2007}, Reichardt \\& White had applied a similar approach than community detection in the framework of \\cite{reichardt2006} to extract roles in networks. In this paper, we assume that the different roles in a network should represent groups of nodes sharing the same behavior within the graph or, in other words, having similar flow patterns. This generalized the notion of communities which can also be described as roles where each node in a role mainly interacts with other nodes in the same role. But many other role interactions may be defined like, for example, a leader-follower model on social network interactions or a block cycle model for food webs. In this paper, we present a pairwise node similarity measure designed to derive such role models. This similarity measure compares the neighborhood patterns of every node and is expected to be high for any pair of nodes sharing analogous flow properties. Since computing the exact pairwise similarity is computationally expensive, we propose a low rank iterative scheme that approximates the similarity score and allows to analyze large networks. We will first present the similarity measure defined as the fixed point solution of a converging sequence. We will then introduce our low rank approximation and briefly demonstrate its convergence. Finally, we will apply the similarity measure and our low rank approximation to random graphs containing a structural block distribution of nodes, and show that they successfully extract the different roles within this kind of graph. We will also exhibit some evidences that analyzing the evolution of the low rank similarity measure can reveal the number of roles in the network. Lastly, we will show that the performances of both measures are quantitatively equivalent hence justifying the application of our low rank iterative scheme in practical contexts.\n\n\\section{Node-to-Node similarity}\nWe consider a weighted and directed graph $G_A(V,E)$, with $V$ the set of vertices and $E$ the set of edges, associated to its adjacency matrix $A\\in\\mathbb{R}^{n\\times n}$ where $A_{i,j}\\neq0$ if $(i,j)\\in E$ for $i,j \\in V$. Our similarity measure should reveal nodes having similar behaviors in the network which we will identify by the neighborhood patterns of each node. We define a neighborhood pattern of length $\\ell$ for a node as a sequence of length $\\ell$ of incoming (I) and outgoing (O) edges starting from the node, which we will call the \\emph{source} node. For example, the neighborhood patterns of length $1$ consist in exactly one edge and end up either in a parent (I) or in a child (O) of the source. If we consider neighborhood patterns of length $2$, then $4$ different types of nodes can be reached: the parent of a parent (I-I), the child of a parent (I-O), the parent of a child (O-I) or the child of a child (O-O). One can easily see that when the length of the neighborhood patterns is increased by $1$ the number of reachable nodes, which we will call the \\emph{target} nodes, is doubled.\n\nOur similarity measure reflects that a pair of nodes is highly similar if they have many neighborhood patterns in common, or in other words, if they can reach many targets with neighborhood patterns of the same kind and length. For example, using the patterns of length $1$, two source nodes will be more similar if they have many common parents (I) or many common children (O). \\fref{fig:pattern} shows all the possible common neighborhood patterns, up to length $3$, where the source nodes are represented as dark circles and each target node as a light gray square. One can compute the number of common target nodes for every pair of source nodes using neighborhood patterns of length $1$ as $$N_1 = AA^T + A^TA,$$ where the first term gives the number of common children (O) and the second term gives the number of common parents (I). Similarly, the number of common target nodes for neighborhood patterns of length $2$ is given by $$N_2 = AAA^TA^T + AA^TAA^T + A^TAA^TA + A^TA^TAA,$$\nwhere the different terms corresponds to the neighborhood patterns (O-O), (O-I), (I-O) and (I-I), respectively.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\framebox{\\parbox{0.85\\columnwidth}{\n\t\t\\includegraphics[width=0.85\\columnwidth]{pattern_fig.pdf} \n\t}}\n \\caption{All the different neighborhood patterns, up to length $3$, captured by the similarity measure $S_{i,j}$ \\eref{eq:S} with the source nodes $i$ and $j$ represented as dark circles and the target node represented as a light gray square.}\n\t\\label{fig:pattern}\n\\end{figure}\n\nOur pairwise node similarity measure $S\\in\\mathbb{R}^{n\\times n}$, previously introduced in \\cite{Denayer2012}, is then defined as\n\\begin{equation}\n\\label{eq:S_np}\nS = \\sum_{\\ell=0}^{\\infty} \\beta^{2(\\ell-1)}N_l,\n\\end{equation}\nwhere $\\beta\\in\\mathbb{R}$ is a scaling parameter. Hence our similarity measure computes the weighted sum of the number of common target nodes using neighborhood patterns of any length, and the contribution of the number of common targets using neighborhood patterns of length $\\ell+1$ is weighted by $\\beta^{2\\ell}$. \n\nOne can define an iterative sequence \n\\begin{equation}\n\\label{eq:S}\nS_{k+1} = \\Gamma_A\\left[I + \\beta^2S_k\\right],\n\\end{equation}\nwhere $\\Gamma$ is a linear operator,\n$$\\Gamma_A:\\mathbb{R}^{n\\times n}\\rightarrow\\mathbb{R}^{n\\times n}:\\Gamma_A[X] = AXA^T + A^TXA,$$\nsuch that\n$$S_{k+1} = \\Gamma_A\\left[I\\right] + \\dots + \\left(\\beta^{2}\\right)^{k}\\Gamma_A^{k+1}\\left[I\\right]+\\left(\\beta^2\\right)^{k+1}\\Gamma_A^{k+1}\\left[S_0\\right]$$\nwhere $\\Gamma_A^k[.]$ corresponds to applying $k$ times the operator $\\Gamma_A$. Hence, our similarity measure $S$ can be computed as the fixed point solution of the iterative sequence \\eref{eq:S}.\n\nOur similarity measure $S$ \\eref{eq:S_np} can be seen as a generalization of the measure proposed by {Cooper and Barahona} \\cite{Cooper2011,Diaz2013} for which the pairwise similarity $S^{CB}$ only compares the total number of paths originating or leading to a node, without comparing the targets or the sources of those paths. Furthermore, this similarity score $S^{CB}$ does not consider all types of neighborhood patterns, as represented in \\fref{fig:pattern}, but only restricts the measure to direct paths (represented in the first row of the panels $\\ell=2$ and $\\ell=3$ in the figure). While being easily computed, this makes the measure unable to extract a good pairwise similarity score for some particular graphs. For example, if one considers a regular block cycle graph, as represented in \\fref{fig:cycle}, where each role contains the same number of nodes and each node is connected to all the nodes in the following role in the cycle, the pairwise similarity measure $S^{CB}$ is of rank $1$ because all the nodes have a constant number of in\/out neighbors at all distances. This makes the extraction of roles in this network impossible using $S^{CB}$. On the contrary, our similarity measure \\eref{eq:S} produces a fixed point solution $S^*$ of rank equal to the number of roles in the network, with an obvious clustering that reveals the different roles. One can see that any $2$ nodes of the same role in the input graph are isomorphic, while any $2$ nodes of different roles are not. This is accurately represented by our measure $S^*$ but not by $S^{CB}$.\n\nThe similarity measure we propose in this paper might also be compared to the self similarity score introduced by {Blondel} et al.\\ \\cite{Blondel2004}. However, this measure has some drawbacks that are avoided using our iterative scheme \\eref{eq:S}, i.e.\\ the sequence $S_k$ converges for any initial matrix $S_0$ and the fixed point solution is unique. Moreover, it is known that the similarity score of {Blondel} et al.\\ $S^B$ is of rank $1$ when the adjacency matrix $A$ is normal. After scaling, $S^{B}$ is therefore the matrix of all ones as $S^{CB}$, which makes the analysis of the block cycle graph again impossible using this similarity measure.\n\n\\def 0.25\\columnwidth {0.25\\columnwidth}\n\\def \\myframebox [#1] {{#1}}\n\\begin{figure}[t]\n\t\\centering\n\t\\framebox{\\parbox[t][][c]{0.95\\columnwidth}{\n\t\t\\myframebox[\\parbox[t][0.25\\columnwidth][c]{0.2\\columnwidth}\n\t\t\t{\\includegraphics[width=0.2\\columnwidth]{cycle_graph.pdf}}]\n\t\t\\hfill\n\t\t\\myframebox[\\parbox[t][0.25\\columnwidth][c]{0.2\\columnwidth}{\\centering$A$\\\\[2pt]\n\t\t\t\\includegraphics[width=0.2\\columnwidth]{cycle_adjacency.pdf}\n\t\t\t}]\n\t\t\\hfill\n\t\t\\myframebox[\\parbox[t][0.25\\columnwidth][c]{0.2\\columnwidth}{\\centering$S^*$\\\\[2pt]\n\t\t\t\\includegraphics[width=0.2\\columnwidth]{cycle_S.pdf}\n\t\t\t}]\n\t\t\\hfill\n\t\t\\myframebox[\\parbox[t][0.25\\columnwidth][c]{0.2\\columnwidth}{\\centering$S^{CB}=S^{B}$\\\\[2pt]\n\t\t\t\\includegraphics[width=0.2\\columnwidth]{cycle_Sb.pdf}\n\t\t\t}]\n\t}}\n \\caption{From left to right: Block cycle role graph where each block has the same number of nodes and each node is connected to all the nodes in the following block. The large gray filled circles represent the roles and the small white circles represent the nodes of the graph; The adjacency matrix of the block cycle graph; The fixed point pairwise similarity score $S^*$, computed using \\eref{eq:S}, reveals all the different blocks; The pairwise similarity score of Cooper and Barahona $S^{CB}$ and Blondel et al.\\ $S^B$ are rank $1$ and do not exhibit the block structure.}\n\t\\label{fig:cycle}\n\\end{figure}\n\nThe parameter $\\beta$ in \\eref{eq:S} can be tuned to vary the weight of long neighborhood patterns but must be chosen wisely to ensure the convergence of the sequence $S_k$. If we initialize $S_0=0$, the iteration \\eref{eq:S} can be written for $k\\geq 1$ as \n\\begin{equation}\nS_{k+1} = S_1+ \\beta^2\\Gamma_A\\left[S_k\\right],\n\\label{eq:Ss}\n\\end{equation}\nwhere \n\\begin{equation}\nS_1 = AA^T + A^TA,\n\\label{eq:S1}\n\\end{equation}\nand the fixed point solution of \\eref{eq:S} is then given by \n\\begin{equation*}\nS^* = S_1 + \\beta^2\\left(AS^*A^T + A^TS^*A\\right),\n\\end{equation*}\nif the sequence converges. Using a classical property of the Kronecker product, this can be written as \n\\begin{equation*}\nvec(S^*) = \\left[I-\\beta^2\\left(A\\otimes A + \\left(A\\otimes A\\right)^T \\right) \\right]^{-1}vec\\left(S_1\\right)\n\\end{equation*}\nwhere $vec(S)$ denotes the vectorization of the matrix $S$, formed by stacking the columns of $S$ into one single column vector. It follows that, to ensure convergence, one can choose $\\beta$ such that\n\\begin{equation}\n\\beta^2\\leq\\frac{1}{\\rho\\left(A\\otimes A + \\left(A\\otimes A\\right)^T \\right)}\n\\label{eq:beta}\n\\end{equation}\nwhere $\\rho(.)$ denotes the spectral radius. Computing the exact upper bound for the parameter $\\beta$ to ensure convergence might be computationally expensive due to the Kronecker products $A\\otimes A \\in\\mathbb{R}^{n^2\\times n^2}$ if $A$ is non-symmetric. However, one can use an easily computed bound\n\\begin{equation}\n\\beta^2 \\leq \\frac{1}{\\rho\\left((A+A^T)\\otimes (A+A^T)\\right)} = \\frac{1}{\\rho\\left((A+A^T)\\right)^2}\n\\label{eq:beta_bound}\n\\end{equation}\nwhich ensures that the constraint \\eref{eq:beta} is satisfied. However, even if $\\beta$ is small enough to guarantee the convergence of the sequence \\eref{eq:Ss}, it might be impossible to compute the fixed point solution up to a small tolerance because of the increasing computational cost and memory requirement. Indeed, even if $A$ is sparse, the matrix $S_k$ tends to fill in as $k$ increases and each single iteration of \\eref{eq:Ss} is $O(n^3)$. This leads us to define a low-rank projected iteration to approximate the solution of \\eref{eq:Ss}. In the next section, we will introduce the low-rank iteration and briefly demonstrate its convergence.\n\n\\section{Low-rank similarity approximation}\nBecause the full rank fixed point solution of \\eref{eq:S} is often computationally too expensive to extract, we introduce a low-rank approximation of rank at most $r$ of $S^*$. Inspired from the formulation \\eref{eq:Ss}, we define the low rank iterative scheme as\n\\begin{equation}\nS^{(r)}_{k+1} = \\Pi^{(r)}\\left[ S^{(r)}_1+ \\beta^2\\Gamma_A\\left[S^{(r)}_k\\right]\\right] = X_{k+1}\\;X_{k+1}^{T}\n\\label{eq:LR}\n\\end{equation}\nwhere $X_k \\in \\mathbb{R}^{n\\times r}$ and $\\Pi^{(r)}\\left[.\\right]$ is the best low-rank projection on the dominant subspace which can be computed using a truncated singular value decomposition (\\textit{SVD}) of rank at most $r$. $S_1^{(r)}$ is the best low-rank approximation of $S_1$ which can be written as\n$$S_1 = \\left[A\\;\\left|\\right. A^T\\right]\\left[A\\;\\left|\\right. A^T\\right]^T,$$\nwhere $\\left[A\\;\\left|\\right. A^T\\right]$ is the horizontal concatenation of $A$ and $A^T$. This allows us to efficiently compute $S_1^{(r)}$ as\n\\begin{align*}\nS_1^{(r)} &= \\Pi^{(r)}\\left[\\left[A\\;\\left|\\right. A^T\\right]\\left[A\\;\\left|\\right. A^T\\right]^T\\right]\\\\\n\t\t&= U_1\\Sigma_1^2U_1^T = X_1X_1^T\n\\end{align*}\nwhere the columns of the unitary matrix $U_1\\in\\mathbb{R}^{n\\times r}$ span the dominant subspace of dimension at most $r$ of $\\left[A\\;\\left|\\right. A^T\\right]$ and $\\Sigma_1\\in\\mathbb{R}^{r\\times r}$ is a diagonal matrix of the dominant singular values, i.e.\\ $\\left[A\\;\\left|\\right. A^T\\right] \\approx U_1\\Sigma_1V_1^T$. To compute each iterative solution of \\eref{eq:LR}, one can see that\n\\begin{align*}\nS^{(r)}_1+ \\beta^2\\Gamma_A\\left[S^{(r)}_k\\right] &= X_1X_1^T + \\beta^2AX_kX_k^TA^T\\\\\n&\\hspace{25pt}+\\beta^2A^TX_kX_k^TA\\\\\n&=Y_k\\;Y_k^T\n\\end{align*}\nwhere\n$$Y_k = \\left[X_1\\;\\left|\\right.\\beta AX_k\\;\\left|\\right.\\beta A^TX_k\\right],$$\nwhich leads to\n$$X_{k+1}X_{k+1}^T = \\Pi^{(r)}\\left[Y_kY_k^T\\right].$$\nTo efficiently compute $X_{k+1}$, we first apply a \\textit{QR} factorization to $Y_k = Q_kR_k$, then compute a truncated \\textit{SVD} of rank at most $r$ of $R_k$ such that $R_k = \\mathcal{U}_k{\\Omega}_k\\mathcal{V}_k$ and finally compute $$X_{k+1} = Q_k\\mathcal{U}_k\\Omega_k.$$\nOne can prove, using perturbation theory \\cite{stewart1973}, that the iterative scheme \\eref{eq:LR} converges locally to a fixed point solution $S^{(r)}$ if the spectral gap at the $r^{th}$ singular value is sufficiently large. Without going into the details of the demonstration of the convergence, let us mention some interesting results that follow from it. First, we consider the function $$f(S) = S_1^{(r)} + \\beta^2\\Gamma_A\\left[S\\right].$$\nClearly, since $S^{(r)}$ is a fixed point solution of \\eref{eq:LR}, we know that there exist a unitary matrix $U\\in\\mathbb{R}^{n\\times r}$ and a diagonal matrix $\\Sigma\\in\\mathbb{R}^{r\\times r}$ such that $S^{(r)} = U\\Sigma^2U^T$ and\n$$[U\\;V]^T\\;f(S^{(r)})\\;[U\\;V] = \n\\left[\\begin{array}{cc}\n\\Sigma^2 &\\\\\n& \\sigma^2\n\\end{array}\\right]\n$$\nwhere $\\Sigma_{i,i}>\\sigma_{j,j}$ $\\forall i,j$ because we assumed that the fixed point solution has a positive spectral gap at the $r^{th}$ singular value.\n\n\\noindent Then, we consider a small symmetric perturbation $\\Delta$ and, using the linearity of the operator $\\Gamma_A[.]$, one can write that $$f(S^{(r)}+\\Delta) = f(S^{(r)}) + \\beta^2\\Gamma_A\\left[\\Delta\\right]$$ and\n$$\n[U\\;V]^T\\;\\left(f(S^{(r)}) + \\beta^2\\Gamma[\\Delta]\\right)\\;[U\\;V] = \n\\left[\\begin{array}{cc}\nE_{11}&E_{21}^T\\\\\nE_{21}&E_{22}\n\\end{array}\\right].\n$$\nSince $U$ is in general not an invariant subspace of $f(S^{(r)}+\\Delta)$, $E_{21}$ will be non-zero.\n\\noindent However, we know from \\cite{stewart1973} that there exists a unitary matrix $Q$ such that $UQ$ is an invariant subspace of $f(S^{(r)}+\\Delta)$ if $$0\\leq 4\\beta^2\\left\\|\\Gamma\\left[\\Delta\\right]\\right\\|_F \\leq \\Sigma_{k,k}^2-\\sigma_{1,1}^2.$$ \nIf $\\left\\|\\Delta\\right\\|_F$ is sufficiently small, the rotation matrix $Q$ will not perturb too much the singular values of $f(S^{(r)})$, so $UQ$ will not only be an invariant but also the dominant subspace of $f(S^{(r)}+\\Delta)$, hence the local convergence of the low-rank iterative scheme is guaranteed for sufficiently small $\\beta$. This leads to the following bound for the distance between $S^{(r)}$ and the projection of $f(S^{(r)}+\\Delta)$\n\\begin{equation*}\n\\norm{S^{(r)}-\\Pi^{(r)}\\left[f(S^{(r)}+\\Delta)\\right]}_F \\leq\\gamma\\norm{\\Delta}_F\n\\end{equation*}\nwhere $\\gamma < 1$ if $$\\beta^2 < \\frac{1}{\\norm{A\\otimes A + A^T\\otimes A^T}_2\\left(\\frac{4\\norm{\\Sigma^2}}{\\Sigma_{k,k}^2-\\sigma_{1,1}^2}+1\\right)}$$\nwhich shows the existence of $\\beta$ such that the iteration \\eref{eq:LR} converges.\n\n\\noindent In the next section, we will apply our low-rank iterative scheme to {Erd\\H{o}s-R\\'{e}nyi} random graphs and demonstrate that it allows to successfully extract roles in those networks.\n\n\\section{Numerical experiments}\n\nWe applied our similarity measure to extract roles in {Erd\\H{o}s-R\\'{e}nyi} random graphs containing a block structure. To build such graphs, we first choose a directed role graph $G_B(V_B,E_B)$, i.e.\\ each node in $G_B$ defines a role that we would like to identify. Some of the role graphs that we considered are represented in the first column of each panel of \\fref{fig:rank}. As previously, in the role graphs, the large gray filled circles represent the different roles and the small white circles represent the nodes of the graph. The role graph in the first panel corresponds to a community structure where nodes in a role interact mainly with other nodes in the same role. This kind of role graph often occurs when considering human interactions in online social networks for example \\cite{Kumpula2009} but has been observed in many other networks \\cite{Fortunato2010}. The second panel represents a block cycle role graph, already presented in \\fref{fig:cycle}, where each node interacts mainly with nodes in the following role in the cycle. This role graph might represent the behavior of animals in some particular food webs. In the third and fourth panels, the role graphs were simply chosen as representative examples for more complex role interactions without precise real life example in mind.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\parbox{0.98\\textwidth}{\n\t\\centering\n\\includegraphics[height=0.85\\textheight]{rank_analysis.pdf} \n\t}\n \\caption{Evolution of the adjacency matrix, the extracted roles and the low rank similarity measure $S^{(r)}$ for different random graphs. Each panel corresponds to a chosen role graph represented in the first column. Each panel is divided in $4$ sections corresponding to different values of $p_{in}$ and $p_{out}$. In each section, we present one realization of the adjacency matrix, then the extracted role assignment for each node and finally the evolution $\\norm{S^*-S^{(r)}}_F$ and $\\norm{S^{(r)} - S^{(r+1)}}_F$ for increasing values of $r$.\n}\n\t\\label{fig:rank}\n\\end{figure*}\n\nOnce the role graph $G_B$ has been chosen, we build a random graph $G_A(V_A,E_A)$ where each node in $G_A$ has a corresponding role in $G_B$. That is, for each node $i\\in V_A$, we select a role $R(i)\\in V_B$. Then, we add the edges in $E_A$ using $2$ probability parameters. For every pair of nodes $i,j\\in V_A$, we add the edge $(i,j)\\in E_A$ with probability $p_{in}$ if there is an edge between the corresponding roles in $G_B$, i.e.\\ $\\left(R(i),R(j)\\right)\\in E_B$. If there is no edge between the corresponding roles in $G_B$, the edge is still added with a probability $p_{out}$.\nIf $p_{in}$ is much larger than $p_{out}$, then the role graph $G_B$ is accurately representing the different roles in the graph $G_A$ and it is expected that the pairwise similarity $S^*$ between the vertices $V_A$ should allow the extraction of those roles. On the other hand, if $p_{out}$ is much larger than $p_{in}$, then the different roles in $G_A$ are more closely represented by the complement graph of $G_B$ represented by the adjacency matrix $\\mathbf{1}\\mathbf{1}^T-B$. However, the role structure is still strongly existing in this complement graph and it is expected that the similarity measure $S^*$ should still be able to differentiate them. It is when the $2$ probabilities $p_{in}$ and $p_{out}$ are close to each other that extracting the different roles becomes challenging but, at the same time, the graph becomes closer to a uniform {Erd\\H{o}s-R\\'{e}nyi} graph which is known to be free of any structure.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\parbox{1.95\\columnwidth}{\n\t\\centering\n\\includegraphics[width=1.9\\columnwidth]{qualityAnalysis.pdf} \n\t}\n \\caption{Average normalized mutual information between the exact role structure and the extracted role structure using the full rank and the low rank similarity measure on different role graphs.\n}\n\t\\label{fig:quality}\n\\end{figure*}\n\nEach of the panels of \\fref{fig:rank} is divided into $4$ sections corresponding to different values of $p_{in}$ and $p_{out}$ for a single role graph, as follows\n\\def1.5{1.5}\n$$\n\\begin{array}{|c|c|c|}\n\\hline\n\\multirow{2}{*}{\\quad} & p_{in} = 0.9 \/ p_{out} = 0.1 & p_{in} = 0.8 \/ p_{out} = 0.2 \\\\ \\cline{2-3}\n\t\t\t\t& p_{in} = 0.7 \/ p_{out} = 0.3 & p_{in} = 0.6 \/ p_{out} = 0.4 \\\\\n\t\t\t\t\\hline\n\\end{array}\n$$\n\n\n\n\\noindent In each section of a panel of \\fref{fig:rank}, we first present the adjacency matrix of one realization of the random graphs $G_A$ generated. For visual clarity, the matrices have been permuted such that nodes in the same role are next to each other. Then, we represent the role assignment of each node extracted using our low rank similarity matrix $S^{(r)}$ for $r=10$. Based on the similarity measure $S^{(r)}$, we extract the role assignment of each node using the community detection algorithm presented in \\cite{Browet2013}. Indeed, within each role, we expect nodes to be highly similar in their neighborhood patterns. This should lead to a similarity graph, whose weighted adjacency matrix is the similarity matrix $S^{(r)}$, with groups of highly connected nodes in each role, hence containing a community structure. Since the algorithm produces hierarchical communities, we present each level of roles when different levels of clustering were extracted from the similarity graph.\n\n\\noindent The last plot in each section of a panel in \\fref{fig:rank} represents the evolution of $S^{(r)}$ for increasing values of $r$. That is, we compute the norm of the difference between the full rank and each low rank solutions, $\\norm{S^*-S^{(r)}}_F$, and between consecutive low rank solutions, $\\norm{S^{(r)} - S^{(r+1)}}_F$, for increasing values of $r$. This should reveal the minimal rank required for $S^{(r)}$ to be a qualitatively good approximation of $S^*$.\n\nThe results of \\fref{fig:rank} clearly show that the different roles within each network can be well extracted using the low rank similarity graph up to some high level of noise. In the first role network, each community is correctly extracted until $p_{in} = 0.6$ and $p_{out} = 0.4$. However, even if the network is really noisy for those parameters, as represented by the adjacency matrix, the first and the third communities are pretty well clustered and the second community is mainly split in $2$. The same observation might be done for the block cycle role graph for which all the roles are perfectly extracted for the first three probability parameters. Again, in the last section, the second and third roles are essentially split in $2$ different clusters but there is only a few nodes with inappropriate role assignments in the final level of clustering. Those first $2$ role graphs have a strong role structure and adding any edge in the role graph would not alter it, i.e.\\ this will not create isomorphic roles. This explains why the role structures are correctly extracted even for high level of noise.\n\nThe third role graph is less strongly defined because if one edge is added from the second block to the first block, the second and the third roles would become isomorphic. Indeed, we observe that some nodes are incorrectly clustered from the second role to the third role for $p_{in}=0.7$ and $p_{out}=0.3$. This might also explain why the second and third roles are grouped together in the final level of clustering for the first two probability parameters but this might also be due to some resolution limit of the community detection algorithm \\cite{Schaub2012,Fortunato2007}. For high level of noise, clustering the pairwise similarity matrix does not provide an accurate result, however, the adjacency matrix clearly indicates that the role structure is very weak.\n\nIn the last role graph composed of $4$ distinct blocks, the results are again reasonably good. Except for an additional merge in the last level of clustering for $p_{in}=0.9$ and $p_{out}=0.1$, all the roles are correctly extracted, and even for the last probability parameters, leading to a high level of noise, each role tends to be correctly extracted. There is only a small number of nodes incorrectly classified for the first and last blocks and the second and third blocks are only bisected as previously observed.\n\nWe also observe that the evolution of the low rank similarity matrix $S^{(r)}$ might be used to reveal the number of roles in the networks. Indeed, when the different roles in the networks are strongly defined, we observe an abrupt variation in the decay of the norm of the differences $\\norm{S^*-S^{(r)}}_F$ and $\\norm{S^{(r)}-S^{(r+1)}}_F$. This abrupt variation indicates that we do not need to consider larger values of the rank to extract qualitatively good roles in the networks, since the gain in precision for the similarity measure starts to decrease very slowly afterwards. What is also interesting is that this abrupt variation always occurs when the rank hits the exact number of roles in the networks. When the networks are highly noisy, we do not observe such an abrupt variation which could indicate that the clustering of the nodes according to the similarity matrix will not produce relevant results. Observing the evolution of the low rank similarity matrix could become a strong indicator of the quality of the extracted roles for real networks when the exact block structure is not known.\n\nFinally, we compare quantitatively the extracted clusters using the full rank similarity $S^*$ and the low rank similarity measure $S^{(r)}$. For each of the different role graphs previously introduced, we compute the normalized mutual information (\\textit{NMI}) \\cite{Danon2005} between the exact role structure and the extracted role assignments using $S^*$ or $S^{(r)}$ and the community detection algorithm. The NMI ranges in $[0,1]$ and is large if the two distributions are similar. More precisely, for each role graph, we generate $20$ random graphs for each couple of probability parameters $p_{in}$ and $p_{out}$ in $[0,1]$ with a discretization step size of $0.05$, and we compute the average NMI on those $20$ realizations of the {Erd\\H{o}s-R\\'{e}nyi} random graphs. The results are presented in \\fref{fig:quality}. As expected, we observe that the extracted roles are accurate when either $p_{in} >> p_{out}$ or the opposite. As we mentioned previously, the third role graph seems harder to recover due to either a resolution limit phenomenon or to the almost isomorphic behavior of two of the role nodes. Nevertheless, we observe that the low rank similarity matrix $S^{(r)}$ produces almost identical results than the full rank similarity $S^*$. This leads us to conclude that, if the rank is sufficiently large, one can always use our low rank pairwise similarity measure to extract role structures in networks. The low rank similarity matrix will always be easier to compute and will produce highly similar results.\n\n\\section{Conclusion}\n\nIn this paper, we present a pairwise similarity measure between the nodes of a graph that allows the extraction of roles or block structures within the graph. Those roles generalized the concept of communities often studied in the literature. Then, we present a low rank iterative scheme to approximate the pairwise similarity measure and prove its convergence when the parameter $\\beta$ is sufficiently small. We applied the similarity measure and its low rank approximation to {Erd\\H{o}s-R\\'{e}nyi} random graphs containing a block structure and showed that, if the noise level is not too large and the block structure correctly represents the different roles of the nodes in the network, our similarity measure and its low rank approximation accurately extract these blocks. We also show that analyzing the evolution of the low rank similarity measure might reveal the number of roles in the networks and also might indicate if the extracted cluster are relevant. Finally, we demonstrate that the pairwise similarity measure and the low rank approximation produce very similar results, hence justifying the use of the low rank approximation in practical examples when computing the full rank measure is computationally too expensive. In future works, we plan to apply our low rank similarity measure to other kinds of random graphs, e.g.\\ scale-free networks. We will also apply our measure to real networks like food webs, international exchange networks or words graphs to automatically uncover similar type of words in the construction of sentences, known as ``tagging'' in natural language processing. We will also analyze the behavior of our similarity measure against weighted networks.\n\n\n\n \n\n\\addtolength{\\textheight}{-12cm} \n \n \n \n \n \n\n\n\n\n\n\n\n\n\\section*{ACKNOWLEDGMENT}\n\nWe would like to thank N. Boumal, R. Jungers and J. Hendrickx for their useful comments regarding the subject.\\\\\n\n\\noindent This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its author(s).\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\n\n\n\n\nArtificial General Intelligence (AGI) requires an intelligent agent to solve a wide variety of real-world tasks.\nLearning to solve these tasks efficiently involves sharing knowledge between tasks, and systematic generalization from relatively few samples. \nIn contrast, agents trained with Reinforcement Learning (RL) frequently fall short in this regard: they rely on excessive amounts of data~\\cite{francois-lavet2018introduction} and are unable to generalize beyond their initial training regime~\\cite{leike2017ai}. %\n\n\n\nModel-based RL promises to alleviate this problem by using a general (non task-specific) world model that captures the latent structure of the environment.\nThis more abstract knowledge about the world is expected to be useful for many (even novel) tasks and facilitate simulation and planning.\nIt is unclear what constitutes a good model, and frequently models are either engineered~\\cite{deavilabelbute-peres2018endtoend} or obtained by training a deep (recurrent) neural network to predict future states of the world~\\cite{schmidhuber1990making,ha2018recurrent}. \nIn the latter case, an underlying assumption is that the learned representations of such a network present suitable abstractions for transfer and planning, analogous to the versatility of features learned by a deep convolutional image classifier.\nHowever, the limited success of learned models for model-based RL in these domains raises doubts about the validity of this assumption.\n\nIn this work, we argue that, rather than learning a single monolithic model that handles all situations in all environments, what is needed is a flexible system which dynamically infers a suitable model on the fly. %\nA human playing the game of \\emph{Space Invaders} uses a mental model that revolves around space ships and aliens, without simultaneously also considering all other aspects of the real world that are relevant for other tasks.\nHumans are also quick to adapt their model to new information by adding or removing additional assumptions.\nFor example, reading the manual of a game before playing greatly increases first-episode performance~\\cite{tsividis2017human}. %\n\nInitially, it may appear that in arguing for a dynamic model we have mostly made the task of model-learning harder: we now require learning many different models that fit specific situations. \nWhy then would we expect such a model to perform any better or even work at all?\n\n\n\n\n\n\n\n\n\n\n\\paragraph{Objects}\nObjects are the key piece to this puzzle, in that they facilitate the modular reuse of prior knowledge and the combinatorial construction of novel models.\nIt is well-established that objects play a central role in human cognition, both for internal reasoning and as the basis for communicating about the world.\nIndeed, objects are widely considered to be core knowledge~\\cite{spelke2007core}, and infants learn about objects already within their first year of life~\\cite{munakata1997rethinking}.\n\nRL methods that leverage the combinatorics of objects and relations have shown similar benefits in terms of systematic generalization~\\cite{zambaldi2019deep}, sample efficiency~\\cite{diuk2008objectoriented}, and transferring skills and knowledge across domains~\\cite{kansky2017schema}. %\nRecently, there appears to be an emerging consensus that objects are important in learning intelligent agents~\\cite{lake2017building}, while it remains unclear in how to fully realize this potential.\nThe discrete and compositional nature of objects seems at odds with many of the core tenets of connectionism, and they are unlikely to emerge naturally in neural networks.\nReconciling the two is a difficult problem and requires careful thought to ensure a synergistic integration. %\n\n\n\n\n\n\n\n\\paragraph{The Binding Problem}\n\nHow then should we think about objects?\nWhy do they not simply emerge in neural networks, what is missing, and how can this be addressed?\nMany of these questions have been raised and debated in theoretical neuroscience and have become known as the binding problem: How does the brain bind features together into objects while keeping them separate from other objects.\nInspired by this literature (cf. \\citet{treisman1999solutions, vondermalsburg1995binding}) we will focus on three main challenges in incorporating objects in connectionist models of the world: segregation, representation, and composing, which we discuss in the next sections.\n\nSegregation is about object discovery, i.e. given a set of observations, what are good candidates for representational objects and how can they be extracted.\nRepresentation is about storing this representational content in neural networks, and as we will find, plain fully-connected feedforward networks are ill-equipped to solve this task.\nFinally, composition is about using representational objects efficiently in a way that ensures combinatorial generalization (\\emph{systematicity}; \\citet{niklasson1994being}).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Representation}\n\\label{sec:representation}\n\n\n\n\nWhat are good object representations? \nIf objects are to serve as the primitives for compositional reasoning, it is important that their representations support that end.\nHere we argue for three main requirements:\\footnote{Keeping space limitations in mind we will not attempt at exhaustively listing all requirements. Instead we focus on those that we believe to be most important.}\n\n\\begin{description}\n \\item[Universal] Each object representation should be able to represent any object regardless of position, class or other properties. \n It should facilitate generalization, even to unseen objects (zero-shot generalization), which in practice means that its representation should be distributed and disentangled.\n \\item[Multi-Object] It should be possible to represent multiple objects simultaneously, such that they can be related and composed but also transformed individually. \n This only needs to cover a small number of objects at the same time (e.g. $7\\pm2$; \\citealt{miller1956magical}), since there is an intractable number of possible objects. \n Instead, objects should be swapped in and out of this working memory on demand.\n \\item[Common Format] All objects should be represented in the same format, i.e. in terms of the same features. \n This makes representations comparable, provides a unified interface for compositional reasoning and allows the transfer of knowledge between objects. %\n\\end{description}\n\n\nIt is easy to see how regular representations of fully-connected neural networks fall short in this regard:\nWhen representing multiple objects, they can either reuse the same features for all objects simultaneously, thus superimposing representations which leads to ambiguities (\\{red, square\\} + \\{blue, triangle\\} = \\{red, blue, square, triangle\\}).\nAlternatively, they can allocate a different set of features per object which violates common format.\nWithout any architectural bias in the form of weight sharing, useful multi-object representations are thus unlikely to emerge naturally in a neural network.\nIn what way can this problem be addressed?\n\nWeight sharing, as it is used, for example, in ConvNets and RNNs, is a step in the right direction.\nWe call these approaches ``slot-based'' because they provide several slots that all share weights and can thus be used to represent objects in a common format.\nIn the case of RNNs there is one slot per time-step~\\cite{eslami2016attend}, while in ConvNets there is one slot per spatial location in the image~\\cite{santoro2017simple}.\nNote that both are in slight violation of universality because they tie a slot to a specific time step or location, while RNNs additionally do not \\emph{simultaneously} represent multiple objects. \nWe can extend the idea of representational slots and consider a setting in which each object has its own universal slot and all slots share a common format (instance slots, c.f.~\\cref{fig:slots}).\nWhile this constitutes a good object representation, it raises another problem: if all slots are identical and share weights, then how do they not end up all representing the same object?\nSolving this conundrum requires a dynamic information routing process that goes beyond simple feed-forward processing (see~\\cref{sec:segregation}). \n\nThere are two, less developed, alternatives to slot-based approaches that have the potential to meet our requirements: \n\\emph{Augmentation} approaches keep a single set of features but augment each feature to include some extra grouping information. \nExamples include complex-valued activations (e.g. \\citet{reichert2013neuronal}) or spiking networks that encode grouping via synchronization (e.g. \\citet{lane1998simple}).\n\\emph{Embedding} approaches carefully embed multi-object representations in a higher-dimensional space (e.g. \\emph{Tensor Product Representations};~\\citealt{smolensky1990tensor}).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{approach_replicate.pdf}\n\\caption{Different types of \\emph{slot-based} representation strategies.}\n\\label{fig:slots}\n\\vspace{-10pt}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Segregation}\n\\label{sec:segregation}\n\nIt can be difficult to provide precise boundaries or definitions even for concrete objects like a tree, a mountain, and a river.\nMatters become even worse with slightly more abstract objects like a hole, a shadow, or a street corner.\nClearly, sensory information does not come pre-structured into objects, yet we so effortlessly and consistently perceive them. %\nHow can we aid our agents in developing an equally general understanding of objects?\nWe address this by focusing on the role of objects as computational primitives in a compositional reasoning system, namely as abstract patterns of the input data that are modular, dynamic, and consistent:\n\n\n\n\n\\begin{description}\n \\item[Modular] Objects subdivide the input into parts with strong internal coherence while being mostly independent of each other given some task under consideration.\n This division can be thought of as a form of clustering by mutual predictability and helps minimize the error that results from treating them as independent entities. \n \n \n \\item[Dynamic] Objects are task-dependent, i.e. there is no one fixed definition of objects that applies to all tasks.\n For example, objects can be part-whole hierarchies whose parts are objects themselves: a stack of chairs can be viewed as a single object (the stack) or as multiple objects (the individual chairs). \n It necessitates top-down feedback: interaction between the up-stream problem solving and down-stream segregation to obtain a dynamic definition of objects.\n \\item[Consistent] \n Representational objects often ``refer to'' physical objects in the real world (although this does not need to be the case), and their usefulness depends on the reliability of that link. \n The output of the segregation process must thus be stable and consistent to ensure that the results from internal reasoning can be mapped back onto the environment.\n Consistency is also important in communication (different agents should agree on objects), and in the absence of information, e.g. as a result of occlusion.\n\\end{description}\n\n\n\n\nModularity rules out standard convolutional neural networks as a means to learn object representations given by the representational content at each spatial slot.\nEach convolutional layer with a kernel size exceeding $1 \\times 1$ creates dependencies between local spatial neighborhoods. \nThrough depth, the representational content of upper layers encode information from all spatial positions and are no longer modular: a change affecting a single object in the input image affects the representations at \\emph{all} spatial locations in the upper layers. \n\nDynamicity implies that we can not treat segregation as a pre-processing step that extracts objects from input data.\nThis rules out the use of large quantities of labeled data to pre-train an image segmenter, or the use of domain-specific engineering as is commonly found in generative models that essentially encode a fixed definition of object.\nMoreover, human labor is an expensive resource that we can not spend exhaustively to overcome all possible situations.\n\n\nWe conclude that to a large extend object learning must be \\emph{unsupervised} through a specialized mechanism that allows for the possibility to incorporate top-down feedback.\nTwo promising approaches from the literature are \\emph{attention} and \\emph{differentiable clustering}.\n\nAttention mechanisms are used to selectively attend to a subset of the image, i.e. parts that correspond to a single object~\\cite{schmidhuber1991learning, eslami2016attend}.\nIn this way, attention restricts the information intake and ensures that the resulting representations are modular. \nTop-down feedback can be incorporated by granting control of the attention window to the agent that learns to solve some task~\\cite{mnih2014recurrent}.\nA downside is that objects are processed in an iterative fashion, which may make it more challenging to reason about multiple objects simultaneously~\\cite{kosiorek2018sequential}.\n\nAn alternative mechanism is differentiable clustering, which seeks to partition the input in a number of segments while learning the similarity function.\nIndividual segments are disjoint and result in modularity, while the iterative nature of these clustering procedures allow top-down feedback to be incorporated~\\cite{greff2017neural, greff2019multiobject}.\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\\section{Composition}\n\\label{sec:composition}\n\n\n\n\n\n\nLet us now assume that representation and segregation have been addressed, and we have available a set of relevant independent objects represented in a common format.\nNote that when used correctly, these object representations can already make tasks like performing basic feature comparisons very easy.\nFor example, a function that receives a pair of objects as input and compares their size-related features could easily be learned, and would almost automatically generalize to arbitrary pairs of objects.%\n\nIn contrast, combinatorial generalization is not a given for more complex relational reasoning. \nWhile it also involves learning general functions that accept objects as their arguments, one has to take extra care in being able to flexibly assign the right objects to their corresponding function arguments, as well as in learning about different structural forms that imply different ways of generalizing~\\cite{kemp2008discovery}.\nThese then imply the following requirements:\n\\begin{description}\n\\item[General Relations]\nRelations differ both in their meaning and in the patterns of generalization that they imply. \nA general reasoning system, therefore, has to be able to instantiate many different types of relations, which necessitates a general representational form.\n\n\\item[Dynamic Binding] \nIn order to construct a model for a specific situation, the system needs the flexibility to freely combine objects and relations into an arbitrary structure.\nBoth the structure of relations and the associated objects (\\emph{variable binding}; \\citealt{browne2000connectionist}) have to be inferred dynamically during run-time.\n\n\\item[Role-filler Independence] The content of objects should be independent of their structural roles~\\cite{hummel2003symbolicconnectionist}.\nThat is, any object can take part in any relation, and the interpretation of the whole is determined by both the parts and the structure.\nThis is related to \\emph{common format} and is the key to compositionality that enables the powerful systematic generalization that is characteristic of many symbolic systems.\n\n\\end{description}\n\nOne approach is to implement complex relational reasoning in a sequential fashion. \nAt each step, an object associated with a particular role is processed and the resulting intermediate computation is stored, to be combined in the next step.\nWhile it is clear that a plain RNN can perform this type of computation, the dual role of intermediate representations in representing objects and intermediate computation suggests a very specific function that may be hard to learn~\\cite{graves2014neural}.\nAlternatively, by combining an RNN with a suitable memory mechanism (eg.~\\citet{Das:92,mozer1993connectionist,reed2015neural,graves2016hybrid}) or fast\nweights (eg.~\\citet{Schmidhuber:92ncfastweights,Schmidhuber:93ratioicann,schlag2018learning}) it may be more easy to learn general functions of this kind.\n\n\nAn alternative approach is to embed objects, and intermediate representations as nodes in a (directed) graph and let computation take place along its edges.\nThese computation graphs can implement arbitrary relationships, including recursive computation by re-applying the same function successively.\nGraph Networks~\\cite{battaglia2018relational} structure neural network computations according to this underlying graph and perform relational reasoning through repeated message-passing between the nodes in the graph. %\nCompositionality is achieved through weight-sharing, i.e. by learning a general function that operates on (pairs of) nodes following their topological relationship.\nHowever, while graph networks have been successfully applied in the domain of physical reasoning (e.g. \\citet{battaglia2016interaction, vansteenkiste2018relational}), a remaining challenge is in dynamically inferring the right structure (i.e. dynamic binding).\n\nWhile graph networks appear most promising in addressing the challenges of composition, one other approach deserves a mention.\n\\emph{Embedding approaches}, such as Poincar\\'{e} embeddings~\\cite{nickel2017poincare}, generalize Euclidean representations to other spaces that more suited in modeling certain types of relations, in this case: hierarchical relationships.\nHowever, the feature representations are essentially adapted to reflect the underlying relation during training, which implies fixed roles and binding during inference.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe have argued that feature representations alone are inadequate abstractions for planning, reasoning, and for systematically transferring knowledge to novel situations.\nTo meet the diverse challenges on our quest towards AGI an agent needs to be able to dynamically construct new models about its environment on the fly while reusing as much prior knowledge as possible. \nWe have argued that objects (dynamically bound features) are adequate building blocks to quickly and flexibly compose such task-specific models.\nAlthough our examples were centered around vision, we believe that the notion of objects applies equally to other domains like audio, tactile and even abstract thought.\nBy focusing on their role as compositional primitives, we have identified some inductive biases that we believe are necessary for objects to arise within a connectionist system.\nThey can be categorized into three areas: representation, segregation, and composition of objects.\n\nAmong these three we find that segregation is most frequently neglected and deserves more attention.\nCommon approaches rely either on some combination of pre-processing pipelines, supervision, or highly engineered generative models of objects.\nMeanwhile, the few approaches that tackle this challenge in a holistic and unsupervised way are brittle and have not yet been scaled to real-world data.\nDeveloping better methods for tackling the segregation problem within the framework of connectionism is going to be a central challenge on the way towards AGI.\nSimilarly, we would like to stress the importance of integrating solutions to all three aspects into a single system.\nThe potential of objects as modular building blocks can only be realized in full if they are both informed by learned representations, and by feedback from the composite model. \n\nAnother important direction is the integration of objects with other critical cognitive mechanisms such as attention and memory.\nBecause objects are optimized to be modular, they naturally aggregate features that need to be processed together, but which can be separated from other information.\nThis makes them ideal primitives for attention and for storage and retrieval from long-term memory.\nAttention, in turn, can simplify a task by filtering out irrelevant information and can guide the processing required for more complex reasoning chains.\nSuch a reasoning process could then also query objects from memory on demand to be compared to or integrated with the current model. \n\nWith this short essay, we hope to draw attention to the intricacies of objects and inspire others to think critically about their integration in connectionist models. \n\n \n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\nThis research was funded by SNF grant 200021\\_165675\/1.\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\n\n\n\n\n\nArtificial General Intelligence (AGI) requires an intelligent agent to solve a wide variety of real-world tasks.\nLearning to solve these tasks efficiently involves sharing knowledge between tasks, and systematic generalization from relatively few samples. \nIn contrast, agents trained with Reinforcement Learning (RL) frequently fall short in this regard: they rely on excessive amounts of data~\\cite{francois-lavet2018introduction} and are unable to generalize beyond their initial training regime~\\cite{leike2017ai}. %\n\n\n\nModel-based RL promises to alleviate this problem by using a general (non task-specific) world model that captures the latent structure of the environment.\nThis more abstract knowledge about the world is expected to be useful for many (even novel) tasks and facilitate simulation and planning.\nIt is unclear what constitutes a good model, and frequently models are either engineered~\\cite{deavilabelbute-peres2018endtoend} or obtained by training a deep (recurrent) neural network to predict future states of the world~\\cite{schmidhuber1990making,ha2018recurrent}. \nIn the latter case, an underlying assumption is that the learned representations of such a network present suitable abstractions for transfer and planning, analogous to the versatility of features learned by a deep convolutional image classifier.\nHowever, the limited success of learned models for model-based RL in these domains raises doubts about the validity of this assumption.\n\nIn this work, we argue that, rather than learning a single monolithic model that handles all situations in all environments, what is needed is a flexible system which dynamically infers a suitable model on the fly. %\nA human playing the game of \\emph{Space Invaders} uses a mental model that revolves around space ships and aliens, without simultaneously also considering all other aspects of the real world that are relevant for other tasks.\nHumans are also quick to adapt their model to new information by adding or removing additional assumptions.\nFor example, reading the manual of a game before playing greatly increases first-episode performance~\\cite{tsividis2017human}. %\n\nInitially, it may appear that in arguing for a dynamic model we have mostly made the task of model-learning harder: we now require learning many different models that fit specific situations. \nWhy then would we expect such a model to perform any better or even work at all?\n\n\n\n\n\n\n\n\n\n\n\\paragraph{Objects}\nObjects are the key piece to this puzzle, in that they facilitate the modular reuse of prior knowledge and the combinatorial construction of novel models.\nIt is well-established that objects play a central role in human cognition, both for internal reasoning and as the basis for communicating about the world.\nIndeed, objects are widely considered to be core knowledge~\\cite{spelke2007core}, and infants learn about objects already within their first year of life~\\cite{munakata1997rethinking}.\n\nRL methods that leverage the combinatorics of objects and relations have shown similar benefits in terms of systematic generalization~\\cite{zambaldi2019deep}, sample efficiency~\\cite{diuk2008objectoriented}, and transferring skills and knowledge across domains~\\cite{kansky2017schema}. %\nRecently, there appears to be an emerging consensus that objects are important in learning intelligent agents~\\cite{lake2017building}, while it remains unclear in how to fully realize this potential.\nThe discrete and compositional nature of objects seems at odds with many of the core tenets of connectionism, and they are unlikely to emerge naturally in neural networks.\nReconciling the two is a difficult problem and requires careful thought to ensure a synergistic integration. %\n\n\n\n\n\n\n\n\\paragraph{The Binding Problem}\n\nHow then should we think about objects?\nWhy do they not simply emerge in neural networks, what is missing, and how can this be addressed?\nMany of these questions have been raised and debated in theoretical neuroscience and have become known as the binding problem: How does the brain bind features together into objects while keeping them separate from other objects.\nInspired by this literature (cf. \\citet{treisman1999solutions, vondermalsburg1995binding}) we will focus on three main challenges in incorporating objects in connectionist models of the world: segregation, representation, and composing, which we discuss in the next sections.\n\nSegregation is about object discovery, i.e. given a set of observations, what are good candidates for representational objects and how can they be extracted.\nRepresentation is about storing this representational content in neural networks, and as we will find, plain fully-connected feedforward networks are ill-equipped to solve this task.\nFinally, composition is about using representational objects efficiently in a way that ensures combinatorial generalization (\\emph{systematicity}; \\citet{niklasson1994being}).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Representation}\n\\label{sec:representation}\n\n\n\n\nWhat are good object representations? \nIf objects are to serve as the primitives for compositional reasoning, it is important that their representations support that end.\nHere we argue for three main requirements:\\footnote{Keeping space limitations in mind we will not attempt at exhaustively listing all requirements. Instead we focus on those that we believe to be most important.}\n\n\\begin{description}\n \\item[Universal] Each object representation should be able to represent any object regardless of position, class or other properties. \n It should facilitate generalization, even to unseen objects (zero-shot generalization), which in practice means that its representation should be distributed and disentangled.\n \\item[Multi-Object] It should be possible to represent multiple objects simultaneously, such that they can be related and composed but also transformed individually. \n This only needs to cover a small number of objects at the same time (e.g. $7\\pm2$; \\citealt{miller1956magical}), since there is an intractable number of possible objects. \n Instead, objects should be swapped in and out of this working memory on demand.\n \\item[Common Format] All objects should be represented in the same format, i.e. in terms of the same features. \n This makes representations comparable, provides a unified interface for compositional reasoning and allows the transfer of knowledge between objects. %\n\\end{description}\n\n\nIt is easy to see how regular representations of fully-connected neural networks fall short in this regard:\nWhen representing multiple objects, they can either reuse the same features for all objects simultaneously, thus superimposing representations which leads to ambiguities (\\{red, square\\} + \\{blue, triangle\\} = \\{red, blue, square, triangle\\}).\nAlternatively, they can allocate a different set of features per object which violates common format.\nWithout any architectural bias in the form of weight sharing, useful multi-object representations are thus unlikely to emerge naturally in a neural network.\nIn what way can this problem be addressed?\n\nWeight sharing, as it is used, for example, in ConvNets and RNNs, is a step in the right direction.\nWe call these approaches ``slot-based'' because they provide several slots that all share weights and can thus be used to represent objects in a common format.\nIn the case of RNNs there is one slot per time-step~\\cite{eslami2016attend}, while in ConvNets there is one slot per spatial location in the image~\\cite{santoro2017simple}.\nNote that both are in slight violation of universality because they tie a slot to a specific time step or location, while RNNs additionally do not \\emph{simultaneously} represent multiple objects. \nWe can extend the idea of representational slots and consider a setting in which each object has its own universal slot and all slots share a common format (instance slots, c.f.~\\cref{fig:slots}).\nWhile this constitutes a good object representation, it raises another problem: if all slots are identical and share weights, then how do they not end up all representing the same object?\nSolving this conundrum requires a dynamic information routing process that goes beyond simple feed-forward processing (see~\\cref{sec:segregation}). \n\nThere are two, less developed, alternatives to slot-based approaches that have the potential to meet our requirements: \n\\emph{Augmentation} approaches keep a single set of features but augment each feature to include some extra grouping information. \nExamples include complex-valued activations (e.g. \\citet{reichert2013neuronal}) or spiking networks that encode grouping via synchronization (e.g. \\citet{lane1998simple}).\n\\emph{Embedding} approaches carefully embed multi-object representations in a higher-dimensional space (e.g. \\emph{Tensor Product Representations};~\\citealt{smolensky1990tensor}).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{approach_replicate.pdf}\n\\caption{Different types of \\emph{slot-based} representation strategies.}\n\\label{fig:slots}\n\\vspace{-10pt}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Segregation}\n\\label{sec:segregation}\n\nIt can be difficult to provide precise boundaries or definitions even for concrete objects like a tree, a mountain, and a river.\nMatters become even worse with slightly more abstract objects like a hole, a shadow, or a street corner.\nClearly, sensory information does not come pre-structured into objects, yet we so effortlessly and consistently perceive them. %\nHow can we aid our agents in developing an equally general understanding of objects?\nWe address this by focusing on the role of objects as computational primitives in a compositional reasoning system, namely as abstract patterns of the input data that are modular, dynamic, and consistent:\n\n\n\n\n\\begin{description}\n \\item[Modular] Objects subdivide the input into parts with strong internal coherence while being mostly independent of each other given some task under consideration.\n This division can be thought of as a form of clustering by mutual predictability and helps minimize the error that results from treating them as independent entities. \n \n \n \\item[Dynamic] Objects are task-dependent, i.e. there is no one fixed definition of objects that applies to all tasks.\n For example, objects can be part-whole hierarchies whose parts are objects themselves: a stack of chairs can be viewed as a single object (the stack) or as multiple objects (the individual chairs). \n It necessitates top-down feedback: interaction between the up-stream problem solving and down-stream segregation to obtain a dynamic definition of objects.\n \\item[Consistent] \n Representational objects often ``refer to'' physical objects in the real world (although this does not need to be the case), and their usefulness depends on the reliability of that link. \n The output of the segregation process must thus be stable and consistent to ensure that the results from internal reasoning can be mapped back onto the environment.\n Consistency is also important in communication (different agents should agree on objects), and in the absence of information, e.g. as a result of occlusion.\n\\end{description}\n\n\n\n\nModularity rules out standard convolutional neural networks as a means to learn object representations given by the representational content at each spatial slot.\nEach convolutional layer with a kernel size exceeding $1 \\times 1$ creates dependencies between local spatial neighborhoods. \nThrough depth, the representational content of upper layers encode information from all spatial positions and are no longer modular: a change affecting a single object in the input image affects the representations at \\emph{all} spatial locations in the upper layers. \n\nDynamicity implies that we can not treat segregation as a pre-processing step that extracts objects from input data.\nThis rules out the use of large quantities of labeled data to pre-train an image segmenter, or the use of domain-specific engineering as is commonly found in generative models that essentially encode a fixed definition of object.\nMoreover, human labor is an expensive resource that we can not spend exhaustively to overcome all possible situations.\n\n\nWe conclude that to a large extend object learning must be \\emph{unsupervised} through a specialized mechanism that allows for the possibility to incorporate top-down feedback.\nTwo promising approaches from the literature are \\emph{attention} and \\emph{differentiable clustering}.\n\nAttention mechanisms are used to selectively attend to a subset of the image, i.e. parts that correspond to a single object~\\cite{schmidhuber1991learning, eslami2016attend}.\nIn this way, attention restricts the information intake and ensures that the resulting representations are modular. \nTop-down feedback can be incorporated by granting control of the attention window to the agent that learns to solve some task~\\cite{mnih2014recurrent}.\nA downside is that objects are processed in an iterative fashion, which may make it more challenging to reason about multiple objects simultaneously~\\cite{kosiorek2018sequential}.\n\nAn alternative mechanism is differentiable clustering, which seeks to partition the input in a number of segments while learning the similarity function.\nIndividual segments are disjoint and result in modularity, while the iterative nature of these clustering procedures allow top-down feedback to be incorporated~\\cite{greff2017neural, greff2019multiobject}.\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\\section{Composition}\n\\label{sec:composition}\n\n\n\n\n\n\nLet us now assume that representation and segregation have been addressed, and we have available a set of relevant independent objects represented in a common format.\nNote that when used correctly, these object representations can already make tasks like performing basic feature comparisons very easy.\nFor example, a function that receives a pair of objects as input and compares their size-related features could easily be learned, and would almost automatically generalize to arbitrary pairs of objects.%\n\nIn contrast, combinatorial generalization is not a given for more complex relational reasoning. \nWhile it also involves learning general functions that accept objects as their arguments, one has to take extra care in being able to flexibly assign the right objects to their corresponding function arguments, as well as in learning about different structural forms that imply different ways of generalizing~\\cite{kemp2008discovery}.\nThese then imply the following requirements:\n\\begin{description}\n\\item[General Relations]\nRelations differ both in their meaning and in the patterns of generalization that they imply. \nA general reasoning system, therefore, has to be able to instantiate many different types of relations, which necessitates a general representational form.\n\n\\item[Dynamic Binding] \nIn order to construct a model for a specific situation, the system needs the flexibility to freely combine objects and relations into an arbitrary structure.\nBoth the structure of relations and the associated objects (\\emph{variable binding}; \\citealt{browne2000connectionist}) have to be inferred dynamically during run-time.\n\n\\item[Role-filler Independence] The content of objects should be independent of their structural roles~\\cite{hummel2003symbolicconnectionist}.\nThat is, any object can take part in any relation, and the interpretation of the whole is determined by both the parts and the structure.\nThis is related to \\emph{common format} and is the key to compositionality that enables the powerful systematic generalization that is characteristic of many symbolic systems.\n\n\\end{description}\n\nOne approach is to implement complex relational reasoning in a sequential fashion. \nAt each step, an object associated with a particular role is processed and the resulting intermediate computation is stored, to be combined in the next step.\nWhile it is clear that a plain RNN can perform this type of computation, the dual role of intermediate representations in representing objects and intermediate computation suggests a very specific function that may be hard to learn~\\cite{graves2014neural}.\nAlternatively, by combining an RNN with a suitable memory mechanism (eg.~\\citet{Das:92,mozer1993connectionist,reed2015neural,graves2016hybrid}) or fast\nweights (eg.~\\citet{Schmidhuber:92ncfastweights,Schmidhuber:93ratioicann,schlag2018learning}) it may be more easy to learn general functions of this kind.\n\n\nAn alternative approach is to embed objects, and intermediate representations as nodes in a (directed) graph and let computation take place along its edges.\nThese computation graphs can implement arbitrary relationships, including recursive computation by re-applying the same function successively.\nGraph Networks~\\cite{battaglia2018relational} structure neural network computations according to this underlying graph and perform relational reasoning through repeated message-passing between the nodes in the graph. %\nCompositionality is achieved through weight-sharing, i.e. by learning a general function that operates on (pairs of) nodes following their topological relationship.\nHowever, while graph networks have been successfully applied in the domain of physical reasoning (e.g. \\citet{battaglia2016interaction, vansteenkiste2018relational}), a remaining challenge is in dynamically inferring the right structure (i.e. dynamic binding).\n\nWhile graph networks appear most promising in addressing the challenges of composition, one other approach deserves a mention.\n\\emph{Embedding approaches}, such as Poincar\\'{e} embeddings~\\cite{nickel2017poincare}, generalize Euclidean representations to other spaces that more suited in modeling certain types of relations, in this case: hierarchical relationships.\nHowever, the feature representations are essentially adapted to reflect the underlying relation during training, which implies fixed roles and binding during inference.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe have argued that feature representations alone are inadequate abstractions for planning, reasoning, and for systematically transferring knowledge to novel situations.\nTo meet the diverse challenges on our quest towards AGI an agent needs to be able to dynamically construct new models about its environment on the fly while reusing as much prior knowledge as possible. \nWe have argued that objects (dynamically bound features) are adequate building blocks to quickly and flexibly compose such task-specific models.\nAlthough our examples were centered around vision, we believe that the notion of objects applies equally to other domains like audio, tactile and even abstract thought.\nBy focusing on their role as compositional primitives, we have identified some inductive biases that we believe are necessary for objects to arise within a connectionist system.\nThey can be categorized into three areas: representation, segregation, and composition of objects.\n\nAmong these three we find that segregation is most frequently neglected and deserves more attention.\nCommon approaches rely either on some combination of pre-processing pipelines, supervision, or highly engineered generative models of objects.\nMeanwhile, the few approaches that tackle this challenge in a holistic and unsupervised way are brittle and have not yet been scaled to real-world data.\nDeveloping better methods for tackling the segregation problem within the framework of connectionism is going to be a central challenge on the way towards AGI.\nSimilarly, we would like to stress the importance of integrating solutions to all three aspects into a single system.\nThe potential of objects as modular building blocks can only be realized in full if they are both informed by learned representations, and by feedback from the composite model. \n\nAnother important direction is the integration of objects with other critical cognitive mechanisms such as attention and memory.\nBecause objects are optimized to be modular, they naturally aggregate features that need to be processed together, but which can be separated from other information.\nThis makes them ideal primitives for attention and for storage and retrieval from long-term memory.\nAttention, in turn, can simplify a task by filtering out irrelevant information and can guide the processing required for more complex reasoning chains.\nSuch a reasoning process could then also query objects from memory on demand to be compared to or integrated with the current model. \n\nWith this short essay, we hope to draw attention to the intricacies of objects and inspire others to think critically about their integration in connectionist models. \n\n \n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\nThis research was funded by SNF grant 200021\\_165675\/1.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{sec:introduction}Introduction}\n\n\nIn eddy viscosity modelling, the specific relationship between the Reynolds stress tensor and the mean velocity gradient tensor is the primary subject of interest. Many relationships have been proposed, from simple mixing length models to the sophisticated elliptic blending $k$-$\\varepsilon$ model. Despite the well-known limitations of the eddy viscosity models for turbulent flows, most industrial simulations still rely on these types of turbulence closures within the Reynolds-averaged Navier-Stokes (RANS) framework because of their computational simplicity and efficiency \\cite{Witherden2017}. While the inherent assumption of a local dependence in eddy viscosity modelling is nearly impossible to avoid, improving the functional relationship between the Reynolds stress tensor and the mean velocity gradient tensor remains critically important for accurate industrial simulations \\cite{CFD2030}. \n\n\nTraditionally, turbulence closure models are formulated using heuristic and physics-based arguments. Common examples include the analogy drawn between turbulent stresses and viscous stresses, or the model transport equations for turbulent kinetic energy (TKE) $k$ and a length-scale determining variable such as $\\varepsilon$ (TKE dissipation rate). However, a new approach in turbulence modelling is the physics-\\textit{informed} data-driven approach \\cite{Duraisamy2020}, which leverages the capabilities of machine learning. Rather than simplifying the underlying physics to close the RANS equations, the functional relationship between the mean flow gradients and Reynolds stresses is determined from high-fidelity datasets. Machine learning is well-suited to ``discover'' this (complex) functional relationship using a data-driven approach.\n\nWithin the framework of the application of machine learning for data-driven closure modelling, a wide range of methods have been presented and nearly all of them have demonstrated promising results for simple flows~\\cite{Duraisamy2019a,Brunton2020,Kutz2017,Wang2017}. Several machine learning and injection frameworks have been proposed~\\cite{Duraisamy2021,Zhu2020,Chang2018,Jiang2021}, including open-loop~\\cite{Bhushan2021} and iterative frameworks~\\cite{Liu2021}. These methods have been demonstrated to improve RANS results in canonical flows~\\cite{Srinivasan2019,Yin2020,Fang2020,Zhang2019b}, flow over airfoils~\\cite{Zhu2019,Matai2019}, combustion~\\cite{Nikolaou2020}, and transonic flows~\\cite{Tan2021}. Several investigations~\\cite{Kaandorp2018,Kaandorp2020,Song2019,Zhang2019a} have used machine learning to determine the coefficients of a tensor basis expansion for the anisotropy tensor, based on the seminal work of Ling {\\em et al.}~\\cite{Ling2016}. Other frameworks include the optimal eddy viscosity approach proposed by Wu {\\em et al.}~\\cite{Wu2018}, or the Reynolds force vector approach advocated by Cruz {\\em et al.}~\\cite{Cruz2019}. However, the issue of ill-conditioning was recently shown by Brener {\\em et al.}~\\cite{Brener2021} to affect a number of these modelling frameworks. When the RANS equations are ill-conditioned, small errors in the Reynolds stress tensor can be amplified, resulting in large errors in the prediction of the mean velocity field~\\cite{Wu2019}. This error amplification is a major issue for physics-informed data-driven closures, as some amount of model error in predicting the Reynolds stress tensor is expected. Brener {\\em et al.}~\\cite{Brener2021} showed that the optimal eddy viscosity approach proposed by Wu {\\em et al.}~\\cite{Wu2018} is an effective strategy to address the ill-conditioning problem.\n\n\nWhile previous investigations have presented optimal eddy viscosity-based frameworks that improve the conditioning of the RANS equations, these formulations provide no guarantee of a non-negative eddy viscosity. For practical purposes, the eddy viscosity should always be non-negative---a negative eddy viscosity prediction has the potential to destabilize convergence in an iterative RANS solver. A new formulation of the optimal eddy viscosity is proposed in the present work, which guarantees a non-negative optimal eddy viscosity. The conditioning of the new formulation is analyzed, and the suitability for use in a physics-informed data-driven turbulence closure model is assessed by using a neural network to predict this optimal eddy viscosity. An additional neural network is used to predict the remaining (non-linear) portion of the Reynolds stress tensor. The prediction of the velocity field is greatly improved after injecting the neural network-predicted Reynolds stress tensor into the mean momentum equations. \n\n\nThe present work is the first to propose a stability constraint on the optimal eddy viscosity and to implement this constraint into a neural network. Along with proposing a neural network architecture for predicting the optimal eddy viscosity, the present work also aims to demonstrate the ability of deep learning augmentation to improve the results of a sophisticated physics-based turbulence closure model. While previous studies have improved results from the simpler $k$-$\\varepsilon$ \\cite{Ling2016} and $k$-$\\omega$ models \\cite{Kaandorp2020}, the model augmented in the present work is the $k$-$\\varepsilon$-$\\phi_t$-$f$ model \\cite{Hanjalic2004},\nwhich is an improved reformulation of the $v2$-$f$ model \\cite{Durbin1991}. This model contains three turbulent transport equations for $k$, $\\varepsilon$, and $\\phi_t$, along with an elliptic equation for the damping variable $f$.\n\n\nThe present work is structured as follows: in Section~\\ref{sec:methodology}, we motivate and discuss the proposed optimal eddy viscosity formulation, and injection process. The choice of the network topology and the associated hyperparameter tuning for the customization of deep neural networks for the task of turbulence closure modelling is then described in Section~\\ref{sec:deeplearning}. Section~\\ref{sec:results} presents the model predictions for an unseen test case. An \\textit{a priori} analysis of the model predictions is given in Section~\\ref{sec:apriori}. After injecting the model predictions into a RANS solver, the converged mean fields are analyzed in an \\textit{a posteriori} sense in Section~\\ref{sec:aposteriori}. Having demonstrated the accuracy improvements gained through machine learning augmentation, Section~\\ref{sec:interpret} presents an interpretation and discussion of the data-driven closure. Conclusions and recommendations for further study are given in Section~\\ref{sec:conclusion}.\n\n\\section{Methodology}\\label{sec:methodology}\n\nThe Reynolds-averaged continuity and momentum equations for an incompressible, isothermal, steady-state mean velocity field $\\vec U = (U,V,W)$ and pressure field $p$ is given by\n\\begin{eqnarray}\n\\nabla \\cdot \\vec{U} = 0\\ ,\\\\\n\\nabla\\cdot (\\vec{U}\\vec{U}) = - \\nabla p + \\nu \\nabla^2 \\vec{U} - \\nabla \\cdot \\tau\\ ,\\label{eq:momentum}\n\\end{eqnarray}\nwhere $\\nu$ is the molecular kinematic viscosity of the fluid and $\\tau \\equiv \\overline{\\vec{u}'\\vec{u}'}$ is the Reynolds stress tensor. Here, mean values are denoted using an overbar and $\\vec{u}' \\equiv (\\vec{u} - \\vec{U})$ is the fluctuating velocity ($\\vec{u}$ is the total instantaneous turbulent velocity).\n\nThese equations are hereinafter referred to as the RANS equations. The unclosed term (Reynolds stress tensor), $\\tau$, is the subject of turbulence closure modelling. The common modelling approach is to approximate the anisotropic part of $\\tau$ analogously to the viscous stress term, so\n\\begin{equation}\na \\equiv \\tau - \\frac{1}{3}\\text{tr}(\\tau)I \\approx -2 \\nu_t S \\ , \\label{eq:anisotropy}\n\\end{equation}\nwhere $a$ is the anisotropy tensor and $I$ is the identity tensor. Equation~(\\ref{eq:anisotropy}) is known as the eddy viscosity approximation, where $\\nu_t$ is the eddy viscosity, and $S$ is the mean strain-rate tensor (symmetric part of the mean velocity gradient tensor) defined by\n$$S \\equiv \\tfrac{1}{2}\\left(\\nabla \\vec{U} + \\nabla \\vec{U}^\\text{T}\\right)\\ ,$$\nwhere the superscript $T$ denotes matrix transposition.\n\n\nThis approximation for $\\tau$ permits the eddy viscosity to be treated implicitly, and added to the molecular viscosity $\\nu$ in Eq.~(\\ref{eq:momentum}). Numerically, this has a stabilizing effect and therefore this approach is widely used in RANS modelling. Additionally, the term $\\nabla \\cdot \\tfrac{1}{3}\\text{tr}(\\tau)I$ (which is equal to $\\nabla \\cdot \\tfrac{2}{3}kI$) is typically absorbed into the isotropic term $\\nabla p$. The resulting isotropic term becomes the gradient of the ``modified pressure'' given by $p' = p + \\tfrac{2}{3}k$. \n\nWhile this approximation for $\\tau$ is common in traditional turbulence modelling, the approach in data-driven turbulence modelling has not been uniform. Liu {\\em et~al.}~\\cite{Liu2021} and Brener {\\em et al.}~\\cite{Brener2021} used the eddy viscosity approximation as follows:\n\\begin{eqnarray}\n\\tau \\approx -2 \\nu_t S \\label{eq:eddyviscosity_tau}. \n\\end{eqnarray}\nHowever, a deficiency with the approximation given by Eq.~(\\ref{eq:eddyviscosity_tau}) is that $\\tau$ (whose non-zero trace is $2k$) can never be completely aligned with $S$ (whose trace is necessarily zero owing to the incompressibility of the flow). Therefore, when formulating an ``optimal eddy viscosity approach'' (Section \\ref{sec:optimal}), it is more appropriate to use the more conventional approximation given in Eq.~(\\ref{eq:anisotropy}).\n\n\\subsection{Optimal eddy viscosity}\\label{sec:optimal}\nWithin the commonly used eddy viscosity approach, closing the RANS equations is achieved using a single constitutive coefficient $\\nu_t$. However, calculating $\\nu_t$ in a way that minimizes the error in the mean flow field is a complex task, and has led to the development of hundreds of RANS closure models. In the present work, we propose to use a neural network to directly estimate $\\nu_t$, effectively eliminating the need for any additional turbulent scalar transport equations. This approach was also applied by Wu {\\em et al.}~\\cite{Wu2018} and Liu {\\em et al.}~\\cite{Liu2021}.\n\nThe optimal eddy viscosity $\\nu^*_t$ is obtained by minimizing the error in the approximation of the anisotropy tensor as being directly proportional to the mean strain-rate tensor. More specifically, the optimal eddy viscosity is obtained as the solution of the following least-squares approximation problem:\n\\begin{eqnarray}\n\\nu^*_t = \\text{arg min}_{\\nu_t} \\| a - (-2\\nu_t S) \\|\\ ,\\label{eq:leastsquares}\n\\end{eqnarray}\nwhere $\\nu^*_t$ is the optimal eddy viscosity in a least-squares sense ($\\|\\ \\cdot\\ \\|$ denotes the Euclidean norm). Equation~(\\ref{eq:leastsquares}) has the following closed-form analytical solution: \n\\begin{eqnarray}\n\\nu^*_t= -\\dfrac{1}{2}\\dfrac{a:S}{S:S}\\label{eq:analyticalnutopt}\\ ,\n\\end{eqnarray}\nwhere the colon operator denotes double contraction. More specifically, for two Cartesian tensors $A$ and $B$ of rank two, their double contraction is given by $A:B\\equiv A_{ij}B_{ij} = \\text{tr}(AB^T) = {\\rm tr}(BA^T)$ (Einstein summation convention implied on repeated indices), where $AB$ denotes the matrix product of $A$ and $B$. For the {\\em special} case where both $A$ and $B$ are symmetric tensors (as in the current case), $A:B = {\\rm tr}({AB}) = {\\rm tr}({BA})$.\n\nWhile optimal, the formulation in Eq.~(\\ref{eq:analyticalnutopt}) has the disadvantage that $\\nu^*_t$ can become negative (especially when the eigenframes of $a$ and $S$ are poorly aligned). When treated implicitly in Eq.~(\\ref{eq:momentum}), a negative eddy viscosity can result in a combined negative effective viscosity, greatly destabilizing the numerical solution. Therefore, we propose the following formulation for the estimation of the optimal eddy viscosity:\n\\begin{eqnarray}\n\\nu^\\dagger_t = \\text{arg min}_{\\nu_t\\geq 0} \\|a - (-2 \\nu_t S)\\|\\ ,\\label{eq:nnlsnut}\n\\end{eqnarray}\nwhere $\\nu^\\dagger_t$ is the optimal eddy viscosity obtained using a non-negative least-squares approximation. The formulation in Eq.~(\\ref{eq:nnlsnut}) guarantees that the injected eddy viscosity will not destabilize the iterative solution, at least in terms of reducing the diagonal dominance of the (assembled) coefficient matrix resulting from the discretization of the mean momentum transport equation. This formulation also allows an non-negativity constraint to be enforced on the output of a predictive model, further guaranteeing that any erroneous predictions at testing time do not destabilize the iterative solution.\n\nIn summary, the present optimal eddy viscosity formulation ($\\nu^\\dagger_t$) is the non-negative least-squares fit for $a\\approx -2 \\nu_t S$. This formulation harmonizes the conventional eddy viscosity approximation used in turbulence modelling, which is in terms of the anisotropy tensor, and promotes iterative stability by guaranteeing a non-negative effective viscosity.\n\n\n\\subsection{Non-linear part of the Reynolds stress tensor}\\label{sec:aperp}\nThe formulation in Eq.~(\\ref{eq:nnlsnut}) optimizes the \\textit{linear} eddy viscosity approximation for $a$. An important question that arises is: how accurate is this linear approximation itself?\n\nTo investigate this question, we consider a case of turbulent flow over periodic hills, simulated using both RANS~\\cite{McConkey2021b} and direct numerical simulation (DNS)~\\cite{Xiao2020}. Specifically, the $\\alpha=1.2$ case is considered, where $\\alpha$ is the slope steepness factor. Figure~\\ref{fig:Eddy_viscosity_error_a_theta_histograms} shows the distributions of the error in $a$ after applying a linear eddy viscosity approximation for this canonical flow. Since this flow is two-dimensional, there are only three independent components of $a$: namely, $a_{xx}$, $a_{xy}$, and $a_{yy}$ (owing to the fact that $a$ is traceless). In the present work, the subscript $\\theta$ is used for quantities obtained from a high-resolution DNS simulation, and the subscript $\\psi$ indicates a quantity obtained from a RANS closure model. Even though the linear eddy viscosity approximation has been optimized on high-fidelity data for $a=-2\\nu^*_{t\\theta}S_\\theta$ and $a=-2\\nu^\\dagger_{t\\theta}S_\\theta$, errors on the order of $100\\%$ are present in all components of $a$. In particular, the principal components of $a$ are severely under predicted, even with an optimized eddy viscosity. Compared to the base RANS simulation, an optimal eddy viscosity approach applied to high-fidelity DNS data does not result in substantial accuracy gain.\n\\begin{figure}\n\\includegraphics[]{Eddy_viscosity_error_a_theta_histograms.pdf}\n\\caption{\\label{fig:Eddy_viscosity_error_a_theta_histograms} Error distribution in each anisotropy component after invoking the linear eddy viscosity approximation $a=-2\\nu_t S$. Relative error is calculated by $\\Delta a_{ij} = (a_{ij}-a_{ij\\theta})\/a_{ij\\theta}$. $\\nu^*_{t\\theta}$ is the optimal eddy viscosity calculated using Eq.~(\\ref{eq:analyticalnutopt}), and $\\nu^\\dagger_{t\\theta}$ is the optimal eddy viscosity calculated using Eq.~(\\ref{eq:nnlsnut}). The subscript $\\theta$ indicates a quantity from DNS, and the subscript $\\psi$ indicates a quantity from a RANS simulation using the $\\phi$-$f$ model (Section~\\ref{sec:phif}). The RMSE for each component of $a$ is shown in each plot.}\n\\end{figure}\n\nFigure~\\ref{fig:R2_R_contours} shows two fit quality metrics for the approximation $a\\approx -2 \\nu^\\dagger_t S$. In Fig.~\\ref{fig:R2_R_contours}(a), the local $R^2$ value for the non-negative least squares fit $a_{ij\\theta} \\approx -2 \\nu^\\dagger_{t\\theta} S_{ij\\theta}$ is shown. While the $R^2$ values are higher near the bulk flow above the separated region, the $R^2$ values are generally poor near the walls, in the separated region, and during reattachment. Figure~\\ref{fig:R2_R_contours}(b) shows the eddy viscosity fit quality metric proposed by Thompson {\\em et al.}~\\cite{Thompson2010} which is expressed as\n\\begin{eqnarray}\\label{eq:thompson}\nR_i = 1 - \\frac{2}{\\pi}\\cos^{-1}\\left( \\sqrt{\\frac{{\\rm tr}(4\\nu_{t\\theta}^{\\dagger 2} S^2_\\theta)}{{\\rm tr}(a^2_\\theta)}}\\right)\\ .\n\\end{eqnarray}\nThis second fit quality metric generally agrees with the $R^2$ value in the bulk flow, in supporting a reasonable quality of the linear eddy viscosity approximation. However, for the separation over the left hill, and the acceleration over the right hill, higher values of the $R_i$ metric suggest a better quality in the optimal linear eddy viscosity fit. Nevertheless, visualizing both of these quality metrics demonstrates that there is substantial room for improvement in terms of representing $a$ using only a linear eddy viscosity approximation for this separated flow.\n\n\n\\begin{figure}\n\\includegraphics[]{R2_R_contours.pdf}\n\\caption{\\label{fig:R2_R_contours} Fit quality metrics after invoking the linear eddy viscosity approximation $a = -2 \\nu^\\dagger_{t\\theta} S_\\theta$: (a) $R^2$ value and (b) eddy viscosity model fit factor $R_i$ [Eq.~(\\ref{eq:thompson})].}\n\\end{figure}\n\n\nThe errors in each component of $a$ arising from the linear eddy viscosity approximation are shown in Fig.~\\ref{fig:a_error_contours}. The linear eddy viscosity approximation fails significantly for all components of $a$ during separation of the flow, a well-known deficiency~\\cite{Pope2000}. Noticeable errors also exists in $a_{yy}$ during reattachment of the flow along the bottom wall. Though Thompson {\\em et al.}'s $R_i$ value suggests a good approximation for $a$ as the flow accelerates over the right hill, the largest errors in the shear component $a_{xy}$ occur here.\n\n\\begin{figure*}\n\\includegraphics{a_error_contours.pdf}\n\\caption{\\label{fig:a_error_contours} \\textit{A priori} contours of error in each component of $a$ after invoking the linear eddy viscosity approximation $a=-2\\nu^\\dagger_{t\\theta}S_\\theta$.}\n\\end{figure*}\n\nWhile the linear eddy viscosity approximation, even after optimization, was found to be deficient in approximating $a$, it was of interest to see whether these errors would affect the accuracy of the mean velocity field. Figure~\\ref{fig:Conditioning_error_linear} shows the errors in the mean field components after injecting the optimal eddy viscosity implicitly into the RANS equation, and allowing the solution to converge around the fixed $\\nu^\\dagger_{t\\theta}$. Figure~\\ref{fig:Conditioning_error_linear}(a) shows that significant errors occur during separation and reattachment along the bottom wall in the streamwise velocity component $U$. \n\n\\begin{figure}\n\\includegraphics[]{Conditioning_error_linear.pdf}\n\\caption{\\label{fig:Conditioning_error_linear} A poteriori contours of error in the velocity vector components after injecting the optimal linear eddy viscosity into the RANS equations, without any non-linear terms: (a) error in the $x$-component and (b) error in the $y$-component.}\n\\end{figure}\n\nEven though the solution in Fig.~\\ref{fig:Conditioning_error_linear} was produced by injecting a high-fidelity optimal eddy viscosity into the RANS equations, there are significant errors in the mean flow field. Furthermore, the analysis in Section \\ref{sec:conditioning} shows that conditioning errors with this optimal eddy viscosity formulation are minimal. If the target of the machine learning model is only the optimal eddy viscosity, mean field accuracy improvements are not guaranteed. In fact, the error fields shown in Fig.~\\ref{fig:Conditioning_error_linear} represent the ``upper limit'' that could be achieved using a linear-component-only data-driven closure based on the optimal eddy viscosity as formulated in the present work.\n\nGiven the deficiencies of the linear eddy viscosity approximation, we propose an additional non-linear term in the anisotropy representation; namely, $a^\\perp$ which is obtained from the following proportional\/orthogonal tensor decomposition of $a$ :\n\\begin{eqnarray}\na = -2 \\nu^\\dagger_t S + a^\\perp.\\label{eq:a_decomp}\n\\end{eqnarray}\nIn Eq.~(\\ref{eq:a_decomp}), $a$ has been decomposed into a linear component ($-2 \\nu^\\dagger_t S$) and a non-linear component ($a^\\perp$). After calculating the optimal eddy viscosity using Eq.~(\\ref{eq:nnlsnut}), the non-linear portion of the anisotropy tensor can be calculated using $a^\\perp = a - (-2\\nu^\\dagger_t S)$.\n\n\n\n\n\n\n\\subsection{Conditioning}\\label{sec:conditioning}\n\nA subject that is of immense importance for data-driven turbulence closure modelling is the issue of conditioning within the RANS equations. When the RANS equations are ill-conditioned, small errors in the closure term can be amplified, resulting in large errors in the mean flow field. This issue affects data-driven models because an error in the model prediction of the closure term is almost always expected. Even with an error-free closure term, an accurate converged mean field cannot be guaranteed~\\cite{Brener2021}. \n\nTo demonstrate the importance of the conditioning problem, a conditioning analysis similar to that performed by Brener {\\em et al.}~\\cite{Brener2021} was conducted. Figures~\\ref{fig:conditioningU} and \\ref{fig:conditioningV} show the errors in the mean velocity field components for two injection experiments. The flow considered here is a turbulent flow at $Re_H=5600$ over periodic hills\\cite{Xiao2020}. In both Figs~\\ref{fig:conditioningU}(a) and (b), the RANS equations have been closed using a highly accurate $a$ from a DNS simulation undertaken by Xiao {\\em et al.}~\\cite{Xiao2020}. In Figures~\\ref{fig:conditioningU}(a) and \\ref{fig:conditioningV}(a), $a_\\theta$ has been injected as an explicit source term into the RANS equations. In Figures~\\ref{fig:conditioningU}(b) and \\ref{fig:conditioningV}(b), the decomposition proposed in the present work has been injected into the RANS equations in accordance to the following scheme: $\\nu^\\dagger_{t\\theta}$ is treated implicitly and $a^\\perp_\\theta$ is treated as an explicit source term in the discretized mean momentum equations.\n\nDespite both systems being closed by the same quantity arising from a high-fidelity simulation, the solution in Figs~\\ref{fig:conditioningU}(a) and \\ref{fig:conditioningV}(a) has a root mean square error (RMSE) that is an order of magnitude larger than the solution in Figs~\\ref{fig:conditioningU}(b) and \\ref{fig:conditioningV}(b). It should be noted that additional differences between the two solutions may arise from the need to use first-order schemes to stabilize the solution for the fully explicit propagation.\n\nThe low error observed after injecting the full anisotropy representation [Eq.~(\\ref{eq:a_decomp})] from a DNS simulation into the RANS equation highlights the merits of the present approach. As discussed in Section~\\ref{sec:deeplearning}, since the quantities $\\nu^\\dagger_{t\\theta}$ and $a^\\perp_\\theta$ are the training labels, injecting these quantities into the RANS equations provides the ``upper limit'' that the data-driven model could achieve. For this reason, we recommend that data-driven closure frameworks be evaluated in terms of the conditioning errors that arise even after a perfect model prediction ($\\tilde a=a_\\theta$).\n\n\n\n\n\\begin{figure}\n\\includegraphics[]{Conditioning_error_U.pdf}\n\\caption{\\label{fig:conditioningU} \\textit{A posteriori} contours of error in the $x$-velocity component after injecting the Reynolds stress anisotropy tensor from DNS into the RANS equations in two different ways: (a) fully explicit injection of $a_\\theta$ and (b) the injection framework used in the present work.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[]{Conditioning_error_V.pdf}\n\\caption{\\label{fig:conditioningV} \\textit{A posteriori} contours of error in the $y$-velocity component after injecting the Reynolds stress anisotropy tensor from DNS into the RANS equations in two different ways: (a) fully explicit injection of $a_\\theta$ and (b) the injection framework used in the present work.}\n\\end{figure}\n\n\\subsection{Injection procedure}\\label{sec:injection}\n\nWe use an open-loop, data-driven framework referred to by Ho and West\\cite{Ho2021} as a \"one-time correction\" model. To this purpose, our framework involves making a fixed correction to the closure term, and then allowing the mean field to converge around this fixed correction. A qualitative description of the injection process in terms of the commonly used residual plot is shown in Figure~\\ref{fig:injection}. The correction is estimated using a converged RANS simulation obtained from a base turbulence model as an input. This framework is in contrast to an iterative framework, where the closure term correction is updated repeatedly as the velocity field evolves. For example, the iterative framework used by Liu {\\em et al.}~\\cite{Liu2021} calls the machine learning model during each iteration. While open-loop frameworks are limited to steady flows, the design and use of iterative correction frameworks remains an open question. For example, the converged solution from an iterative framework can depend on the initial flow field~\\cite{Ho2021}.\n\n\\begin{figure}\n \\centering\n \\includegraphics{qualitative_plot.pdf}\n \\caption{\"Qualitative\" residual plot for the machine learning corrective framework used in the present study. After injection, no further updates to the turbulence equations (e.g., the $k$ and $\\varepsilon$ transport equations) are required.}\n \\label{fig:injection}\n\\end{figure}\n\nMotivated by the conditioning analysis of Brener {\\em et al.}~\\cite{Brener2021}, and our own analysis shown in Figs~\\ref{fig:conditioningU} and \\ref{fig:conditioningV}, we make two closure corrections in the present work: namely, one using $\\nu^\\dagger_t$ and another using $a^\\perp$. We correct the eddy viscosity to be the optimal eddy viscosity, formulated in Eq.~(\\ref{eq:nnlsnut}), and treat it implicitly in the momentum equation through an effective viscosity. Having established the importance of including the non-linear part of the anisotropy tensor (Section~\\ref{sec:aperp}), we also include the divergence of $a^\\perp$ as a source term in the momentum equation. \n\nAfter the correction, the RANS equations are solved iteratively until the mean fields converge around the fixed correction terms. The goal is that this converged solution will be more accurate than the solution produced by the base turbulence model. No update is required for the turbulent transport equations, as the eddy viscosity is directly corrected in our framework.\n\n\n\\subsection{The $\\phi$-$f$ model}\\label{sec:phif}\nIn the present work, we use the $\\phi$-$f$ model implemented in OpenFOAM v2006 as the base RANS model~\\cite{Laurence2005,Openfoam}. The $\\phi$-$f$ model is a reformulated version of the $v^2$-$f$ model, where $\\phi$ is the ratio $v^2\/k$, and $f$ is a damping scalar. By adding a carefully constructed transport equation for the streamline-normal Reynolds stress component ($v^2$), this model accounts for the wall-blocking effects on the Reynolds stresses. The construction of the $v^2$ equation includes the effects of pressure strain on the streamline-normal Reynolds stress, a key contributor to wall-blocking effects. These wall-blocking effects are observed as unequal scaling (anisotropy) of the Reynolds stress components, which are not captured in a two-equation model. Often, the use of wall damping functions is required in two-equation models to correctly capture near-wall scaling. Laurence {\\em et al.}'s~\\cite{Laurence2005} $\\phi$-$f$ model is a reformulated version of a $v^2$-$f$ model variant due to Lien and Kalitzin~\\cite{Lien2001}, which is, in turn, an improved version of Durbin's~\\cite{Durbin1991} original $v^2$-$f$ model. Reformulating the $v^2$ and $f$ equation in terms of $\\phi$ results in a more robust model. For example, this model has reduced stiffness in terms of the near-wall damping singularity \\cite{Laurence2005}. The $\\phi$-$f$ model estimates the eddy viscosity as\n\\begin{equation}\n \\nu_t = C_\\mu \\phi k T \\ ,\n\\end{equation}\nwhere $C_\\mu$ is a model coefficient, and $T$ is the turbulent time scale.\n\nDetails on the OpenFOAM implementation of this model are given in the OpenFOAM documentation \\cite{Openfoam}. The original $\\phi$-$f$ model equations can be summarized as follows:\n\\begin{equation}\n \\frac{\\partial k}{\\partial t} + U_i \\frac{\\partial k}{\\partial x_i} = P - \\varepsilon + \\frac{\\partial}{\\partial x_j}\\left[ \\left(\\nu + \\frac{\\nu_t}{\\sigma_k}\\right) \\frac{\\partial k}{\\partial x_j}\\right]\\ ;\\label{eq:k}\n\\end{equation}\n\\begin{equation}\n \\frac{\\partial \\varepsilon}{\\partial t} + U_i \\frac{\\partial \\varepsilon}{\\partial x_i} = \\frac{C_{\\varepsilon_1} P}{T} - \\frac{C_{\\varepsilon_2}\\varepsilon}{T} + \\frac{\\partial}{\\partial x_j}\\left[ \\left(\\nu + \\frac{\\nu_t}{\\sigma_\\varepsilon}\\right) \\frac{\\partial \\varepsilon}{\\partial x_j}\\right]\\ ;\\label{eq:epsilon}\n\\end{equation}\n\\begin{eqnarray}\n \\frac{\\partial \\phi}{\\partial t} + U_i \\frac{\\partial \\phi}{\\partial x_i} = f - P \\frac{\\phi}{k}+\\frac{2\\nu_t}{k\\sigma_k}&&\\frac{\\partial \\phi}{\\partial x_j}\\frac{\\partial k}{\\partial x_j} \\nonumber\\\\ && + \\frac{\\partial}{\\partial x_j}\\left[ \\left(\\frac{\\nu_t}{\\sigma_k}\\right) \\frac{\\partial \\phi}{\\partial x_j}\\right]\\ ;\\label{eq:phi}\n\\end{eqnarray}\nand,\n\\begin{eqnarray}\nL^2 \\frac{\\partial^2 f}{\\partial x_j} - f = \\dfrac{1}{T}(C_{f_1} &&-1) \\left[\\phi - \\dfrac{2}{3}\\right] \\nonumber\\\\ &&- C_{f_2} \\dfrac{P}{k} - 2 \\dfrac{\\nu}{k}\\dfrac{\\partial \\phi}{\\partial x_j}\\dfrac{\\partial k}{\\partial x_j} - \\nu \\dfrac{\\partial^2 \\phi}{\\partial x_j}\\ .\\label{eq:f}\n\\end{eqnarray}\nEquations~(\\ref{eq:k}), (\\ref{eq:epsilon}), and (\\ref{eq:phi}) are model transport equations for $k$, $\\varepsilon$, and $\\phi$, respectively. Written in index notation, $U_i = \\vec{U}$. Equation~(\\ref{eq:f}) is an elliptic relaxation equation for $f$, which is a scalar predicting near-wall damping effects.\n\nThe turbulent time scale $T$ and length scale $L$ are given, respectively, by\n\\begin{eqnarray}\n T = \\text{max}\\left(\\frac{k}{\\varepsilon},C_T\\sqrt{\\frac{\\nu}{\\varepsilon}}\\right)\\label{eq:T} \\ ,\\\\\n L = C_L\\text{max}\\left(\\frac{k^{3\/2}}{\\varepsilon}, C_\\eta \\left(\\frac{\\nu^3}{\\varepsilon}\\right)^{1\/4}\\right) \\ .\\label{eq:L}\n\\end{eqnarray}\nFinally, the model constants assume the values summarized \nas follows: $C_\\mu = 0.22$; $C_{\\varepsilon_1}=1.4(1.0+0.05 \\sqrt{1.0\/\\phi})$; $ C_{\\varepsilon_2}=1.9$; $C_T = 6.0$; $C_L = 0.25$; $C_{f_1} = 1.4$; $ C_{f_2} = 0.3$; $C_\\eta = 110.0$; $\\sigma_k = 1.0$; and, $\\sigma_\\varepsilon = 1.3$.\n\n\\subsection{Numerical methods}\nAll simulations in the present work use OpenFOAM v2006. For the base RANS simulation, the PIMPLE algorithm was used to achieve a converged solution. Then, a modified PIMPLE solver, which accepts the corrected $\\nu^\\dagger_t$ and $a^\\perp$, was used to inject these corrections into the RANS equations and iteratively solve for the mean fields. While all flows in the present work reach a steady state solution, the unsteady algorithm was used to stabilize the iterative solution. \n\nFor discretizing the RANS equations, a second-order upwind scheme was used for the convective terms, and a second-order central difference scheme was used for the diffusion terms. For the convective terms in the base turbulence model transport equations, a first-order scheme was used. \n\nA wall-resolved mesh ($y^+ \\leq 1$) was used for all simulations. Further details on the mesh, domain, and boundary conditions are provided in McConkey {\\em et al.}'s~\\cite{McConkey2021b} description of the machine learning dataset. Cyclic boundary conditions were used for all flow variables at the inlet and outlet. At the top and bottom walls, the boundary conditions were fixed-zero for velocity, and zero-gradient for pressure.\n\n\n\\section{Deep learning procedure for predicting the Reynolds stress tensor}\\label{sec:deeplearning}\n\n\\subsection{Dataset}\nTo test the proposed decomposition and injection framework, a series of flow over periodic hills was chosen. This flow features separation over the left hill, reattachment along the bottom wall, and acceleration over the right hill. The dataset presented for data-driven turbulence modelling by McConkey {\\em et al.}~\\cite{McConkey2021b}, includes these cases, based on the DNS simulations of Xiao {\\em et al.}~\\cite{Xiao2020}. The periodic hills portion of the dataset consists of five cases, each with 14,751 data points, for a total of 73,755 data points. At each data point, the complete set of RANS input features and DNS fields are provided, so that the model can learn a mapping from the RANS features to the DNS labels. The RANS fields come from a converged solution using the $k$-$\\varepsilon$-$\\phi_t$-$f$ model \\cite{Laurence2005} in OpenFOAM v2006. The five cases are generated by parametrically varying the hill steepness $\\alpha$ and the overall length of the geometry. The five cases correspond to $\\alpha = 0.5$, 0.8, 1.0, 1.2, and $1.5$, as shown in Fig.~\\ref{fig:geometry_phll}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{PHLL_geom_annotated.eps}\n\\caption{\\label{fig:geometry_phll} The geometry and flow variables used for the five periodic hills cases in the present work (Reproduced from McConkey {\\em et al}.~\\cite{McConkey2021b}).}\n\\end{figure}\n\n\n\n\\subsection{Model architecture}\n\n\nThe model consists of two independent neural networks: namely, EVNN, and apNN. The objective of EVNN is to predict the optimal eddy viscosity $\\nu^\\dagger_t$, whereas that of apNN is to predict the non-linear part of the anisotropy tensor, $a^\\perp$. Both models are feed-forward, fully connected neural networks. The EVNN has one output neuron, and the apNN has three output neurons (one for each independent component of $a^\\perp$).\n\nBecause $\\nu^\\dagger_t$ is non-negative, the stability of the EVNN predictions during training and injection can benefit by enforcing a non-negative prediction. For the EVNN, the activation function used for the output neuron was a simple exponential function $f(x) = e^x$, where $x$ is the input value to the layer. This activation function guarantees a non-negative eddy viscosity prediction by the EVNN, regardless of the input activation layer (viz., the choice of the exponential function as the activation function induces an inductive bias of the network output). This stability constraint is possible due to the formulation of $\\nu^\\dagger_t$, which guarantees a non-negative training label. For the apNN, the three output neurons utilize a linear activation function. For the hidden layers, the scaled exponential linear unit (SELU) activation function was used. Combined with the LeCun initialization of the network weights, a SELU-activated, fully connected, feed-forward neural network is self-normalizing~\\cite{Klambauer2017}, eliminating the need for batch normalization layers~\\cite{Geron}.\n\nThe neural networks were implemented using the high-level Keras application programming interface (API) in TensorFlow \\cite{keras}.\nThe hyperparameters used for both networks were identical: namely, both networks have $L=14$ hidden layers, with $n=30$ neurons in each layer, resulting in a network architecture with an depth-to-width aspect ratio of $L\/n \\approx 0.467$. The EVNN has 13,141 trainable parameters (weights and biases) with a single output, and the apNN has \\text{13,203} trainable parameters with three outputs. For traditional turbulence modelling, tuning this number of parameters is a nearly impossible task. However, for deep learning, tuning these parameters is relatively ``straightforward''---the number of learnable parameters is often orders of magnitude greater than those used in the EVNN and the apNN. Nevertheless, despite the large number of model parameters used in these types of overparameterized neural networks (where the number of model parameters far exceed the number of training data), there is mounting empirical (indeed, practical) evidence that these networks can be successfully trained to provide excellent predictions---perhaps supporting the notion that it is not the number of model parameters, but the representation learning provided by the subtle correlations in the intralayer and interlayer interactions of the neural units in the network that ultimately determines the predictive skill and generalizability of the model in deep learning.\n\n\\subsection{Input feature selection}\nA wide range of input features have been used in previous data-driven closure frameworks. The input features for a data-driven closure are generally required to possess the same invariance properties as the underlying Navier-Stokes equations: reflection, rotation, and Galilean transformation invariance. Methods used to generate the input feature set vary from a purely heuristic selection, to a systematic generation of an integrity basis~\\cite{Wu2018}. In the present work, we use a combination of these two methods. Three heuristic scalars are selected as input features given by\n\\begin{eqnarray}\nq_1 &&\\equiv \\text{min}(2.0,\\sqrt{k} y_\\perp\/(50\\nu))\\ , \\nonumber\\\\\nq_2 &&\\equiv k\/\\varepsilon \\|S\\|\\ , \\nonumber\\\\\nq_3 &&\\equiv \\|\\tau\\|\/k\\ .\n\\end{eqnarray}\nHere, $y_\\perp$ is the wall-normal distance. $q_1$ is the wall distance based Reynolds number, $q_2$ is the ratio of the mean strain timescale to a turbulent time scale, and $q_3$ is the ratio of the total Reynolds stress magnitude to the diagonal Reynolds stresses. These Galilean invariant scalars were used previously by Kaandorp and Dwight~\\cite{Kaandorp2020}.\n\nIn addition to these scalars, we modify the procedure presented by Wu {\\em et al.}~\\cite{Wu2018} to significantly augment the input feature set. In Wu {\\em et al.}'s procedure, four gradient tensors were selected: $S$, $R$,$\\nabla k$, and $\\nabla p$. The TKE and pressure gradient vectors were converted to second-order tensors, by casting their components into an anti-symmetric matrix. Then, the method outlined by Spencer and Rivlin~\\cite{Spencer1962} was used to generate an integrity basis for the space spanned by these four second-order tensors. This integrity basis consists of 47 tensors. Finally, a scalar input feature was extract from each tensor, by taking the first invariant (viz., the trace) of each integrity basis tensor.\n\nWhile this procedure extracts a large number of input features from the flow, a number of issues arise that have received limited attention in the literature. Firstly, the terms related to $\\nabla p$ in the integrity basis tend to be numerically unstable in our experience, possibly due to the normalization used by Wu {\\em et al.}~\\cite{Wu2018}. In their frameworks, Wang {\\em et al.}~\\cite{Wang2017} and Kaandorp and Dwight~\\cite{Kaandorp2020} omitted these terms, reducing the number of invariants down to 16. Secondly, many of these invariants are zero for two-dimensional flows. These zero invariants arise from either a direct result of the $z$-direction gradients being zero, or from the incompressibility of the flow. Lastly, this procedure only uses the first invariant $I_1$ (or, trace) of each integrity basis tensor---more information can be extracted from this basis by including further invariants in the input feature set. We have presented an analysis of Wu {\\em et al.}'s~\\cite{Wu2018} integrity basis in Appendix \\ref{ap:invariants}, showing expressions for each of the basis tensor invariants for two-dimensional flows.\n\nTo circumvent the issue of unstable terms related to $\\nabla p$ being unusable as input features, we replace $\\nabla p$ in Wu {\\em et al.}'s basis with $\\nabla v^2$. This replacement is possible since our base turbulence model is the $\\phi$-$f$ model (Section~\\ref{sec:phif}). We found that terms including $\\nabla v^2$ were much more stable than the terms including $\\nabla p$, and therefore more features could be included. It is noted that $\\nabla v^2$ provides additional information compared to $\\nabla k$, because $v^2$ measures the degree of wall-blocking effects and anisotropy in the flow. Additionally, we augment the 47 original first invariants by also taking the second invariant $I_2$ of a tensor $A$ ($I_2 = \\tfrac{1}{2}[(\\text{tr}(A))^2 - \\text{tr}(A^2)]$). For a three-dimensional flow, this effectively doubles the number of basis tensor invariants that can be used as input features. However, as shown in Appendix~\\ref{ap:invariants}, many of these invariants are zero for two-dimensional flows. Nevertheless, after eliminating the zero-valued invariants, a set of 29 invariants remained as suitable input features. Table~\\ref{tbl:features} summarizes the input features used.\n\n\\begin{table}[]\n\\caption{Input features for EVNN and apNN.}\\label{tbl:features}\n\\begin{tabular}{cccc}\n\\hline\nNumber & Input feature & Expression & Transformation \\\\ \\hline\n1 & $I_1(B_1)$ & $I_1(S^2)$ & $\\text{log}(|x|+1)$ \\\\\n2 & $I_1(B_3)$ & $I_1(R^2)$ & $\\text{log}(|x|+1)$ \\\\\n3 & $I_1(B_4)$ & $I_1(A_{v2}^2)$ & $\\sqrt[3]{x}$ \\\\\n4 & $I_1(B_5)$ & $I_1(A_k^2)$ & $\\text{log}(|x|+1)$ \\\\\n5 & $I_1(B_7)$ & $I_1(R^2S^2)$ & $\\text{log}(|x|+1)$ \\\\\n6 & $I_1(B_9)$ & $I_1(A_{v2}^2 S)$ & $\\sqrt[3]{x}$ \\\\\n7 & $I_1(B_{10})$ & $I_1(A_{v2}^2 S^2)$ & $\\text{log}(|x|+1)$ \\\\\n8 & $I_1(B_{12})$ & $I_1(A_{k}^2 S)$ & $\\sqrt[3]{x}$ \\\\\n9 & $I_1(B_{13})$ & $I_1(A_{k}^2 S^2)$ & $\\text{log}(|x|+1)$ \\\\\n10 & $I_1(B_{16})$ & $I_1(A_{v2}A_k)$ & $\\text{log}(|x|+1)$ \\\\\n11 & $I_1(B_{21})$ & $I_1(A_{v2}RS)$ & $\\text{log}(|x|+1)$ \\\\\n12 & $I_1(B_{25})$ & $I_1(A_{v2}^2SRS^2)$ & $\\text{log}(|x|+1)$ \\\\\n13 & $I_1(B_{29})$ & $I_1(A_{k}^2RS)$ & $\\text{log}(|x|+1)$ \\\\\n14 & $I_1(B_{33})$ & $I_1(A_{k}SRS^2)$ & $\\text{log}(|x|+1)$ \\\\\n15 & $I_1(B_{34})$ & $I_1(A_{v2}A_kS)$ & $\\sqrt[3]{x}$ \\\\\n16 & $I_1(B_{35})$ & $I_1(A_{v2}A_k S^2)$ & $\\text{log}(|x|+1)$ \\\\\n17 & $I_1(B_{42})$ & $I_1(RA_{v2}A_k)$ & $\\sqrt[3]{x}$ \\\\\n18 & $I_1(B_{43})$ & $I_1(RA_{v2}A_kS)$ & $\\sqrt[3]{x}$ \\\\\n19 & $I_1(B_{45})$ & $I_1(RA_{v2}A_kS^2)$ & $\\sqrt[3]{x}$ \\\\\n20 & $I_1(B_{46})$ & $I_1(RA_kA_{v2}S^2)$ & $\\sqrt[3]{x}$ \\\\\n21 & $I_2(B_1)$ & $I_2(S^2)$ & $\\text{log}(|x|+1)$ \\\\\n22 & $I_2(B_2)$ & $I_2(S^3)$ & $\\text{log}(|x|+1)$ \\\\\n23 & $I_2(B_3)$ & $I_2(R^2)$ & $\\text{log}(|x|+1)$ \\\\\n24 & $I_2(B_4)$ & $I_2(A_{v2}^2)$ & $\\text{log}(|x|+1)$ \\\\\n25 & $I_2(B_5)$ & $I_2(A_{k}^2)$ & $\\text{log}(|x|+1)$ \\\\\n26 & $I_2(B_6)$ & $I_2(R^2S)$ & $\\text{log}(|x|+1)$ \\\\\n27 & $I_2(B_7)$ & $I_2(R^2S^2)$ & $\\text{log}(|x|+1)$ \\\\\n28 & $I_2(B_8)$ & $I_2(R^2SRS^2)$ & $\\text{log}(|x|+1)$ \\\\\n29 & $I_2(B_{16})$ & $I_2(A_{v2}A_k)$ & $\\text{log}(|x|+1)$ \\\\\n30 & $q_1$ & $\\text{min}(2.0,\\sqrt{k} y_\\perp\/(50\\nu))$ & --- \\\\\n31 & $q_2$ & $k\/\\varepsilon \\|S\\|$ & --- \\\\\n32 & $q_3$ & $\\|\\tau\\|\/k$ & --- \\\\\n33 & $\\phi$ & $\\phi$ & --- \\\\ \\hline\n\\end{tabular}\n\\end{table}\nThis input feature set represents one of the richest feature sets used in a data-driven turbulence closure model. While many studies relax the condition of Galilean invariance (most commonly, by including the turbulence intensity), all of our input features are Galilean invariant. Section~\\ref{sec:interpret} analyzes the relative importance of each of these features to the model predictions. \n\n\n\n\\subsection{Label calculation}\nEach neural network has a different set of training labels, based on the desired output. For the EVNN, the goal is to predict the optimal eddy viscosity $\\nu^\\dagger_t$. As shown in Section~\\ref{sec:conditioning}, injecting the optimal eddy viscosity from DNS produced a well-conditioned closure. Therefore, the label field for the EVNN consists of the $\\nu^\\dagger_{t\\theta}$ field. To calculate this field, scikit-learn \\cite{scikit-learn} was used to perform the non-negative least-squares (NNLS) regression fit for $a_\\theta = -2\\nu_tS_\\theta$ at each cell in the dataset.\n\nThe labels for the apNN consist of the three independent components of the non-linear part of the Reynolds stress tensor. Therefore, the three component labels were extracted from the tensor $a^\\perp_\\theta = a_\\theta + 2 \\nu^\\dagger_{t\\theta}S_\\theta$, where $\\nu^\\dagger_{t\\theta}$ are the labels for the EVNN. At injection time, $a^\\perp_{zz}$ can be calculated as $a^\\perp_{zz}=-a^\\perp_{xx} - a^\\perp_{yy}$, because $a^\\perp$ is a traceless tensor.\n\n\\subsection{Pre-processing}\n\\label{sec:preprocessing}\n\nAfter calculating the input feature and label sets, the five periodic hills cases were divided into a training set, a validation set, and a testing set. The training process consists of updating the model weights and biases using an optimization algorithm. The model makes predictions using the training set features and evaluates the predictions using the training set labels. The validation and testing sets are used similarly, but at different times in the process. The validation set is used to evaluate the predictive performance of the neural network throughout the training process in order to assess when the model has converged. The validation set is not used to update the weights and biases of the network. Instead, it helps identify overfitting during the training process. If the training process continues for too long, the model will begin to memorize noise and other undesired details in the training set. Repeated evaluation of the neural network predictive performance using the validation set will allow one to assess the generalization performance of the model during training. The testing set is used to evaluate the final, trained model on an entirely new set of data---it can be used to determine the model (or generalization) error and to evaluate the generalization capability of the fully-trained neural network model. More specifically, the generalization error is perhaps the primary quantitative measure of the success of the fully-trained neural network model providing, as such, the standard for how well the network is really approximating the underlying function for the Reynolds stresses.\n\nThe $\\alpha=1.2$ case was selected as a test set. In terms of the parameter $\\alpha$, this represents an interpolation test case, since the training and validation set contain $\\alpha$ values both above and below 1.2. The training\/validation set consists of 59,004 data points, for the remaining four cases. At this stage, the typical procedure is to shuffle and split all of the remaining data into a training and validation set. However, for training a data-driven turbulence closure neural network model, we propose a different method to better estimate the generalization performance during training. Our recommendation is that the validation set should also consist of distinct cases from the training set. For example, the validation set could be all data points for the $\\alpha=1.0$ test case, and the training data would then consist of the $\\alpha = 0.5$, 0.8, and 1.5 cases.\n\nIt is important to stress that the choice of validation set greatly influences the training process. This issue is normally solved by using $k$-fold cross-validation, where the training data is split into $k$ folds. Each of the folds is then used as a validation set, while the model is trained on the remaining folds. We apply 4-fold cross-validation, where the model is trained four times, each using a remaining case ($\\alpha = 0.5$, 0.8, 1.0, 1.5) as the validation set. The model which exhibits the best generalization performance (defined as the difference between the training and validation error) is then selected for testing.\n\nThe input features and labels are often scaled and transformed in deep learning in order to eliminate scale differences between input features and to ensure well behaved weights and biases during training~\\cite{Geron}. To this purpose, a transformation was selected for each of the input features and labels to ensure a more uniform distribution (histogram) of their values before they are used in training. These transformations are given in Table~\\ref{tbl:features}. The selection of input feature transformations is highly heuristic, and depends on the data used for each study. However, we have generally applied a logarithm transform ($\\hat{x} = \\log (|x|+1)$) to data which is entirely positively- or negatively-valued, and a cube-root transform ($\\hat{x} = x^{1\/3}$) to data which contains both positive and negative values. For the majority of data containing RANS mean fields, the histograms tend to be highly skewed due to domain-specific factors. For example, the behavior of almost all fields in the near-wall region is completely different than the bulk flow region. It was found that transforming the components of $a^\\perp$ was not necessary for sufficient model performance. After transforming the features (Table~\\ref{tbl:features}), the values were scaled to lie in the unit interval [0,1] using scikit-learn's MinMaxScaler~\\cite{scikit-learn}. The $\\nu^\\dagger_{t\\theta}$ labels were transformed using $\\hat{ x} = \\text{log}(|x| +1)$, then scaled using the same method as the features. No transformation was necessary for the $a^\\perp_{\\theta}$ labels. Only scaling was applied to the $a^\\perp_{\\theta}$ label set.\n\n\\subsection{Training}\nFor the training process, the Nadam optimizer in Keras was selected~\\cite{keras}. This optimizer combines the adaptive moment estimation (Adam) optimizer with Nesterov momentum to accelerate training. The learning rate used was $1\\times 10^{-4}$. For the EVNN, the loss function was the mean-squared error, with L2 regularization used. The L2 regularization weight $\\lambda$ was $1\\times 10^{-5}$. L2 regularization adds an additional penalty term to the loss function to constrain the norm squared of the optimized model parameters (or, weights)---the neural network seeks the best compromise between minimizing the errors in the prediction of the output and constraining the magnitude of the neural network weights to be as small as possible (viz., shrinking the weights towards zero). In practice, this regularization is required for some deep learning scenarios. We did not apply any L2 regularization to the apNN loss function, as the generalization performance was found to be sufficient without any additional regularization term to constrain the weights.\n\nThe training process was completed for the four cross-validation folds, as described in Section~\\ref{sec:preprocessing}. After completing the training procedure for each of these folds, the model which provided the best generalization performance was selected. The training loss curves for the selected EVNN and apNN are shown in Figs~\\ref{fig:loss_EVNN} and \\ref{fig:loss_apNN}, respectively. While the apNN loss curve remains relatively stable, the loss curve for the EVNN contained instabilities throughout the training process. Typically, these instabilities are a symptom of the learning rate being too large, but the learning rate was not found to be responsible for the occurrence of these jumps. This indicates that there may be significant discontinuities in the total loss surface consisting of the mean-squared error for prediction of $\\nu_t^\\dagger$ and the L2 regularization (penalty) term involving the weights. After approximately 1000 epochs, the mean-square error (MSE) part of the EVNN loss function remains relatively stable, indicating that no further accuracy is being gained in predicting $\\nu^\\dagger_t$. The remaining drop in the overall loss function comes from the training procedure optimizing the weights in the EVNN. It was found that the optimal number of epochs at which to stop the EVNN training was 2000, which allows an additional 1000 epochs for the optimizer to tune the network weights.\n\nThe loss curve for the apNN (cf.~Fig.~\\ref{fig:loss_apNN}) is simply the MSE of the prediction, because no regularization term for the weights was included in the training process for this neural network. After about 500 epochs, the training loss is smaller than the validation loss, and the training loss continues to decrease monotonically after this point. By assessing the {\\em a priori} and {\\em a posteriori} performance for various selections of the number of epochs to use as the training endpoint (early stopping), it was found that a training endpoint of 1500 epochs yielded the best results for the apNN. \n\n\\begin{figure}\n\\includegraphics[]{training_curve_EVNN.pdf}\n\\caption{\\label{fig:loss_EVNN} Loss curves during training for the EVNN: (a) the total loss function consisting of the mean squared error (MSE) and the L2 regularization (penalty term) of the weights and (b) the MSE portion of the total loss function.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[]{training_curve_apNN.pdf}\n\\caption{\\label{fig:loss_apNN} Loss curve during training for the apNN. The loss function for the apNN consists of the mean squared error (MSE) only (viz., no regularization term was added to the loss function).}\n\\end{figure}\n\n\n\\section{Results}\\label{sec:results}\nThe final step in the deep learning procedure includes a test on a hold-out case---known also as the test case that is set aside and used to evaluate the neural network model only after training is complete. For the periodic hills training set, the $\\alpha=1.2$ case was held out of the training. Testing the network on a hold-out case provides an assessment of the network's ability to generalize by predicting a previously unseen case. The results for the hold-out test case are discussed in the following sections. A tilde is used to denote a quantity predicted by the neural networks (e.g., $\\tilde \\nu^\\dagger_t$ is a prediction of the optimal eddy viscosity provided by the neural network).\n\nFor the open-loop framework used here, there are two stages of evaluation: namely, {\\em a priori} and {\\em a posteriori}. In Section~\\ref{sec:apriori}, the model predictions are evaluated before injecting into the modified PIMPLE solver ({\\em a priori} assessment). In Section~\\ref{sec:aposteriori}, the resulting mean flow fields and combined anisotropy tensor ($a$) are evaluated after the iterative solution has converged around the fixed predictions ({\\em a posteriori} assessment). The framework used here treats the linear component of $a$ implicitly, so the combined prediction $\\tilde a$ is not available in an {\\em a priori} sense---it must be evaluated {\\em a posteriori}.\n\n\\subsection{{\\em A priori} assessment}\\label{sec:apriori}\nPrior to injecting the model predictions into the RANS equations, the predictions on the $\\alpha=1.2$ test case were analyzed for accuracy. The objective of the EVNN is to accurately predict $\\nu^\\dagger_t$. The mean-squared error is calculated by (for $N$ data points)\n\\begin{equation}\n{\\rm MSE} = \\frac{1}{N}{\\sum_{i=1}^N (\\tilde \\nu^\\dagger_{ti} - \\nu^\\dagger_{t\\theta i})^2}\\ .\n\\end{equation}\nFor the $\\alpha=1.2$ test case, the MSE in the prediction of $\\nu^\\dagger_t$ is $1.89\\times 10^{-7}$ m$^4$~s$^{-2}$. Figure \\ref{fig:nut_pred_hist} compares $\\tilde \\nu^\\dagger_t$ to $\\nu^\\dagger_{t\\theta}$. Figure~\\ref{fig:nut_pred_hist}(a) shows that while the predictions generally fall along the (ideal) line $\\tilde \\nu^\\dagger_t = \\nu^\\dagger_{t\\theta}$ at smaller values of $\\nu^\\dagger_t$, some scatter in the predictions is observed at larger values. The error distribution for the prediction of $\\nu^\\dagger_t$ [Fig.~\\ref{fig:nut_pred_hist}(b)] is roughly symmetric, indicating that there is not a significant over or under prediction tendency for the EVNN. The large prediction errors in Fig.~\\ref{fig:nut_pred_hist} demonstrate the importance of model conditioning---these errors will always be present at model testing time.\n\nFigure~\\ref{fig:nut_pred_contours} shows the spatial contours of $\\tilde \\nu^\\dagger_t$ and $\\nu^\\dagger_{t\\theta}$.\nThis figure shows that the EVNN is able to capture trends in the spatial variation of $\\nu^\\dagger_t$ well. The large values for $\\nu^\\dagger_t$ in the center of the domain are predicted well by the EVNN. In general, the neural network predictions match the ground truth (DNS data), with some minor discrepancies in the high $\\nu^\\dagger_t$ region above the first crest. While Fig.~\\ref{fig:nut_pred_hist} presents a pessimistic view of the ability of the EVNN to accurately predict $\\nu^\\dagger_t$, the contours in Fig.~\\ref{fig:nut_pred_contours} provide a more optimistic view.\n\n\n\\begin{figure}\n\\includegraphics[]{nut_pred_hist.pdf}\n\\caption{\\label{fig:nut_pred_hist} \\textit{A priori} prediction accuracy for the EVNN on the $\\alpha=1.2$ test case: (a) plot of the predicted value (ordinate) vs the ground-truth value obtained from DNS (abscissa) and (b) distribution of the $\\nu^\\dagger_t$ prediction errors. }\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[]{nut_pred_contours.pdf}\n\\caption{\\label{fig:nut_pred_contours} \\textit{A priori} contours of $\\nu^\\dagger_t$ for the test case: (a) ground-truth value from DNS and (b) value predicted by the EVNN.}\n\\end{figure}\n\n\nFigures~\\ref{fig:ap_pred_hist} and \\ref{fig:ap_pred_contour} show that the apNN predicts the non-linear part of the anisotropy tensor well. The MSEs associated with the prediction of each component of $a^\\perp$ are ${\\rm MSE}_{a^\\perp_{xx}}=2.31\\times 10^{-12}$, MSE$_{a^\\perp_{xy}}=9.40\\times 10^{-13}$, and MSE$_{a^\\perp_{yy}}=9.56\\times 10^{-13}$ (in units of m$^4$~s$^{-4}$). Spatial trends in the magnitude of $a^\\perp$ are accurately predicted by the apNN (cf.~Fig.~\\ref{fig:ap_pred_contour}). The individual points for the various components generally lie all along the ideal line $\\tilde{a}^\\perp=a^\\perp_\\theta$ (shown as the diagonal dashed lines in Fig.~\\ref{fig:ap_pred_hist}). A perusal of Fig.~\\ref{fig:ap_pred_hist} shows that the diagonal (normal stress) components of $a^\\perp$ are generally better predicted using apNN than the off-diagonal (shear stress) components. Indeed, the error distributions for the prediction of the normal stress components are more markedly peaked around zero (error) than that for the shear stress component [cf.~Figs.~\\ref{fig:ap_pred_hist}(a) and (f) with Fig.~\\ref{fig:ap_pred_hist}(d)].\n\n\n\\begin{figure}\n\\includegraphics[]{ap_pred_hist.pdf}\n\\caption{\\label{fig:ap_pred_hist} \\textit{A priori} prediction accuracy for the apNN on the $\\alpha=1.2$ test case. The prediction accuracy for each component of $a^\\perp$ is shown: (a), (c), (e) plots of the predicted value (ordinate) vs the ground-truth value obtained from DNS (abscissa) and (b), (d), (f) distribution of the $a^\\perp$ prediction errors. }\n\\end{figure}\n\\begin{figure*}\n\\includegraphics[]{ap_pred_contours.pdf}\n\\caption{\\label{fig:ap_pred_contour} \\textit{A priori} contours of each component of $a^\\perp$ for the test case: (a), (b), (c) ground-truth value obtained from DNS and (d), (e), (f) value predicted by the apNN.}\n\\end{figure*}\n\nComparing the predictions by the EVNN (Figs~\\ref{fig:nut_pred_hist} and \\ref{fig:nut_pred_contours}) to the predictions by the apNN (Figs~\\ref{fig:ap_pred_hist} and \\ref{fig:ap_pred_contour}), we see that the apNN generally out-performs the EVNN in terms of predicting their respective targets. This outcome agrees with our experience training and testing several times with these models---$\\nu^\\dagger_t$ is harder to predict than \n$a^\\perp$. This discrepancy indicates that the information content embodied in the RANS fields for the prediction of $a^\\perp$ is larger than that for the prediction of $\\nu^\\dagger_{t\\theta}$.\n\n\n\\subsection{{\\em A posteriori} assessment}\\label{sec:aposteriori}\nThe fields predicted by the EVNN and apNN were injected into the RANS equations. A similar procedure as Section \\ref{sec:conditioning} was utilized, whereby a modified PIMPLE solver in OpenFOAM v2006 was employed to inject the predicted fields from the EVNN ($\\tilde \\nu_t^\\dagger$) and apNN ($a^\\perp$) into the RANS equations, and iteratively reach a converged solution for the mean velocity field. \n\nFigure~\\ref{fig:U_mag_contours} shows the contours of the velocity magnitude for the converged solution. The converged results agree well with the flow fields coming from the highly resolved DNS simulation. In particular, the size of the separated region is well predicted, as well as several other important features of the flow: namely, boundary layers along the top and bottom wall, reattachment of the flow, and acceleration over the rightward hill. \n\nExamining several slices of the velocity field in Figs~\\ref{fig:U_samples} and \\ref{fig:V_samples}, we see that both the $U$ and $V$ velocity components are captured well compared to the DNS fields. While the base RANS model (the green line in Figs~\\ref{fig:U_samples} and \\ref{fig:V_samples}) fails to predict the sharp transition in the $U$ profiles above the separated region, the ML augmented solution predicts these transitions accurately. Furthermore, the $U$ profiles after reattachment show much better agreement with the DNS solution than the base RANS model. Along the top wall, the ML augmented solution slightly underpredicts the $U$ profile, while the base RANS model overpredicts the $U$ profile.\n\nWhile the test flow here is dominated by the streamwise flow, the $V$ profiles in Fig.~\\ref{fig:V_samples} shed light on key areas that ML augmentation improves the solution. The base RANS models under-predicts the bulge in $V$ within the separated region, thereby indicating an over-prediction of separation. The tendency of commonly used RANS models to over-predict separation is widely known. Another key area of improvement in the solution concerns the acceleration of the flow over the right hill---the RANS model greatly under-predicts the $V$ component along the right hill, while the corrected solution exhibits a much more accurate estimate compared to DNS. \n\n\n\n\\begin{figure}\n\\includegraphics[]{U_mag_contours.pdf}\n\\caption{\\label{fig:U_mag_contours} \\textit{A posteriori} contours of velocity magnitude after injecting the model predictions into the RANS equations: (a) DNS flow field and (b) result predicted by the present study. Here, $\\| \\tilde U \\|$ is the magnitude of the velocity vector arising from the final converged solution, after injecting the model predictions for $\\nu^\\dagger_t$ and $a^\\perp$.}\n\\end{figure}\n\n\n\n\\begin{figure*}\n\\includegraphics[]{U_samples.pdf}\n\\caption{\\label{fig:U_samples} \\textit{A posteriori} Samples of the $U$ velocity component along several lines throughout the flow field.}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[]{V_samples.pdf}\n\\caption{\\label{fig:V_samples} \\textit{A posteriori} Samples of the $V$ velocity component along several lines throughout the flow field.}\n\\end{figure*}\n\n\nFigure \\ref{fig:U_V_error_histograms} compares the error distributions in $U$ and $V$ from the base RANS model to the ML augmented solution. The base RANS model displays a tendency to over-predict the $U$ component, demonstrated by a left skewed $U$ error histogram. While the augmented solution contains smaller error magnitudes, it also contains this same skewed tendency. The $V$ histograms are both roughly symmetric, with the augmented solution being slightly more skewed than the base RANS model.\n\n\\begin{figure}\n\\includegraphics[]{U_V_error_histograms.pdf}\n\\caption{\\label{fig:U_V_error_histograms} Error distribution for the $U$ and $V$ velocity components after injecting the neural network model predictions for $\\nu_t^\\dagger$ and $a^\\perp$ into the RANS equations: (a) error in the $U$ velocity component and (b) error in the $V$ velocity component.}\n\\end{figure}\n\nSince the linear part of $a$ continues to evolve as the solution converges, the full anisotropy tensor is an {\\em a posteriori} quantity. Figure~\\ref{fig:a_mag_contours} compares the predicted anisotropy tensor magnitude to the DNS value. The prediction for $a$ generally agrees well with $a_\\theta$. A slight increase in the magnitude of $a$ as the flow accelerates over the right hill is captured, as well as the large increase in $a$ as the flow separates over the left hill. This accurate prediction of $a$ confirms the assumption in the present open loop framework that if $a^\\perp$ and $\\nu^\\dagger_t$ are accurately predicted, $\\vec{U}$ can evolve such that $\\vec{U}\\approx\\vec{U}_\\theta$, and therefore $S\\approx S_\\theta$. Finally, given $a^\\perp \\approx a^\\perp_\\theta$, $\\nu^\\dagger_t \\approx \\nu^\\dagger_{t\\theta}$, and $S\\approx S_\\theta$, we have $a \\approx a_\\theta$, which is the result shown in Fig.~\\ref{fig:a_mag_contours}.\n\n\\begin{figure}\n\\includegraphics[]{a_mag_contours.pdf}\n\\caption{\\label{fig:a_mag_contours} \\textit{A posteriori} contours of the anisotropy tensor magnitude: (a) ground-truth value obtained from DNS and (b) value predicted by the present study.}\n\\end{figure}\n\n\nThe conditioning problem for data-driven closures is often described as an amplification of errors in the closure term. It was of interest to determine whether this amplification was present in the solution after correcting $\\nu^\\dagger_t$ and $a^\\perp$. Figure~\\ref{fig:Rel_error_comparison_hist} compares the relative errors in the converged velocity field to the relative errors in the closure term, $a$. The errors in $U$ are \\textit{reduced} compared to the errors in $a$. This reduction indicates that a well-conditioned closure formulation can also benefit the solution by suppressing errors in the closure term, leaving more room for errors in the machine learning model predictions. Clearly, the issue of conditioning deserves more attention in data-driven turbulence modelling. We recommend that a similar conditioning analysis be completed each time a new decomposition of the closure term is presented, or each time a new injection framework is proposed. \n\n\\begin{figure}\n\\includegraphics[]{Rel_error_comparison_hist.pdf}\n\\caption{\\label{fig:Rel_error_comparison_hist} \\textit{A posteriori} comparison of conditioning error in $U$ to conditioning errors in $a$. Relative error is calculated in an identical manner to Fig.~\\ref{fig:Eddy_viscosity_error_a_theta_histograms}.}\n\\end{figure}\n\n\nAn important flow phenomenon in the periodic hills test case is separation. For the $\\alpha=1.2$ test case, the flow completely separates over the left hill, and reattaches along the bottom wall. The reattachment point is identified by the change in sign of the wall shear stress $\\nu \\frac{\\partial U_n}{\\partial n}$, where $n$ is the wall-normal direction, and $U_n$ is the velocity parallel to the wall. Figure~\\ref{fig:WSS} shows the predicted wall shear stress along the reattaching region of the bottom wall.\n\nThe reattachment point $x_r$ in Fig.~\\ref{fig:WSS} is the location where the wall shear stress changes signs. The base RANS model predicts a delayed reattachment point, a result of over-predicting the separation. The ML corrected velocity field reattaches slightly earlier than the DNS field. Nevertheless, the predicted reattachment point is much closer after applying the ML augmentation. The relative error in $x_r$ for the base RANS model is 20\\%, while the error in the ML corrected solution is significantly reduced to 8.9\\%.\n\n\n\\begin{figure}\n\\includegraphics[]{WSS.pdf}\n\\caption{\\label{fig:WSS} Wall shear stress for a portion of the bottom wall. The area considered is shown in the top right corner of the figure. The reattachment position (i.e., the location at which the wall shear stress changes sign) is summarized in the bottom right corner.}\n\\end{figure}\n\n\\subsection{Interpreting the data-driven corrective closure model}\\label{sec:interpret}\nGiven the rich input feature set used by the two neural networks in the present work, it was of interest to determine which features were found to be important by the model. In deep learning, this analysis is referred to as \"interpretability\"---taking a look inside the neural network black box. For this purpose, Shapley additive explanation (SHAP) values are commonly used to interpret machine learning models. The SHAP package by Lundberg and Lee~\\cite{Lundberg2017} was used to collect the SHAP values for a range of $\\nu^\\dagger_t$ predictions by the EVNN. The EVNN was selected for an interpretability analysis, because it is the principle neural network in the present model. The eddy viscosity prediction is a critical injected component. SHAP values are used to determine the relative incremental contribution of each input feature on the prediction, while accounting for non-linear interactions between features. \n\n\\begin{figure}\n\\includegraphics[]{shap_hist.pdf}\n\\caption{\\label{fig:shap_hist} Mean absolute SHAP values for each feature over the entire test case. This value is used to measure the importance of each feature. The values are sorted from highest relevance ($\\phi$) to lowest relevance ($I_1(B_{21})$). The SHAP values in this plot were calculated using the EVNN.}\n\\end{figure}\n\nSummation of the SHAP values for all predictions can be used to determine the global relative importance of each input feature. Figure~\\ref{fig:shap_hist} shows the relative importance of each input feature used in the EVNN. On a global basis, the use of $\\phi$ as an input feature (made possible by the use of the $\\phi$-$f$ model) was highly valuable, as the EVNN ranked it as the most important feature. Furthermore, the heuristic scalars ($q_1$, $q_2$, $q_3$) also ranked relatively high in terms of importance. The rest of the input features consists of various combinations of the strain rate, rotation rate, $k$ gradient, and $v^2$ gradient. The additional invariant $I_2$ used to extract additional features in the present work also added useful input features. Interestingly, there is no general correlation between the usefulness of $I_1$ and $I_2$ for a given basis tensor---the feature $I_1(B_7)$ ranked third, but $I_2(B_7)$ ranked twenty-fifth. The opposite is true for $B_5$---the second invariant ranked much higher than the first invariant.\n\nFigure~\\ref{fig:local_shap} shows the relative local SHAP values for the EVNN at several important points in the flow. The SHAP values for the eight most important features from Fig.~\\ref{fig:shap_hist}, normalized by the maximum value at each location, are exhibited in this figure. The locations are annotated in Fig.~\\ref{fig:local_shap}(a), and are summarized as follows: (b) the bulk flow region above the separated region; (c) the upper wall above the reattached and developing flow; (d) the accelerating flow above the right hill; (e) the lower wall just after reattachment; and, (f) the separated region. A blue colour indicates a positive influence on the $\\nu^\\dagger_t$ prediction, while a red colour indicates a negative influence. The sign indicates the direction each input feature \"nudges\" the output prediction, while the magnitude indicates the amount of influence.\n\nFigure~\\ref{fig:local_shap} shows that while some regions of the flow may feature similar phenomena (e.g., (b) and (g)), the relative importance of each feature is distinct. Though locations (c), (d), and (e) are all near the boundary layer, the magnitude of the $q_1$ importance is relatively low. Though $q_1$ is an input feature designed to detect near-wall regions, $q_1$ does not strongly influence $\\tilde \\nu^\\dagger_t$ near the wall. At the top wall (c), the anisotropy measure $\\phi$ is the most important feature. On the contrary, as the flow develops along the bottom wall at (e), $\\phi$ is less important than some higher-order gradients of the mean field. In the separated region (f), the ratio of total to normal Reynolds stress ($q_3$) is significantly more important than the other input features. This observation justifies the heuristic basis for including $q_3$. As the flow recirculates along the bottom wall, it is possible that the EVNN has learned to associate a particular change in $q_3$ with separation. The anisotropy measure $\\phi$ was not found to be as important in the separated region, but was important for the near-wall points (c) and (e).\n\n\n\n\n\n\n\\begin{figure}[!h]\n\\includegraphics[]{local_shap.pdf}\n\\caption{\\label{fig:local_shap} Relative feature importance at six locations in the flow: (a) locations in the flow domain and (b)--(g) bar plots showing the relative feature importance for the eight most important features in Fig.~\\ref{fig:shap_hist}. The relative influence on the prediction is calculated by normalizing the SHAP values by the highest SHAP value within the top eight features at the given location.}\n\\end{figure}\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nApplying machine learning to develop data-driven turbulence closures is a promising method to improve the accuracy of RANS simulations. However, a major issue for such models is the conditioning of the RANS equations after injecting a prediction for the closure term. If the framework is ill-conditioned, any errors made in the closure term could be amplified, and result in large errors in the converged mean field.\n\nThe decomposition of $\\tau$ (and, in turn, $a$) proposed in this work was shown to have good conditioning, through an analysis similar to Brener {\\em et al.}'s~\\cite{Brener2021}. By injecting the training labels (calculated from DNS), rather than the model predictions, the resulting mean field provides the \"upper limit\" achievable at testing time. We demonstrated that decomposing $a$ into an optimal eddy viscosity ($\\nu^\\dagger_t$) and a remaining non-linear part ($a^\\perp$) produced a well-conditioned closure. We also motivated the inclusion of the non-linear part of $a$ through evaluating the best prediction by a purely linear model.\n\nWe impose the often relaxed requirement of Galilean invariance on all input features. The input feature set was augmented by using an additional invariant to the one previously used in the literature. The minimal integrity basis for the strain rate, rotation rate, $k$ gradient, and $p$ gradient consists of 47 tensors, and the first invariant yields a scalar for each of these basis tensors. For a complex three-dimensional flow, using the second tensor invariant can effectively double the number of input features arising from this tensor basis to 94. For the two-dimensional flow considered here, we showed that many of these basis tensors are zero (Appendix~\\ref{ap:invariants}), an outcome not previously discussed in detail. However, the input feature set used in the present work is still one of the richest feature sets applied for data-driven turbulence modelling to date. To determine which input features were found to be important, an interpretability framework (SHAP) was also applied to explain the predictions for the optimal eddy viscosity. Using this framework, it was found that the three heuristic scalars $q_1$, $q_2$ and $q_3$ ranked high along with $\\phi$ (normalized wall-normal Reynolds stress) for the prediction of the optimal eddy viscosity.\n\n\nUsing one neural network to predict $\\nu^\\dagger_t$ and another to predict $a^\\perp$, we demonstrated that these quantities are predictable with reasonable accuracy for a periodic hills dataset. Even after introducing model error into the predictions, the resulting mean fields agree well with the DNS results. {\\em A posteriori} predictions of both the velocity field and the anisotropy tensor were demonstrated to agree well with the DNS results. Furthermore, the framework proposed here has a \\textit{smaller} error in the velocity field than the closure prediction. This result is significant because it shows that under certain conditions, the RANS equations can suppress rather than amplify errors in the closure prediction.\n\nIn the present work, we also demonstrated that machine learning augmentation can be used to improve even a sophisticated RANS model. The base turbulence model used was the $\\phi$-$f$ model, a type of $v^2$-$f$ model with three transport equations and one elliptic relaxation equation. The majority of previous studies has focused on simpler turbulence closure models, such as the standard $k$-$\\varepsilon$ model or the $k$-$\\omega$ model. This result indicates that machine learning augmentation has the potential to improve the results from even more sophisticated and recent RANS models, such as the elliptic blending (EB) $k$-$\\varepsilon$-lag model~\\cite{Lardeau2016}. Since the field $\\phi$ was found to be a highly useful feature, we conclude that more sophisticated models may benefit even more from machine learning augmentation than the simpler RANS models. The wider set of mean fields for forming a complex closure relationship may provide a richer description of the flow, yielding more information for a machine learning model to use in predicting quantities of interest. This idea could be extended to develop an augmented Reynolds stress transport model, where an even broader set of mean fields are available.\n\nFuture work includes further optimizing the non-linear part of the anisotropy tensor $a^\\perp$. While the present work imposes strict invariance quantities for the input features, the direct prediction of the components of $a^\\perp$ means that the outputs do not possess the same invariance quantities. Possible options for remedying this discrepancy include using a tensor basis neural network (TBNN) for $a^\\perp$ to construct neural network for $a^\\perp$ that is invariant with respect to coordinate transformations (e.g., rotations, reflections). Further work also includes applying this framework to a broader set of flows. While the periodic hills dataset used here sufficiently demonstrated the merits of this framework, using a wider range of flows could provide additional useful information, especially when interpreting the closure model. For example, an analysis of the importance of various features for distinct flows could guide the input feature selection for a sophisticated three-dimensional machine learning closure model.\n\n\n\\begin{acknowledgments}\nR.M. is supported by the Ontario Graduate Scholarship (OGS) program, and the Natural Sciences and Engineering Research Council of Canada (NSERC). The computational resources for this work were supported by the Tyler Lewis Clean Energy Research Foundation (TLCERF) and the Shared Hierarchical Academic Research Computing Network (SHARCNET).\n\\end{acknowledgments}\n\n\\section*{Data Availability Statement}\n\n\n\n\nThe data that support the findings of this study are openly available in \"A curated dataset for data-driven turbulence modelling\" at doi.org\/10.34740\/kaggle\/dsv\/2637500.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nDynamical localization of Floquet states in time-dependent and chaotic Hamiltonian systems\nis a phase coherent effect arising from quantum interferences. Quantum kicked rotor \nis a paradigmatic model for quantized chaotic systems which displays localization effects.\nQuantum localization in kicked rotor (KR) continues to attract attention\nin a variety of contexts ranging from metal-insulator transitions \\cite{chabe, garcia}, coherent control \\cite{gong},\nentanglement measures \\cite{santh}, quantum resonances \\cite{fish1, max, beh, dana1}, quantum ratchets \\cite{schanz, dana} to quantum transport \\cite{jones}\nand decoherence effects \\cite{sarkar, lutz}. Most of such studies have focussed on KR as a model for time-dependent potential exhibiting classically chaotic\ndynamics and quantum mechanical localization. For sufficiently strong non-linearity,\nKR displays chaotic classical dynamics and it is associated with diffusive growth of mean energy\nwith time. In the corresponding quantum regime, this unbounded energy growth\nis strongly suppressed by localization arising from destructive quantum interferences \\cite{kr}.\nThis effect in KR has been shown to be analogous to Anderson localization for electronic\ntransport in crystalline solids \\cite{fishman, stockmann, reichl}.\n\nOne significant property shared by the quantum KR and Anderson model is\nthe exponential decay of their eigenstates. In the one dimensional Anderson model all\nthe eigenstates are exponentially localized in position representation \\cite{anderson1, anderson2}, \ni.e, $\\psi(x) \\sim e^{-x\/x_l}$ where $x_l$ is the localization length. In the KR\nsystem, eigenstates are exponentially localized in the momentum representation \\cite{dima}.\nThe latter has been experimentally realized in microwave ionization of hydrogen atoms and \nin cold atomic cloud in optical lattices \\cite{moore, delande, fromhold}.\n\nKicked rotor can thus be regarded as a representative dynamical system from two distinct points of view.\nFirstly, in the classical sense, it belongs to a class of chaotic systems that obeys \nKolmogorov-Arnold-Moser (KAM) theorem \\cite{jurgen}. This effectively implies that, upon\nvariation of a chaos parameter, the system makes a smooth transition from regular to\npredominantly chaotic dynamics. Secondly, in the quantum mechanical regime,\nKR is a paradigmatic example of dynamical localization and the associated exponential\nprofile of its Floquet states. In the last one decade, many other facets of chaos and localization\nin variants of KR have been studied that have provided results different\nfrom this standard scenario \\cite{bala, italo, pragya, klappauf, garcia}.\n\nOne class of important variant is to place the KR in a singular\npotential. Presence of singularity in the potential violates one of the conditions\nfor the applicability of KAM theorem and leads to a scenario in which abrupt,\nrather than smooth, transition\nfrom integrability to chaotic dynamics becomes possible. Such abrupt transition to chaos\nis a feature of non-KAM systems and is seen, for instance, in the\nkicked particle in an infinite potential well \\cite{sankar, hu}. The quantum eigenstates of this\nsystem had been reported to display localization and its profile is {\\sl not}\nexponential but was claimed to have power-law type decay in the unperturbed basis.\nA more systematic study in Ref. \\cite{garcia} incorporated singularity in the KR\nthrough a tunable potential term $V(q;\\alpha)$ such that it becomes singular at some special\nvalue of tunable parameter $\\alpha= \\alpha_s$. It was shown, through numerical simulations,\nthat if $\\alpha=\\alpha_s$ in the potential, then all the eigenstates of the system are\npower-law localized. Indeed, it was even suggested that KR when acted upon by\na singular potential would display eigenstate localization with power-law profile\nin contrast to the exponential profile obtained in the context of standard KR \\cite{hu, liu}.\nThis suggestion has not yet been numerically tested in a variety of chaotic\nHamiltonian systems and general analytical results in support for this claim remains\nan open question.\n\nIn this paper, we examine the question whether the presence of non-analytic potential\nin a kicked rotor would {\\it generically} imply power-law profile for its eigenstates\nin the quantum regime. To address this question, we consider the dynamics of a periodically kicked particle\nplaced in a stationary finite potential well of height $V_0$. This is primarily a non-KAM system\nand its unusual classical and quantum transport properties, reflective of its\nnon-KAM nature, were recently reported in Ref. \\cite{paul}. This system subsumes two limiting cases; it\nis the standard KR (a KAM system) in the absence of finite well potential, i.e, $V_0 = 0$ and\nif $V_0 \\to \\infty$, then it becomes a kicked rotor system placed in an infinite well\n(a non-KAM system) and has been studied in Refs. \\cite{sankar, hu}.\nHence, this is a suitable test bed to understand the transition in the nature of\nFloquet states as $V_0$ is varied from the\nlimit of a KR system (analytic potential) to that of a system with \nsingular potential. Further, this can lead to a better understanding of the quantum manifestations of\nclassical chaos non-KAM systems. \n\n\n\\begin{figure}\n\\includegraphics*[width=3.0in]{fig1.eps}\n\\caption{(Color online) Schematic of the stationary\npotential, $V_{sq}(\\theta)$ with $V_0$ as the potential height, $b$ and $w$ as barrier and well width. $A$ and $B$ represents the positions at which periodic boundary conditions are applied. I and II denotes the regions below and above $V_0$.}\n\\label{fig1}\n\\end{figure}\n\n\n\n\nUsing the context of this system based on KR, we show in this paper that the\npresence of singularity in the potential does not always guarantee\npower-law localization of Floquet states.\nSingular potentials are associated with power-law localized Floquet states provided\nthat the Floquet states span an energy band in which the singularity is effectively\nfelt by the particle. Further, it is demonstrated that the spectral\nfluctuations properties such as the level spacing distributions for this\nsystem depends on the energy range being considered. Hence, spacing distributions\ndo not characterize the system at all the energy scales.\n\nIn section 2, we introduce the model of kicked particle in a finite barrier and in\nsection 3 we report results on the decay profile of the Floquet states and relate\nit to the decay of the Floquet matrix and to the effective singularity ``felt\" by the\nkicked particle at various energy scales. In section 4, we obtain a tight binding\nform for our system to deduce the non-exponential nature of Floquet state decays.\nFinally, in section 5, we discuss the manifestation of potential singularity in \nthe averaged quantities derived from Floquet states.\n\n\n\n\n\n\n\\section{Kicked particle in finite barrier}\n\n\n\nThe dimensionless Hamiltonian of a periodically kicked particle in a finite well\npotential \\cite{paul} is\n\\begin{equation}\n\\begin{aligned}\nH &= \\frac{p^2}{2}+V_{sq}(\\theta)+k~\\cos(\\theta) \\sum_{n=-\\infty}^{\\infty} \\delta(t-n)\\\\\n &= H_0 + V(\\theta) \\sum_{n=-\\infty}^{\\infty} \\delta(t-n).\n\\end{aligned}\n\\label{ham1}\n\\end{equation}\nIn this, $V(\\theta)=k~\\cos(\\theta)$ and $V_{sq}(\\theta)$ is the square well potential shown in Fig. \\ref{fig1} and can\nbe represented as\n\\begin{equation*}\nV_{sq}(\\theta)=V_0[\\Theta(\\theta-R\\pi)-\\Theta(\\theta-R\\pi-b)],\n\\end{equation*}\nwhere $V_0$ is the potential height and $b$ is the barrier width,\n$R=w\/\\lambda$ is the ratio of well width to the wavelength of the kicking field and $k$ is the kick \nstrength. Throughout this work, we have set $\\lambda=2\\pi$, $b$ and $w$ are constrained by\n$b+w=2\\pi$. Periodic boundary conditions are applied at positions A and B shown in Fig. \\ref{fig1}.\n\n\n\n\n\t\t\t \nLet $E_n$ and $|\\psi_n\\rangle$ represent the energy and the eigenstate of\nthe unperturbed system such that $H_0|\\psi_n \\rangle = E_n |\\psi_n \\rangle$.\nFurther, $|\\psi_n\\rangle$ can be written as a superposition of all momentum\nstates $|l\\rangle$, i.e. $|\\psi_n \\rangle=\\sum_l a_{nl}|l\\rangle$, where $a_{nl}$ represents the \nexpansion coefficient. Then any general initial state can be expressed\nin the energy basis state representation as $|\\Psi\\rangle=\\sum_n b_n|\\psi_n\\rangle$.\nThe mean energy in the state $|\\Psi(t)\\rangle$ can be obtained as\n\n \\begin{equation}\n \\begin{aligned}\n E(t) &= \\langle\\Psi(t)|\\hat{H_0}|\\Psi(t)\\rangle \\\\\n &= \\sum_m E_m |b_m(t)|^2.\n \\end{aligned}\n \\end{equation}\nThe quantum map that connects the state $|\\Psi(N+1)\\rangle$ at time $N+1$ with the \nstate $|\\Psi(N)\\rangle$ can be obtained by evolving the Schroedinger equation\n$|\\Psi(N+1)\\rangle=\\widehat{U}|\\Psi(N)\\rangle$,\nwhere $\\widehat{U}$ is the Floquet operator, \n\\begin{equation}\n\\widehat{U}=e^{-H_0\/\\hbar_s}e^{-iV(\\theta)\/\\hbar_s},\n\\end{equation}\nand $\\hbar_s$ is the scaled Planck's constant. In the energy representation,\nthe elements of the Floquet operator are given by\n\\begin{equation}\nU_{nm}=\\sum_{p,p'} a_{np}^{*} a_{mp'} i^{|p-p'|}J_{|p-p'|}\\left(\\frac{k}{\\hbar_s}\\right),\n\\label{umat}\n\\end{equation}\nwhere $J_{|p-p'|}(\\cdot)$ is the Bessel function of order $|p-p'|$.\n\nThe eigenvalue equation governing the Floquet operator is $\\widehat{U} |\\phi\\rangle = e^{i \\omega} |\\phi\\rangle$\nin which $|\\phi\\rangle$ represents a Floquet state and $\\omega$ is its quasi-energy. The \nFloquet operator is an unitary operator and hence the eigenvalues lie\non a unit circle. Further the quasi-energy state, $|\\phi\\rangle$, can be decomposed as a \nsuperposition of all energy states $|\\psi_n\\rangle$, i.e., \n$|\\phi\\rangle = \\sum_n c_n |\\psi_n\\rangle$, where $|c_n|$ is the\nprobability density of finding the particle in state $|\\psi_n\\rangle$.\n\n\n\n\nIn order to analyse the localization properties of the\nFloquet states, Floquet matrix of order $N$ is numerically diagonalized to determine the\nquasi-energies and the Floquet vectors. In this work, $N=10035$ and we have ensured \nthat for the choice of parameters used in this paper, the system is classically chaotic \\cite{supplement}.\nFloquet states for standard KR are generally known to be exponentially localized in\nmomentum space $|\\psi(p)|^2 \\sim \\exp(-p\/\\xi)$ characterized by a localization length $\\xi$. In\ncontrast to that, Floquet states of a kicked particle in a periodic potential well is localized over\nenergy basis state $|\\psi_n\\rangle, n=1,2,\\dots$. In the subsequent sections,\nit is shown that the system in Eq. \\ref{ham1} exhibits a transition from exponential\nto power-law localization as the parameters $V_0$ and $k$ are varied.\n\t\t\t\n\n \n\\section{Floquet states}\t\t\n\n\n\\begin{figure}\n\\centering\n\\includegraphics*[width=3.3in]{fig2.eps}\n\\caption{(Color online) Decay of Floquet states over the unperturbed basis states, averaged\nover all the Floquet states. The parameters are $b=1.4\\pi, \\hbar_s = 1.0$, (a) (KRIW limit) \n$V_0=5000.0, k=0.25$, and (b) (KR limit) $V_0=0.5, k=4.25$. In (a) black line is a linear fit and in (b) black curve corresponds to KR.}\n\\label{avgdFloquet states}\n\\end{figure}\n\n\t \nIn this section, we will mainly focus on the average spectral properties of the\nFloquet states which governs the dynamics in the quantum regime.\nFor the finite well represented by Hamiltonian in Eq. \\ref{ham1}, the nature of\nFloquet state decay profile, in general, will depend on the choice of parameters, namely,\nkick strength $k$ and potential height $V_0$.\nFig. \\ref{avgdFloquet states} has been obtained by averaging over $10035$\nFloquet states $|\\phi\\rangle(=\\sum_n c_n |\\psi_n\\rangle)$ for each set of\nparameters. Prior to averaging, each Floquet state was shifted by $n_{max}$, i.e. $v = n-n_{max}$, where $n_{max}$\ncorresponds to $n$ for which $|c_n|^2$ is maximum.\n\nIf $V_0 > 0$, the Hamiltonian in Eq. \\ref{ham1} is a non-KAM system due to the presence of\nsingularities in $V_{sq}(\\theta)$. Based on numerical simulations of kicked systems with singular\npotentials, it was argued that their Floquet states display power-law decay over the\nunperturbed basis \\cite{sankar, hu, liu, garcia, jose}. Further, a new universality class has been proposed in Ref. \\cite{garcia} based on the\npresence of classical singularity and power-law localization.\nTo discuss the results, in the light of this proposal, two limiting cases can be identified;\n(i) $0< V_0 < 1$ (KR limit) and\n(ii) $V_0 \\gg 1$ (KR in infinite well (KRIW) limit).\nIn the KR limit, notwithstanding the singularity in the potential, the Floquet states can be \nexpected to be qualitatively closer to that of KR.\nIn particular, if kick strength $k \\gg 1$, all the Floquet states display exponential decay profile.\nOn the other hand, in the KRIW limit, the potential height is large ($V_0 \\gg 1$) and is \nqualitatively closer to the kicked infinite well system \\cite{sankar, hu}. In this limit, even for small kick strengths \n$k<1$, it is known that all the Floquet states show power-law decay over the \nunperturbed basis \\cite{hu, liu}. Both these limits are illustrated in Fig. \\ref{avgdFloquet states}.\n\nIn Fig. \\ref{avgdFloquet states}(a), the decay of the averaged Floquet state in the KRIW limit is\nshown for $V_0=5000.0$ and $k=0.25$. It is consistent with a power-law form $P(v) \\sim v^{-\\gamma}$,\nwhere $v > 0$ and $\\gamma \\approx 2.5$, in agreement with the value reported in Ref. \\cite{hu} and the deviation observed can be attributed to the finite height of well.\nOn the other hand, averaged Floquet state in the KR limit for $V_0 = 0.5$\nand $k = 0.25$ shown in Fig. \\ref{avgdFloquet states}(b) displays exponential decay, $P(v) \\sim \\exp(-v\/l)$, where $l$ is the\nlocalization length. This is the standard dynamical localization scenario but is generally\nnot associated with non-KAM systems.\nBoth these decay profiles in Fig. \\ref{avgdFloquet states} can be understood if the relation\nbetween singular potential and power-law localization can be restated in the following\nmanner. For this purpose, let $\\epsilon_{if}= \\{E_i, E_{i+1}, E_{i+2}, \\dots E_f\\} $ \ncollectively represent the energies of a set of states of $H_0$ lying in the energy band $(E_f-E_i)$\nbetween two states with quantum numbers $f$ and $i$\nin the unperturbed system. The classical singularities are associated with quantum\npower-law localization of a set of Floquet states mostly lying in the energy range $\\epsilon_{if}$,\nprovided $\\epsilon_{if} < V_0$. Thus, Floquet states will display power-law localization\nonly if they effectively ``feel\" the non-smooth potential. This requires that\nenergy scale $\\epsilon_{if}$ be less than that representing $V_0$.\n\nIt must be emphasised that the Hamiltonian in Eq. \\ref{ham1}\nis classically a non-KAM system if $V_0 > 0$, for all values of $k>0$.\nHence, with $V_0=5000.0$ and for kick strength as small\nas $k=0.25$ the system is classically chaotic \\cite{supplement} and the corresponding quantized\nsystem displays power-law localized profile of the Floquet states (Fig. \\ref{avgdFloquet states}(a)).\nIn this case, most of the $10035$ Floquet states used for averaging are such that $\\epsilon_{if} < V_0$.\nOn the other hand, in the\ncase of Fig. \\ref{avgdFloquet states}(b), even though it is still a non-KAM system with\nsingular potential, the Floquet states mostly straddle energy scales $\\epsilon_{if}$ larger than\n$V_0$ and are not affected by the shallow singular potential and hence\nthe exponentially localized Floquet states are obtained.\n\n\\begin{figure}\n\\centering\n\\includegraphics*[width=3.3in]{fig3.eps}\n\\caption{(Color online) Averaged decay of the matrix elements of the Floquet operator $\\widehat{U}$\nas a function of $m$. The parameters are $b=1.4\\pi, \\hbar_s = 1.0$, (a) (KRIW limit) \n$V_0=5000.0, k=0.25$, and (b) (KR limit) $V_0=0.5, k=4.25$, (c) $V_0=100.0, k=4.25$,\n(d) $V_0=5000.0, k=4.25$. $n_c$ represents the crossover point from exponential to power-law profile. All black curves corresponds to KR.}\n\\label{mele}\n\\end{figure}\n\n\\subsection{Matrix element decay}\nIt is known that exponential localization in the KR is associated with\nthe exponential decay of the matrix elements of the corresponding Floquet operator.\nFurther, from Ref. \\cite{sankar, hu}, it is also known that in the case of KR in the\ninfinite well, a non-KAM system, the matrix elements of $\\widehat{U}$ display a \npower-law decay after a bandwidth $\\eta\\propto k$.\nHence, it is natural to enquire how the decay of matrix elements changes\nits character as $V_0>>1$ approaches the limit $V_0 \\to 0$. In the unperturbed basis, the\nmatrix elements are $U_{nm}$ as given by Eq. \\ref{umat}.\nThis is illustrated in Fig. \\ref{mele} which shows $M_m = \\langle |U_{nm}| \\rangle_n$\nas a function of $m$, with $m>n$, in log-log plot.\n\nFigure \\ref{mele}(a,b) shows $M_n$ as log-log plot for the same choice of parameters as in \nFig. \\ref{avgdFloquet states}(a,b). Figure \\ref{mele}(a) corresponds to KRIW limit and\nshows a short regime of exponential decay followed by an asymptotic power-law decay.\nIn Fig. \\ref{mele}(b), $V_0=0.5$ corresponding to the KR limit and the decay of $M_n$\nlargely follows that of KR except for $n>>1$ where it decays as a power-law.\nIn general, the following features are observed. In the limit as $V_0 \\to \\infty$,\nthe decay is of power-law form. In the opposite limit of $V_0 \\to 0$, the decay\nis exponential in nature. In general, for any intermediate $V_0$, i.e., $0 < V_0 < \\infty$, \nan initial exponential decay is followed by an asymptotic power-law decay whose\nslope is approximately 2.7.\nIf $V_0 < \\infty$, the initial exponential decay is always present. \nThe exponential decay sharply changes over\nto a power-law decay at $n=n_c$ as shown by dotted vertical lines in Fig. \\ref{mele}.\nFor any fixed value of kick strength $k$, as $V_0$ varies from 0 $\\to \\infty$, then $n_c$\nchanges from $\\infty \\to 0$. It is also to be noted that for fixed $V_0$, as $k$\nincreases, $n_c$ also increases.\n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics*[width=6.2 in]{fig4.eps}\n\\caption{(Color online) Floquet states of Hamiltonian in Eq. \\ref{ham1} with parameters $b=1.4\\pi$, $\\hbar_s=1.0$, \n$V_0=5000.0$, $k=4.25$. The three figures differ in how Floquet states were averaged over;\n(a) averaged over all the computed states, (b) averaged over states with $\\mu< 1$, \n(c) averaged over states with $\\mu \\gg 1$. In (a,b) black curve represent best fit line and in (c) black curve corresponds to KR.}\n\\label{avgdrm}\n\\end{figure*}\n\n\n\\subsection{Energy scales}\n\nIn this section, we show how singularity of the potential and energy scales associated\nwith the Floquet states determine the localization structure of these states.\nAs discussed in Sec. 2, $c_{rj} = \\langle \\phi_r | \\psi_j \\rangle$,\nwhere $|\\psi_j\\rangle$ is an eigenstate of $H_0$ with\nassociated energy $E_j$. Let $c_{max} = \\mbox{max}(|c_{r,1}|^2, |c_{r,2}|^2, \\dots |c_{r,N}|^2)$\nrepresent the largest overlap of $r$-th Floquet state with $|\\psi_j\\rangle$. The energy\nassociated with $|\\psi_j\\rangle$, and hence with $c_{max}$, is denoted by $E_{max}$.\nThen, an effective parameter $\\mu=E_{max}\/V_0$ can be identified to distinguish two regimes,\nnamely, (i) $\\mu \\le 1$ and (ii) $\\mu >> 1$. Physically, $\\mu \\le 1$ corresponds to\nFloquet states (in energy basis) mostly confined to the potential height $V_0$ and $\\mu >>1$\ncorresponds to those Floquet states that have significant overlap with states lying\nin the energy scales far greater than $V_0$.\n\nIn Fig. \\ref{avgdrm}(a), $\\ln \\langle c_{v} \\rangle$ is shown by averaging\nover all the computed Floquet states for $V_0=5000.0$ and $k=4.25$. As discussed\nearlier, the Floquet state profile is a combination of initial exponential decay\nfollowed by a power-law decay. However, if average is taken only over those states\nthat satisfy the condition $\\mu \\le 1$, then the resulting profile is shown\nin Fig. \\ref{avgdrm}(b).\nIn this case $k=4.25$ and all the Floquet states are confined\nto an energy scale well below $V_0$. Hence, these set of Floquet states can be\nexpected to ``feel\" the presence of singularity in the potential. In this\nregime, we observe a power-law profile for the averaged Floquet states as displayed in Fig. \\ref{avgdrm}(b).\n \nHowever if the states are averaged subject to the condition that $\\mu >> 1$, then\nsingularity is not strongly felt by the Floquet states since the bandwidth of their energy\ndistribution is far greater than $V_0$. Effectively, at this\nenergy scale, the singularity becomes insignificant and hence we can expect it to \nlie in the KR limit. Indeed, as seen in Fig. \\ref{avgdrm}(c), \n$\\langle c_{v} \\rangle$ is nearly identical to that of KR (shown as black curve in Fig. \\ref{avgdrm}(c)) at $k=4.25$.\n\nIn general, for the Hamiltonian in Eq. \\ref{ham1}, the localization property of a subset \nof Floquet states in an energy band $\\epsilon_{if}$ depends on the effectiveness of \nthe singularity for the spectral range $\\epsilon_{if}$ under consideration.\nIn a given energy band $\\epsilon_{if}$, if the singularity is effective, then \npower-law localization is obtained and if singularity is weak or absent then \nexponential localization results for the states in $\\epsilon_{if}$. \nAs far as localization of eigenstates of chaotic systems are concerned, it is known \nthat either all the states are exponentially localized (as in KR) or power-law \nlocalized (as in systems with singular potentials) but, to the best of our\nknowledge, combinations of these localization profiles have not been reported before.\nIn the next section, we transform the Hamiltonian in Eq. \\ref{ham1} to that of a\ntight binding model and show that $V_0\/E$ controls the localization property of eigenvectors.\n\n\n\\section{Tight binding model}\t\t\t\nThe dynamical localization in the quantum KR system was mapped to \nAnderson model for electron transport in a one-dimensional crystalline lattice \\cite{fishman}.\nBy implication, the exponential decay profile of eigenstates in the Anderson model \ntranslates to exponential profile (in the momentum representation) for the \nFloquet states of quantum KR.\nFollowing this mapping technique, in this section, we map the Hamiltonian in Eq. \\ref{ham1}\nto a tight binding Hamiltonian.\nSince Eq. \\ref{ham1} represents a time periodic system, using Floquet-Bloch \ntheorem, we can write the quasi-energy state as\n\\begin{equation}\n\\phi(\\theta,t)=e^{-i\\omega t} u(\\theta,t),\n\\label{eq1}\n\\end{equation}\nwhere, $u(\\theta, t) = u(\\theta, t+1)$. In between two consecutive\nkicks, the Hamiltonian $H_0$ governs the evolution of the particle and is given by,\n\\begin{equation}\n\\phi_n^{-}(t+1)=e^{-iE_n} \\phi_n^{+}(t).\n\\label{eq2}\n\\end{equation}\nIn this, $\\phi_n^{-}(t+1)$ and $\\phi_n^{+}(t)$ are the quasi-energy states just\nbefore the $(t+1)$-th kick and just after $t$-th kick and $E_n$ is the $n$-th energy\nlevel of $H_0$. During the evolution it acquires an extra phase $e^{-iE_n}$.\nBy substituting Eq. \\ref{eq1} in Eq. \\ref{eq2} and using the periodicity of $u(\\theta, t)$, \nwe obtain\n\\begin{equation}\n\\begin{aligned}\nu_n^{-}(\\theta, t+1)=e^{i \\omega} e^{-i E_n} u_n^{+}(\\theta, t).\n\\end{aligned}\n\\label{eq3}\n\\end{equation} \n\t\t\t\nNow the quasi-energy state just after a $t$-th kick can be obtained using a map\n$\\phi^{+}(\\theta, t) = e^{-iV(\\theta)} \\phi^{-}(\\theta, t)$.\nBy using Eq. \\ref{eq1}, this can be written in-terms of $u(\\theta, t)$ as \n\\begin{equation}\nu^{+}(\\theta, t)=e^{-iV(\\theta)} u^{-}(\\theta, t).\n\\label{eq4}\n\\end{equation}\n\nNow, $e^{-iV(\\theta)}$ is expressed in terms of trigonometric function\n$W(\\theta)=-\\tan\\left(\\frac{V(\\theta)}{2} \\right)$ as\n\\begin{equation}\ne^{-iV(\\theta)}=\\frac{1+iW(\\theta)}{1-iW(\\theta)}.\n\\label{eq5}\n\\end{equation}\nThis is used in Eq. \\ref{eq4} to obtain\n\\begin{equation}\n\\frac{u^{+}(\\theta)}{1+i W(\\theta)}=\\bar{u}=\\frac{u^{-}(\\theta)}{1-i W(\\theta)},\n\\end{equation}\nwhere $\\bar{u}$ is defined as $\\bar{u}=[u^{+}(\\theta)+u^{-}(\\theta)]\/2$.\nUsing Eq. \\ref{eq4} and Eq. \\ref{eq5}, the evolution of the quasi-energy state after \none period is\n\\begin{equation}\nu^{+}(\\theta)=e^{-iV(\\theta)} e^{i(\\omega -E_n)} u^{+}(\\theta).\n\\end{equation}\nThis can be written as,\n\\begin{equation}\n(1-i W(\\theta)) \\bar{u}=e^{i(\\omega -H_0)} \\bar{u} (1+i W(\\theta)),\n\\label{and1}\n\\end{equation}\nwhere $\\bar{u}=\\frac{u^{+}}{1+i W(\\theta)}$. \nNow rearrangement of terms leads to\n\\begin{equation}\n\\tan\\left(\\frac{\\omega - H_0}{2} \\right) \\bar{u} + W(\\theta) \\bar{u} = 0.\n\\label{and2}\n\\end{equation}\n\nThe quasi-energy state can be expanded in the unperturbed basis as $|\\bar{u}\\rangle=\\sum_m u_m |\\psi_m\\rangle$\nwhere, $|\\psi_m\\rangle$ are the eigenstates of $H_0$ and $u_m$ is given by,\n\\begin{equation}\nu_m = \\int \\bar{u} \\psi_m(\\theta) d\\theta = \\int \\frac{1}{2} [u^{+}(\\theta) + u^{-}(\\theta)] \\psi_m(\\theta)~d\\theta.\n\\end{equation}\nTaking the inner product of Eq. \\ref{and2} with $|\\psi_m\\rangle$, we will formally obtain\n\\begin{equation}\nT_m u_m + \\sum_l W_{ml} u_l = 0.\n\\label{tight1}\n\\end{equation}\nIn this, $T_m=\\tan(\\frac{\\omega - E_m}{2})$ represents the on-site energy and $W_{ml}$ is the\nhopping strength for a particle to hop from $m$th site to $l$th site and can be written in the\nenergy basis as\n\\begin{equation}\n\\begin{aligned}\nW_{ml} &= \\langle \\psi_m| W(\\theta) | \\psi_l \\rangle \\\\\n&= \\int \\sum_{p,q} a^{*}_{mp} e^{-ip\\theta} W(\\theta) a_{lq} e^{i q \\theta} d\\theta \\\\\n&= \\sum_{p,q} a_{mp}^{*} a_{lq} \\int W(\\theta) e^{-i(p-q) \\theta} d\\theta \\\\\n&= \\sum_{p,q} a_{mp}^{*} a_{lq} W_{p-q},\n\\end{aligned}\n\\end{equation}\nwhere $W_n=\\frac{1}{2 \\pi}\\int_0^{2 \\pi} W(\\theta) e^{-i n \\theta} d\\theta$ is the\nFourier transform of $W(\\theta)$.\nThus, in energy basis, after simple manipulation, Eq. \\ref{tight1} takes the form\n\\begin{equation}\n\\left(T_m + \\sum_{p,q} a_{mp}^{*} a_{mq} W_{p-q} \\right) u_m + \\sum_{p,q,l\\neq m} a_{mp}^{*} a_{lq} W_{p-q} u_l = 0.\n\\label{tight2}\n\\end{equation}\nThis is the tight binding model version of the Hamiltonian in Eq. \\ref{ham1}. In this, \n$(T_m + \\sum_{p,q} a_{mp}^{*} a_{mq} W_{p-q})$ represents the diagonal term and \n$a_{mp}^{*} a_{lq} W_{p-q}$ is the off-diagonal term of the transfer matrix. \nIt does not appear straightforward to analytically prove power-law profile of\nFloquet states\nstarting from Eq. \\ref{tight2}, though it appears fair to expect that\nin this case the decay of Floquet state profile will be different from exponential form. As numerical results\nshow, we obtain power-law localization. Similar results has also been reported in \\cite{cohen}.\n\nHowever, using Eq. \\ref{tight2}, it is possible to make an inference about Floquet state profile\nin the limit $\\mu >> 1$.\nIn this case $E_n \\gg V_0$ and the singularity in the potential becomes\ninsignificant. Effectively, the system behaves as a free particle with energy\n$E_n=\\frac{\\hbar_s n^2}{2}$ and the wave-function of $H_0$ is just the momentum eigenstate,\n$|\\psi_n\\rangle = a_{nn} e^{i n \\theta}$, with $a_{mn}=\\delta_{mn}$.\nThis set of conditions, if applied to Eq. \\ref{tight2}, lead to \n\\begin{equation}\n(T_m + W_0 ) u_m + \\sum_l W_{m-l} u_l = 0.\n\\end{equation}\nThis is just standard KR Hamiltonian transformed to the 1D Anderson model \\cite{fishman}, for which\nall the eigenstates are known to display exponential profile. Hence,\nas seen in Fig. \\ref{avgdrm}(c) for $\\mu \\gg 1$, the observed localization is exponential in nature.\nThus, even in the presence of singular potentials, eigenstate localization is not\ngenerically of power-law form. We reiterate the main result of the paper that the association\nbetween power-law profile of eigenstates and singular potentials needs to take into account\nthe effectiveness of singularity in a given energy band.\n\t\t\t\n\\section{Spectral signatures}\n\n Based on the results presented in Fig. \\ref{avgdrm}, a novel scenario for the\nspectral signatures can be expected. As the regimes $\\mu < 1$ and $\\mu \\gg 1$ are\ntraversed, by considering Floquet states in a suitable energy band $\\epsilon_{if}$, the decay profile \nof Floquet state changes from power-law to exponential form. This would also imply that \na unique spectral signature for the nearest neighbour spacing distribution $P(s)$, such \nas either the Poisson or Wigner distributions, may not exist for the system as a whole. Quite\nunusually, $P(s)$ would depend on the energy band $\\epsilon_{if}$ being considered.\nThus, in the same system for a given choice of parameters, in the limit $\\mu \\gg 1$ (KR limit)\nwe expect Poisson distribution and in the limit $\\mu < 1$ (KRIW limit) we expect P(s) to be \ncloser to Wigner distribution or possibly, a Brody distribution \\cite{brody}. \n\nThe Floquet operator $\\widehat{U}$ being a unitary operator, all the\neigenvalues lie on a unit circle, $\\omega_i\\in[0,2\\pi)$. In this case, level\ndensity is constant $\\left(\\frac{N}{2\\pi} \\right)$ and hence the unfolding of Floquet\nlevels is not necessary. To compute the spacing distribution, we have treated the \neigenvalues of even and odd parity states separately. The nearest neighbour spacing \ndistribution reveals two\ndifferent forms; for $\\mu < 1$ level repulsion is observed in the form of Brody distribution\nand for $\\mu \\gg 1$ level clustering is seen in the form of Poisson distribution \\cite{supplement}.\nThe regime of $\\mu < 1$ corresponds to KRIW limit and power-law decay of Floquet\nstates (see Fig. \\ref{avgdrm}(b)) and is associated with level correlations that\nare intermediate between no correlation and random matrix type level repulsion.\nOn the other hand, the limit of $\\mu \\gg 1$ is KR limit and levels remain uncorrelated\ndue to occurrence of dynamical localization.\nIt must be emphasised that two different level spacing distributions and level correlations \nfor the same system with identical parameters is a novel feature not usually\nencountered in the context of chaotic quantum systems. This unusual spacing distribution\nreinforces the central result of this paper that the relation between potential singularity \nand eigenvector profile is conditioned by energy regime being considered.\n\n\\begin{figure}\n\\centering\n\\includegraphics*[width=3.0in]{fig5.eps}\n\\caption{(Color online) Participation ratio of the Floquet states as a function of $E_{max}$ \nfor $b=1.4\\pi$, $\\hbar_s = 1.0$, $V_0=5000.0$, $k=4.25$ (same set of parameters\nas in Fig. \\ref{avgdrm}). The vertical line is placed at $E_{max}=V_0$ ($\\mu=1$), the height \nof potential barriers. Horizontal black dotted line represents the mean participation ratio for $\\mu>1$.}\n\\label{PR}\n\\end{figure}\n\nThis dichotomy is reflected in the eigenvector statistics as well.\nThis is easily observed by studying the participation ratio (PR) of the Floquet states\nthat provides information about their localization properties. For an eigenstate that resides\nin the infinite dimensional Hilbert space, participation ratio is defined as\n\\begin{equation}\nP=\\sum_{i=1}^{\\infty} |\\psi_i|^4\n\\end{equation}\nwith the condition that $\\sum_i |\\psi_i|^2 = 1.0$, where $\\psi_i$ are components \nof a Floquet state. It is a measure of how many basis states\neffectively participate in making up the eigenstate. If $P \\approx 1$, then\nthe state is strongly localized and implies that one basis state contributes significantly \nto the Floquet state\nwhile the contribution from the rest of the basis are almost negligible. However, if \n$P \\sim \\frac{1}{N}$, then the Floquet state is of extended nature and all the \nbasis states make equal contribution on an average.\nFigure \\ref{PR} displays $P$ for all the $10035$ converged Floquet states as a function\nof energy $E_{max}$ for the identical choice of parameters as in Fig. \\ref{avgdrm}.\nQuite surprisingly, $P$ distinguishes the two regimes, $\\mu < 1$ and $\\mu \\gg 1$. The\nboundary between the two regimes is at $E_{max}=V_0$, the height of potential barriers.\nFor $\\mu \\gg 1$, exponential localization of Floquet states implies that\n$|\\phi\\rangle \\sim e^{-n\/l}$, where $l$ is the localization length. A remarkable result due\nto Izrailev \\cite{kr, chirikov} provides the relation, $l \\approx \\frac{k^2}{2 \\hbar_s^2}$. For our case, this estimate\ngives $l \\approx \\frac{k^2}{2 \\hbar_s^2} = 9.03$\nand this represents the effective number of basis states that goes in constructing\nthe Floquet states. As participation ratio is the inverse of the effective number\nof basis states, it is estimated to be $P \\approx \\frac{2 \\hbar_s^2}{k^2} = 0.11$. As seen in Fig. \\ref{PR},\nthis value closely matches the computed PR in the regime $\\mu \\gg 1$.\n\nFor $\\mu<1$, the mean $P$ is larger compared to that for $\\mu>1$ as shown in Fig. \\ref{PR}.\nThe reason can be traced back to the fact that in the case of infinite well $E_n \\sim n^2$\nand hence levels are spaced far apart.\nThis implies that the Floquet states for $\\mu<1$ has overlap only with a few unperturbed\nbasis states and this effectively increases the value of participation ratio for $\\mu<1$. \nUltimately, this results in a more compact localization.\n\nFinally, all the results discussed in this paper can be summarised in the form of a \n`phase diagram' displayed in Fig. \\ref{sumfig}. For $\\mu <1$, singularity in the\npotential is effective and hence power law profile of the Floquet states is obtained.\nThis regime is indicated by red color in Fig. \\ref{sumfig}.\nHowever, if $\\mu > 1$, singularity is not effectively `felt' by the particle and\nhence exponential profile is obtained. This regime is indicated by black color \nin the figure. Depending on the choice of parameters, regimes\nin which transition occurs between these two Floquet state profiles are also\nobserved. In Fig. \\ref{sumfig}, this is indicated by white color.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics*[width=3.0in]{fig6.eps}\n\\caption{(Color online) Summary of the results presented in this work. In all the cases,\nthe potential $V_{sq}$ is singular. Power-law decay profile\nof the Floquet is obtained whenever $\\mu < 1$ (shown as deep red). Stronger red color\nrepresents power-law profile over larger energy scales. For $\\mu > 1$, exponential localization\nis obtained. Darker grey represent dominant exponential profile over longer energy scales. \nWhite color represents regimes of transition between these profiles. They cannot be classified\nas power-law or exponential with definiteness. Black broken horizontal line corresponds to $V_0=1$} \n\\label{sumfig}\n\\end{figure}\n\n\n\\section{Conclusion}\nIn summary, we have studied a non-KAM system represented by the Hamiltonian in Eq. \\ref{ham1}, \nnamely a periodically kicked particle in a finite potential well of height $V_0$, to primarily understand \nthe nature of its Floquet states. This Hamiltonian can be thought of as representing two limiting\ncases, (i) the standard KR for $V_0=0$ and (ii) KR in infinite potential\nwell for $V_0 \\to \\infty$. It is well known that, for sufficiently large kick strengths, all the \nFloquet states of the KR are localized with an exponential profile \\cite{kr}. Further, it has been\nsuggested that for kicked systems with singularity in their potential, the Floquet states display\npower-law profile \\cite{liu}. We examine the Floquet states of the Hamiltonian in Eq. \\ref{ham1} in the light\nof these results. To understand its Floquet states, we map this problem to that of a tight binding\nmodel. \n\nThe results presented in this work show that the decay profile of the Floquet states\nis not determined by the potential singularity alone, but by the representative energy band \n$\\epsilon_{if}$ of a set of Floquet states relative to the potential height $V_0$. \nThus, we show that if $\\epsilon_{if} > V_0$ then the effect of singularity is weak for those\nset of Floquet states and they display exponential profile. This represents the KR\nlimit of the problem. On the other hand, the condition $V_0 > \\epsilon_{if}$ represents\nFloquet states strongly affected by the singular potential. In this case, we have shown that \nFloquet states have predominantly power-law profile. In the region intermediate between these\ntwo extremes, the Floquet states typically display an initial exponential decay followed by\nan asymptotic power-law decay. The presence of these two contrasting Floquet state profiles\nin Hamiltonian in Eq. \\ref{ham1} leaves its signature in the spectral correlations as well.\nFor identically same set of parameters, depending on the reference energy scale $\\epsilon_{if}$,\nthe spacing distribution turns out to be Poisson distribution ($\\epsilon_{if} > V_0$) or a \ngeneral Brody distribution ($V_0 > \\epsilon_{if}$). Typically, the spacing distribution \nis taken to characterize quantum chaos in a system and it is generally independent of the energy\nband being considered provided it is in the semi-classical limit.\nQuite surprisingly, the semi-classical limit of the system in Eq. \\ref{ham1} lacks a unique \nspacing distribution as it depends on the energy band $\\epsilon_{if}$ being considered.\nKR was experimentally realized in a test-bed of cold atomic cloud in flashing \noptical lattices. Using more than one optical lattice, KR confined to a `potential well'\nhas also been realized. We believe that the results in this work is amenable to experiments\nin a suitable atom-optics set up.\n\n\n\\section{Acknowledgement}\nS. P. would like to acknowledge the University Grants Commission for research fellowship.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nConsider an affine algebraic curve \n$$\nC: F(x,y)=\\sum_{i+j \\leq d} a_{i,j} x^i y^j =0, a_{i,j} \\in {\\bf k},\n$$\ndefined over field ${\\bf k},$ $ {\\rm char}\\, {\\bf k} >0.$\nLet ${\\bf k}[C]$ and ${\\bf k}(C)$ be the algebras of polynomial and rational function of coefficients of the curve $C$.\nThose affine transformations of plane whose preserve the algebraic form of equation $F(x,y)$ generate a group $G$ which is subgroup of the group of plane affine transformations $A(2).$ A function $\\phi(a_{0,0},a_{1,0},\\ldots,a_{d,0}) \\in {\\bf k}(C) $ is called $G$-invariant if \n$\\phi(\\tilde{a}_{0,0},\\tilde{a}_{1,0},\\ldots,\\tilde{a}_{d,0})=\\phi(a_{0,0},a_{1,0},\\ldots,a_{d,0}),$ where $\\tilde{a}_{0,0},$ $\\tilde{a}_{1,0},\\ldots,$ $\\tilde{a}_{d,0}$ are defined from the condition \n$$F(gx,gy)=\\sum_{i+j \\leq d} a_{i,j} (gx)^i (gy)^j=\\sum_{i+j \\leq d} \\tilde{a}_{i,j} x^i y^j,$$ for all $g \\in G.$\nCurves $C$ and $C'$ are said to be $G$-isomorphic if they lies on the same $G$-orbit.\n\nThe algebras of all $G$-invariant polynomial and rational functions denotes by ${\\bf k}[C]^G$ and by ${\\bf k}(C)^G,$ respectively. One way to find elements of the algebra ${\\bf k}[C]^G$ is a specification of invariants of associated ternary form of order $d.$ In fact, consider a vector space $T_d$ generated by the ternary forms $\\sum\\limits_{i+j\\leq d}b_{i,j}x^{d-(i+j)} y^i z^{j},$ $b_{i,j} \\in {\\bf k}$ endowed with the natural action of the group $GL_3:=GL_3({\\bf k}).$ Given $f$ $GL_3$-invariant function of ${\\bf k}(T_d)^{GL_3},$ the specification $f:$ $b_{i,j} \\mapsto a_{i,j}$ or $b_{i,j} \\mapsto 0$ in the case when $a_{i,j} \\notin {\\bf k}(C),$ gives us an element of ${\\bf k}(C)^G.$\n\n\n But $SL_3$-invariants(thus and $GL_3$-invariants) of ternary form are known only for the cases $d \\leq 4,$ see \\cite{Bro}. Furthermore, analyzing of the Poincare series of the algebra of invariants of ternary forms, \\cite{B_X}, we see that the algebras are very complicated and no chance to find theirs mininimal generating sets.\n\nSince ${\\bf k}(T_d)^{GL_3}$ coincides with ${\\bf k}(T_d)^{\\mathfrak{gl}_3}$ it implies that the algebra of invariants is an intersection of kernels of some derivations of the algebra ${\\bf k}(T_d).$ Then in place of the specification of coefficients of the form we may use a \"specification\" of those derivations. \n\nFirst, consider the motivating example.\nLet \n$$\nC_3: y^2+a_0x^3+3a_1x^2+3a_2x+a_3=0, \n$$\nand let $G_0$ be a group generated by the translations $x \\mapsto \\alpha \\tilde{x}+b.$\n It is easy to show that $j$-invariant of the curve $C_3$ equals (\\cite{Sylv86}, p. 46): \n$$\nj(C_3)=6912\\,{\\frac { \\left( a_{{0}}a_{{2}}-{a_{{1}}}^{2} \\right) ^{3}}{{a_{\n{0}}}^{2} \\left( 4\\,{a_{{1}}}^{3}a_{{3}}-6\\,a_{{3}}a_{{0}}a_{{1}}a_{{2\n}}-3\\,{a_{{1}}}^{2}{a_{{2}}}^{2}+{a_{{3}}}^{2}{a_{{0}}}^{2}+4\\,a_{{0}}\n{a_{{2}}}^{3} \\right) }}.\n$$\nUp to constant factor the $j(C_3)$ equal to $\\dfrac{S^3}{T}$ where $S$ and $T$ are the specification of two $SL_3$-invariants of ternary cubic, see \\cite{Stur}, p.173.\n\nFrom another side a direct calculation yields that the following is true: $\\mathcal{D} \\left(j(C_3)\\right)=0$ and ${\\mathcal{H}} \\left(j(C_3)\\right)=0$ where $\\mathcal{D},$ ${\\mathcal{H}}$ denote the derivations of the algebra of rational functions ${\\bf k}(C_3)={\\bf k}(a_0,a_1,a_2,a_3):$\n $$\\mathcal{D}(a_i)=i a_{i-1},{ \\mathcal{H}}(a_i)=(3-i)a_{i}, i=0,1,2,3.$$\n \n From computing point of view, the calculation of $\\ker \\mathcal{D} \\cap \\ker \\mathcal{H} $ is more effective than the calculating of the algebra of invariants of the \n ternary cubic. We will derive further that \n$$ \n\\ker \\mathcal{D}_3 \\cap \\ker H_3={\\bf k}\\left( \\frac{\\left( a_{{0}}a_{{2}}-{a_{{1}}}^{2} \\right) ^{3}}{a_0^3}, \\frac{{a_{3}}\\,a_0^{2} + 2\\,{a_{1}}^{3} - 3\\,{a_{1}}\\,{a_{2}}\\,a_0}{a_0^2} \\right).\n$$\n\nIn the notes we prove that for arbitrary algebraic curve $C$ and its preserved form group $G$ there exist derivations $D_i, i\\leq 6$ of ${\\bf k}(C)$ such that ${\\bf k} (C)^G=\\bigcap\\limits_i \\ker D_i,$ (Theorem 3.2).\n\nIn section 1 we give full description of the algebras of polynomial and rational invariants for the curve $y^2=f(x)$. In section 2 we give exactly form for the derivations of the action of the Lie algebra $\\mathfrak{gl}_3$ on ${\\bf k} (C)$ and give a specification of such action for a curve of the form $y^2+g(x)y=f(x).$ \n\n\n\\section{Invariants of $y^2=f(x).$}\n\nIn some simple cases we may obtain the defining derivation directly.\n\n\nConsider the curve\n$$\nC_d: y^2=a_0x^d+da_1 x^{d-1}+\\cdots +a_d =\\sum_{i=0}^d a_d {d \\choose i} x^{d-i},\n$$\nand let $G$ be the group generated by the following transformations \n$$\nx=\\alpha \\tilde{x}+b, y=\\tilde{y}.\n$$\nThe algebra ${\\bf k}(C_d)^G$ consists of functions $\\phi(a_0,a_1,\\ldots,a_d)$ that have the invariance property\n$$\n\\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)=\\phi(a_0,a_1,\\ldots,a_d).\n$$\nHere $\\tilde{a}_i$ denote the coefficients of the curve $\\tilde{C}_d:$\n$$\n\\tilde{C}_d:\\sum_{i=0}^d a_d {d \\choose i} (\\alpha \\tilde{x}+b)^{d-i}=\\sum_{i=0}^d \\tilde{a}_d {d \\choose i} \\tilde{x}^{d-i}.\n$$\nThe coefficients $\\tilde{a}_i$ are given by the formulas\n\\begin{gather}\\label{1}\n\\tilde{a}_i=\\alpha^{n-i}\\sum_{k=0}^k {i \\choose k} b^k a_{n{-}k}.\n\\end{gather}\nThe following statement holds\n\\begin{te} We have \n$$\n{\\bf k}(C)^G=\\ker \\mathcal{D}_d \\cap \\ker \\mathcal{E}_d,\n$$\nwhere $\\mathcal{D}_d,$ $\\mathcal{E}_d$ denote the following derivations of the algebra ${\\bf k}(C):$\n$$\n\\mathcal{D}_d(a_i)=i a_{i-1}, \\mathcal{E}_d(a_i)=(d-i) a_i.\n$$\n\\end{te}\n\nRecall that a linear map $D:{\\bf k}(C) \\to {\\bf k}(C)$ is called a derivation of the algebra ${\\bf k}(C)$ if $D(f g)=D(f)g+f D(g),$ for all $f,g \\in {\\bf k}(C).$ The subalgebra $\\ker D := \\{ f \\in {\\bf k}(C) \\mid D(f)=0 \\}$ is called the kernel of the derivation $D.$ The above derivation $\\mathcal{D}_d$ is called the basic Weitzenb\\\"ock derivation.\n\\begin{proof}\nActing in classical manner, we differentiate with respect to $b$ both sides of the equality\n$$\\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)= \\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d),$$\nand obtain in this way \n$$\n\\frac{\\partial \\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_0} \\dfrac{\\partial \\tilde{a}_0}{\\partial b}+\\frac{\\partial \\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_1}\\dfrac{\\partial \\tilde{a}_1}{\\partial b}+\\cdots+\\frac{\\partial \\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_d}\\dfrac{\\partial \\tilde{a}_d}{\\partial b}=0.\n$$\nSubstitute $\\alpha=1,$ $b=0$ to $\\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)$ and taking into account that $\\dfrac{\\partial \\tilde{a}_i}{\\partial b}\\Bigl |_{b=0}=i a_{i-1},$ we get:\n$$\n\\tilde{a}_0 \\frac{\\partial \\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_1} +2\\tilde{a}_1 \\frac{\\partial \\phi(\\tilde{a}_0,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_2}+\\cdots d \\tilde{a}_{d-1} \\frac{\\partial \\phi(\\tilde{a}_0,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_d} =0\n$$\nSince the function $\\phi(\\tilde{a}_0,\\ldots,\\tilde{a}_d)$ depends on the variables $\\tilde{a}_i$ in the exact same way as the function $\\phi(a_0,a_1,\\ldots,a_d)$ depends on the $a_i$ then it implies that $\\phi(a_0,a_1,\\ldots,a_d)$ satisfies the differential equation\n\\begin{gather*}\\label{D1}\na_0 \\frac{\\partial \\phi({a}_0,a_1\\ldots,{a}_d)}{\\partial {a}_1} +2a_1 \\frac{\\partial \\phi({a}_0,a_1\\ldots,{a}_d)}{\\partial {a}_2}+d a_{d-1} \\frac{\\partial \\phi({a}_0,a_1\\ldots,{a}_d)}{\\partial {a}_d} =0\n\\end{gather*}\nThus, $\\mathcal{D}_d(\\phi)=0.$\nNow we differentiate with respect to $\\alpha$ both sides of the same equality $$\\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)= \\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d).$$\n$$\n\\frac{\\partial \\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_0} \\dfrac{\\partial \\tilde{a}_0}{\\partial \\alpha}+\\frac{\\partial \\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_1}\\dfrac{\\partial \\tilde{a}_1}{\\partial \\alpha}+\\cdots+\\frac{\\partial \\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_d}\\dfrac{\\partial \\tilde{a}_d}{\\partial \\alpha}=0.\n$$\nSubstitute $\\alpha=1,b=0,$ to $\\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)$ and taking into account $\\dfrac{\\partial \\tilde{a}_i}{\\partial \\alpha}\\Bigl |_{ \\alpha=1 \\,\nb=0}=(d-i) a_{i},$ we get:\n$$\n\\tilde{a}_0 \\frac{\\partial \\phi(\\tilde{a}_0,\\tilde{a}_1,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_0} +(d-1)\\tilde{a}_1 \\frac{\\partial \\phi(\\tilde{a}_0,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_1}+\\cdots + \\tilde{a}_{d-1} \\frac{\\partial \\phi(\\tilde{a}_0,\\ldots,\\tilde{a}_d)}{\\partial \\tilde{a}_{d-1}} =0\n$$\nIt implies that \n $\\mathcal{E}_d(\\phi({a}_0,a_1\\ldots,{a}_d))=0.$\n\\end{proof}\n\nThe derivation $\\mathcal{E}_d$ sends the monom $a_0^{m_0} a_1^{m_1}\\cdots a_d^{m_d}$ to the term $$(m_0 d+m_1 (d-1)+\\cdots m_{d-1}) a_0^{m_0} a_1^{m_1}\\cdots a_d^{m_d}.$$ Let the number $\\omega \\left( a_0^{m_0} a_1^{m_1}\\cdots a_d^{m_d} \\right):= m_0 d+m_1 (d-1)+\\cdots m_{d-1}$ be called the weight of the monom $a_0^{m_0} a_1^{m_1}\\cdots a_d^{m_d}.$ In particular $\\omega(a_i)=d-i.$ \n\n A homogeneous polynomial $f \\in {\\bf k}[C]$ be called isobaric if all their monomial have equal weights. A weight $\\omega(f)$ of an isobaric polynomial $f$ is called a weight of its monomials. Since $\\omega(f)>0,$ then ${\\bf k}[C]^{\\mathcal{E}_d}=0.$ It implies that ${\\bf k}[C]^{G}=0.$ \n \n If $f,g$ are two isobaric polynomials then \n$$\n\\mathcal{E}_d\\left(\\frac{f}{g}\\right)=(\\omega(f)-\\omega(g))\\frac{f}{g}.\n$$\nTherefore the algebra $k(C)^{\\mathcal{E}_d}$ is generated by rational fractions which both denominator and numerator has equal weight.\n\n\nThe kernel of the derivation $\\mathcal{D}_d$ also is well-known, see \\cite{Now}, \\cite{Aut}, and \n$$\n\\ker \\mathcal{D}_d={\\bf k}(a_0,z_2,\\ldots,z_d),\n$$\nwhere \n$$\nz_i:= \\sum_{k=0}^{i-2} (-1)^k {i \\choose k} a_{i-k} a_1^k a_0^{i-k-1} +(i-1)(-1)^{i+1} a_1^i, i=2,\\ldots,d.\n$$\nIn particular, for $d=5$, we get\n$$\n\\begin{array}{l}\nz_2={a_{2}}\\,a_0 - {a_{1}}^{2}\n\\\\\nz_3={a_{3}}\\,a_0^{2} + 2\\,{a_{1}}^{3} - 3\\,{a_{1}}\\,{a_{2}}\\,a_0\n\\\\\nz_4={a_{4}}\\,a_0^{3} - 3\\,{a_{1}}^{4} + 6\\,{a_{1}}^{2}\\,{a_{2}}\\,a_0 - 4\n\\,{a_{1}}\\,{a_{3}}\\,a_0^{2}\n\\\\\nz_5={a_{5}}\\,a_0^{4} + 4\\,{a_{1}}^{5} - 10\\,{a_{1}}^{3}\\,{a_{2}}\\,a_0 + \n10\\,{a_{1}}^{2}\\,{a_{3}}\\,a_0^{2} - 5\\,{a_{1}}\\,{a_{4}}\\,a_0^{3}\n.\n\\end{array}\n$$\nIt is easy to see that $\\omega(z_i)=i (n-1).$ The following element $\\dfrac{z_i^d}{a_0^{i(d-1)}}$ has the zero weight for any $i.$ Therefore the statement holds:\n\\begin{te}\n$$\n{\\bf k}(C_d)^G={\\bf k}\\left( \\frac{z_2^d}{a_0^{2(d-1)}},\\frac{z_3^d}{a_0^{3(d-1)}}, \\cdots, \\frac{z_d^d}{a_0^{d(d-1)}}\\right).\n$$\n\\end{te}\nFor the curve \n$$\nC_d^0: y^2=x^d+da_1 x^{d-1}+\\cdots +a_d =x^d+\\sum_{i=1}^d a_d {d \\choose i} x^{d-i}.\n$$\nand for the group $G_0$ generated by translations $x=\\tilde{x}+b,$ the algebra of invariants becomes simpler:\n$${\\bf k}\\left({C^0}_d\\right)^{G_0}={\\bf k}(z_2,z_3,\\ldots,z_d).$$\n \n\\begin{te}\n$(i)$ For arbitrary set of $d-1$ numbers $j_2,$$j_3,\\ldots,j_d$ there exists a curve $C$ such that \n$z_i(C)=j_i.$ \n\n$(ii)$ For two curves $C$ and $C'$ the equalities $z_i(C)=z_i(C')$ hold for $ 2 \\leq i \\leq d,$ if and only if these curves are $G_0$-isomorphic.\n\\end{te}\n\\begin{proof}\n$(i).$ Consider the system of equations \n$$\n\\left\\{\n\\begin{array}{l}\n{a_{2}} - {a_{1}}^{2}=j_2\n\\\\\n{a_{3}} + 2\\,{a_{1}}^{3} - 3\\,{a_{1}}\\,{a_{2}}=j_3\n\\\\\n{a_{4}} - 3\\,{a_{1}}^{4} + 6\\,{a_{1}}^{2}\\,{a_{2}} - 4\n\\,{a_{1}}\\,{a_{3}}=j_4\n\\\\\n\\ldots \\\\\n\\displaystyle a_d+\\sum_{k=1}^{d-2} (-1)^k {d \\choose k} a_{d-k} a_1^k +(d-1)(-1)^{d+1} a_1^d=j_d\n\\end{array}\n\\right.\n$$\nBy solving it we obtain \n\\begin{gather}\\label{2}\na_n=j_n+\\sum_{i=2}^n {n \\choose i} a_1^k j_{n-k}\n\\end{gather}\nPut $a_1=0$ we get $a_n=j_n$,\\, i.e. the curve \n$$\nC: y^2=x^d+{d \\choose 2} j_2 x^{d-2}+\\cdots +j_d,\n$$\nis desired one.\n\n$(ii).$ We may assume, without loss of generality, that the curve $C$ has the form\n$$\nC: y^2=x^d+{d \\choose 2} j_2 x^{d-2}+\\cdots +j_d.\n$$\nSuppose that for a curve \n$$\nC': y^2=x^d+da_1 x^{d-1}+\\cdots +a_d =x^d+\\sum_{i=1}^d a_d {d \\choose i} x^{d-i}.\n$$\nholds $z_i(C')=z_i(C)=j_i.$\nComparing (\\ref{2}) with (\\ref{1}) we deduce that the curve $C'$ is obtained from the curve $C$ by the translation $x+a_1.$ \n\\end{proof}\n\n\\section{General case and invariants of $y^2+g(x)y=f(x)$}\n\nConsider the vector ${\\bf k}$-space $T_d$ of ternary form of degree $d:$\n$$ \nu(x,y,z)=\\sum_{i+j\\leq d} \\, \\frac{d!}{i! j! (d{-}(i+j))!}a_{i, j}\\, x^{d-(i+j)} y^i z^j,\n$$\nwhere $a_{i, j} \\in {\\bf k}.$\nLet us identify in the natural way the algebra of rational function ${\\bf k}(T_d)$ on the vector space $T_d$ with the algebra of polynomials of the $\\displaystyle \\frac{1}{2} (d+1)(d+2)$ variables. The natural action of the group $GL_3$ on $T_d$ induced the action of $GL_3$ (and the Lie algebra $\\mathfrak{gl_{3}}$) on ${\\bf k}[T_d]$. \nThe corresponding algebra of invariants ${\\bf k}(T_d)^{GL_3}={\\bf k}(T_d)^{\\mathfrak{gl_{3}}}$ is called the algebra of $GL_3$-invariants (or absolute invariants) of ternary form of degree $d.$ The following statement holds:\n\\begin{te}\n$$ {\\bf k}(T_d)^{GL_3}=\\ker D_1 \\cap \\ker D_2 \\cap \\ker \\hat{D}_1 \\cap \\ker \\hat{D}_2 \\cap \\ker E_1 \\cap \\ker E_2 \\cap \\ker E_3$$\nwhere\n$$\n\\begin{array}{ll}\n\\displaystyle \\! \\! D_1(a_{i,j})=i\\,a_{i{-}1,j}, & \\! \\! D_2(a_{i,j})=j\\,a_{i{+}1,j{-}1},\\\\\n\\displaystyle \\! \\! \\hat D_1(a_{i,j})=(n-(i+j))\\,a_{i{+}1,j}, & \\! \\! \\hat D_2(a_{i,j})=i\\,a_{i{-}1,j{+}1},\\\\\n \\hat{D}_3(a_{i,j})=(n-(i+j)) a_{i,j+1}, & \\! \\! D_3(a_{i,j})=j a_{i,j-1}, \\\\\n\\! \\! E_1(a_{i,j})=(n-(2i+j)) a_{i,j}, & \\! \\! E_2(a_{i,j})=i a_{i,j},\n\\end{array}\n$$\n$$E_3(a_{i,j})=j a_{i,j}. $$\n\\end{te}\n\\begin{proof}\nThe Lie algebra $\\mathfrak{gl}_3$ acts on the vector space of ternary form $T_d$ by derivations,namely \n$$\n\\begin{array}{ll}\n\\displaystyle D_1=-y \\frac{\\partial}{\\partial x}, & \\displaystyle D_2=-z \\frac{\\partial}{\\partial y}, \\\\\nE_1=-x\\frac{\\partial}{\\partial x}, &E_2= -y\\frac{\\partial}{\\partial y}, \\\\\n\\displaystyle \\hat D_1=-x \\frac{\\partial}{\\partial y}, & \\displaystyle \\hat D_2=-y \\frac{\\partial}{\\partial z}, \\\\\n \\displaystyle D_3=-z \\frac{\\partial}{\\partial x}, & E_3= -\\displaystyle z \\frac{\\partial}{\\partial z},\n\\end{array}\n$$\n$$\n\\begin{array}{lll}\n \\displaystyle & \\displaystyle \\hat D_3=-x \\frac{\\partial}{\\partial z}. &\n\\end{array}\n$$\nTo extend the actions $\\mathfrak{gl}_3$ to the algebra ${\\bf k}(T_d)$ we use the well-known fact of classical invariant theory that the generic form $u(x,y,z)$ is a covariant. It means that any of above derivation (considered as derivation of ${\\bf k}[T_d,x,y,z]$) must kill the form. \nIn particular, for the derivation $D_1$ we have\n\\begin{gather*}\nD_1(u(x,y,z))=\n\\sum_{i+j\\leq n} \\, \\frac{n!}{i! j! (n{-}(i+j))!}(D_1(a_{i,j}) x^{n-(i+j)} y^i z^j+\n+a_{i,j} D_1(x^{n-(i+j)} y^i z^j))=\\\\\n=D_1(a_{0,1}) x^{n-1} x_3+\\cdots+D_1(a_{0,n}) \\frac{1}{n!} z^n+ \n+\\sum_{\\begin{array}{c} \\mbox{\\small \\it i+j}\\leq n \\\\ i>0 \\end{array} } \\Bigl( D_1(a_{i,j})-i\\,a_{i{-}1,j}\\Bigr) x^{d-(i+j)} y^i z^j.\n\\end{gather*}\nIt following that the equality $D_1(u(x,y,z))=0$ is possible only if all coefficients are equal to zero. Therefore we get $D_1(a_{0,j})=0$ for all $0\\leq j \\leq n,$ and $D_1(a_{i,j})=i\\,a_{i{-}1,j} $ as required. In an exactly similar way we will obtain actions on ${\\bf k}(T_d)$ for the rest derivations. \n\\end{proof}\n\n\\noindent\n{\\bf Corollary.} ${\\rm tr \\,deg}_k\\,\\,{\\bf k}(T_{d})^{GL_3} \\leq \\displaystyle \\frac{1}{2} (d+1)(d+2)-7.$\n\\begin{proof}\nIt follows from the fact that for a nonzero derivation $D$ of polynomial ring $R$ of $n$ variables ${\\rm tr \\,deg} \\ker D \\leq n-1,$ see \\cite{Now}, Proposition 7.1.1.\n\\end{proof}\n\nAn obvious consequence of the theorem is the following: \n\n\\begin{te} For affine algebraic curves $d$\n\n$$\nC: \\sum_{i+j=d} \\frac{d!}{i! j! (d-(i+j))!} a_{i,j} x^{d-(i+j)}y^i=0, a_{d,0}=0, a_{i,j} \\in {\\bf k}, {\\rm char} {\\bf k} >0,\n$$\nthe following holds:\n\n(i) if $a_{d,0} \\neq 0$ and $a_{0,0} \\neq 0$ then\n$$\n{\\bf k}(C)^G=\\ker D_1 \\cap \\ker D_2 \\cap \\ker \\hat{D}_1 \\cap \\ker E_1 \\cap \\ker E_2 , \n$$\n\n(ii) if $a_{d,0} =0 0$ $a_{0,0} \\neq 0$ then\n\n$$\n{\\bf k}(C)^G=\\ker D_2 \\cap \\ker D_3 \\cap \\ker \\hat{D}_1 \\cap \\ker E_1 \\cap \\ker E_2 , \n$$\nwhere\n\\begin{align*}\nD_1(a_{i,j})&=i a_{i-1,j},D_2(a_{i,j})=j a_{i+1,j-1}, \\hat{D}_1(a_{i,j})=(d-(i+j))a_{i+1,j},\\\\\n&\\\\\nE_1(a_{i,j})&=(d-(i+j)) a_{i,j}, E_2(a_{i,j})=i a_{i,j},D_3(a_{i,j})=j a_{i,j-1}.\n\\end{align*}\n\\end{te}\n\\begin{proof}\n$(i)$ Consider the associate projective plane curve in $\\mathbb{P}^2:$\n$$\n\\sum_{i+j=d} \\frac{d!}{i! j! (d-(i+j))!} a_{i,j} X^{d-(i+j)}Y^iZ^j=0, a_{d,0} \\neq 0, a_{0,0} \\neq 0. \n$$\nThe transformations $X \\mapsto \\alpha X+ \\beta Y+ b Z,$ $Y \\mapsto \\gamma X+ \\delta Y+a Z,$ $Z \\mapsto Z$ generate of a subgroup of $GL_3$ which preserve the algebraic form of the equation of the curve. Therefore the algebra of invariants of the curve (and corresponding affine curve) coincides with the intersection of the kernels of the five derivations $ D_1, D_2, \\hat{D}_1, E_1, E_2,$ ($[D_1,D_2]=D_3$).\n\n$(ii).$ For this case the transformations are as follows: $X \\mapsto \\alpha X+ b Z,$ $Y \\mapsto \\gamma X+ \\delta Y+a Z,$ $Z \\mapsto Z $ and we have to exclude the derivation $D_1.$\n\\end{proof}\n\nFor the curve \n\\begin{gather*}\n\\mathcal{C'}_{d}:\\frac{d(d-1)}{2} a_{2,d-2} y^2+\\sum_{i=0}^{d-1}\\frac{ d!}{i!(d-(1+i))!}a_{1,i}x^{d-(i+1)}y+\\sum _{i=0}^{d}{\\frac {d!\\,}{i!\\, \\left( d-i\n \\right) !}a_{{0,i}}{x}^{d-i}}=0,\n\\end{gather*}\nand for the group $G$ generated by $x \\mapsto \\alpha x+a, y \\mapsto \\beta y+b$ we have \n$$\n{\\bf k}(\\mathcal{C'}_{d})^G=\\ker D_2 \\cap \\ker D_3 \\cap \\ker \\hat{D}_1 \\cap \\ker E_1 \\cap \\ker E_2 , \n$$\nand ${\\rm tr \\,deg}_k\\,\\,{\\bf k}(\\mathcal{C'}_{d})^G \\leq 2d-3.$\n\n{\\bf Example.} Let us calculate the invariants of curve $C'_5$\n\\begin{gather*}\n10\\,{y}^{2}+(5\\,a_{{1,0}}{x}^{4}+20\\,a_{{1,1}}{x}^{3}+30\\,a_{{1,2}}{x\n}^{2}+20\\,a_{{1,3}}x+5\\,a_{{1,4}})y=\\\\\n={x}^{5}+5\\,a_{{0,1}}{x}^{4}+10\\,a_{{0,2}}{x}^{3}+10\\,a_{{0,3}}{x}^{2}+5\\,a_{{0,4}}x+a_{{0,5}},\n\\end{gather*}\nwith respect to the group $G_0$ generated by the translations $x=\\tilde{x}+a, y=\\tilde{y}+b.$ Theorem 3.2 implies that $C_5^G=\\ker D_3 \\cap \\ker D_2,$ where the derivations $D_2, D_3$ act by\n\\begin{align*}\n &D_{{2}} \\left( a_{{1,1}} \\right) =0,D_{{2}} \\left( a_{{1,0}} \\right) =0,D_{{2}} \\left( a_{{0,1}} \\right) =-a_{{1,0}},D_{{2}}\n \\left( a_{{1,2}} \\right) =0,D_{{2}} \\left( a_{{0,2}} \\right) =-2\\,a_{\n{1,1}},\\\\&D_{{2}} \\left( a_{{0,3}} \\right) =-3\\,a_{{1,2}},D_{{2}} \\left( \na_{{0,5}} \\right) =-5\\,a_{{1,4}},D_{{1}} \\left( a_{{0,4}} \\right) =-4\n\\,a_{{1,3}},D_{{2}} \\left( a_{{1,3}} \\right) =0,D_{{2}} \\left( a_{{1,4\n}} \\right) =4.\n\\end{align*}\nand \n\\begin{align*}\n&D_{{3}} \\left( a_{{1,2}} \\right) =2\\,a_{{1,1}},D_{{3}} \\left( a_{{1,1}} \\right) =a_{{1,0}},D_{{2}} \\left( a_{{1,0}} \\right) \n=0,D_{{2}} \\left( a_{{0,2}} \\right) =2\\,a_{{0,1}},D_{{3}} \\left( a_{{0\n,1}} \\right) =1,\\\\\n&D_{{2}} \\left( a_{{0,3}} \\right) =3\\,a_{{0,2}},D_{{3}}\n \\left( a_{{0,5}} \\right) =5\\,a_{{0,4}},D_{{2}} \\left( a_{{0,4}}\n \\right) =4\\,a_{{0,3}},D_{{2}} \\left( a_{{1,3}} \\right) =3\\,a_{{1,2}},\nD_{{3}} \\left( a_{{1,4}} \\right) =4\\,a_{{1,3}}.\n\\end{align*}\nBy using the Maple command {\\tt pdsolve()} we obtain that \n$$\n{\\bf k}(C'_5)^{G_0}={\\bf k}(g_1,g_2,g_3,g_4,g_5,g_6,g_7),{\\bf k}[C'_5]^{G_0}={\\bf k}[g_1,g_2,g_3,g_4,g_5,g_6,g_7],\n$$\nwhere\n\\begin{gather*}\ng_1=a_{1,0},\\\\\ng_2={a_{{1,0}}}^{2}a_{{0,2}}+{a_{{1,1}}}^{2}-2\\,a_{{1,1}}a_{{1,0}}a_{{0,1}},\\\\\ng_3=a_{{1,2}}-2\\,a_{{1,1}}a_{{0,1}}+a_{{1,0}}a_{{0,2}},\\\\\ng_4=6\\,{a_{{1,1}}}^{2}a_{{0,1}}a_{{1,0}}-4\\,{a_{{1,1}}}^{3}-3\\,{a_{{1,0}}}\n^{2}a_{{0,2}}a_{{1,1}}-3\\,a_{{1,2}}{a_{{1,0}}}^{2}a_{{0,1}}+3\\,a_{{1,0\n}}a_{{1,1}}a_{{1,2}}+a_{{0,3}}{a_{{1,0}}}^{3},\\\\\ng_5=2\\,{a_{{1,1}}}^{3}-3\\,a_{{1,0}}a_{{1,1}}a_{{1,2}}+a_{{1,3}}{a_{{1,0}}}\n^{2},\\\\\ng_6=3\\,{a_{{1,0}}}^{4}{a_{{0,2}}}^{2}+{a_{{1,0}}}^{4}a_{{0,4}}-12\\,{a_{{1,0\n}}}^{3}a_{{1,1}}a_{{0,1}}a_{{0,2}}-4\\,{a_{{1,0}}}^{3}a_{{1,1}}a_{{0,3}\n}-4\\,{a_{{1,0}}}^{3}a_{{1,3}}a_{{0,1}}-\\\\-12\\,{a_{{1,0}}}^{3}a_{{0,2}}a_{\n{1,2}}+12\\,{a_{{1,1}}}^{2}{a_{{1,0}}}^{2}{a_{{0,1}}}^{2}+24\\,{a_{{1,1}\n}}^{2}{a_{{1,0}}}^{2}a_{{0,2}}+4\\,a_{{1,1}}{a_{{1,0}}}^{2}a_{{1,3}}+36\n\\,a_{{1,1}}{a_{{1,0}}}^{2}a_{{0,1}}a_{{1,2}}-\\\\-24\\,a_{{1,0}}a_{{1,2}}{a_\n{{1,1}}}^{2}-48\\,a_{{1,0}}{a_{{1,1}}}^{3}a_{{0,1}}+24\\,{a_{{1,1}}}^{4},\\\\\ng_7={a_{{1,0}}}^{4}{a_{{0,2}}}^{2}+{a_{{1,0}}}^{3}a_{{1,4}}-4\\,{a_{{1,0}}}\n^{3}a_{{1,1}}a_{{0,1}}a_{{0,2}}+6\\,{a_{{1,0}}}^{3}a_{{0,2}}a_{{1,2}}+4\n\\,{a_{{1,1}}}^{2}{a_{{1,0}}}^{2}{a_{{0,1}}}^{2}-\\\\-4\\,{a_{{1,1}}}^{2}{a_{\n{1,0}}}^{2}a_{{0,2}}-12\\,a_{{1,1}}{a_{{1,0}}}^{2}a_{{0,1}}a_{{1,2}}-4\n\\,a_{{1,1}}{a_{{1,0}}}^{2}a_{{1,3}}+4\\,{a_{{1,0}}}^{2}a_{{0,1}}-4\\,a_{\n{1,1}}a_{{1,0}}+\\\\+12\\,a_{{1,0}}a_{{1,2}}{a_{{1,1}}}^{2}+8\\,a_{{1,0}}{a_{\n{1,1}}}^{3}a_{{0,1}}-8\\,{a_{{1,1}}}^{4}\n\\end{gather*}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzejwa b/data_all_eng_slimpj/shuffled/split2/finalzzejwa new file mode 100644 index 0000000000000000000000000000000000000000..3f8f530c7e3eed07ac38ac4769a25846c5ef6ed2 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzejwa @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\nPhysics of neutrinos has entered into a new stage \nafter establishment of the mass-induced neutrino oscillation due to \nthe atmospheric~\\cite{atm_evidence}, the accelerator \n~\\cite{K2K_evidence, MINOS}, and \nthe reactor neutrino~\\cite{KL_evidence} experiments, \nconfirming the earlier discovery~\\cite{SKatm,solar,KamLAND} \nand identifying the nature of the phenomenon. \nIn the new era, the experimental endeavors will be focused on search for \nthe unknowns in neutrino masses and the lepton flavor mixing, \n$\\theta_{13}$, the neutrino mass hierarchy, and CP violation. \nOn the theory side, various approaches toward understanding physics of \nlepton mixing and the quark-lepton relations are extensively pursuit \n\\cite{moha-smi}, which then further motivate \nprecision measurement of the lepton mixing parameters. \nWe will use the standard notation~\\cite{PDG} of the lepton mixing matrix, \nthe Maki-Nakagawa-Sakata (MNS) matrix~\\cite{MNS}, \nthroughout this paper.\n\n\nIt was recognized sometime ago that there exists problem of \nparameter degeneracy \nwhich would act as an obstacle against precision measurement of \nthe lepton mixing parameters. \nThe nature of the degeneracy can be understood as \nthe intrinsic degeneracy~\\cite{intrinsic}, \nwhich is duplicated by the unknown sign of atmospheric \n$\\Delta m^2$~\\cite{MNjhep01} (hereafter, ``sign-$\\Delta m^2$ degeneracy'' for \nsimplicity) \nand by the possible octant ambiguity of $\\theta_{23}$~\\cite{octant} \nthat exists if $\\theta_{23}$ is not maximal. \nFor an overview of the resultant eight-fold degeneracy, see e.g., \n\\cite{BMW,MNP2}. \n\n\nIn a previous paper~\\cite{T2KK}, we have shown that \nthe identical two detector setting in Kamioka and in Korea with \neach fiducial mass of 0.27 Mton water, which \nreceives the identical neutrino beam from the J-PARC facility \ncan be sensitive to the neutrino mass hierarchy and CP violation \nin a wide range of the lepton mixing parameters, $\\theta_{13}$ \nand the CP phase $\\delta$.\nIt is the purpose of this paper to point out that the same setting \nhas capability of resolving the $\\theta_{23}$ octant degeneracy \nto a value of $\\theta_{23}$ which is rather close to the maximal, \n$\\sin^2 2 \\theta_{23} < 0.97 (0.94)$ at 2 (3) standard deviation \nconfidence level (CL). \nIt is achieved by detecting solar-$\\Delta m^2$ scale oscillation effect in \nthe Korean detector. \nTogether with the sensitivities to resolution of the degeneracy \nrelated to the mass hierarchy and the CP phase discussed in \nthe previous paper, we demonstrate that \nthe Kamioka-Korea two detector setting is capable of solving \nthe total eight-fold parameter degeneracy.\nWe stress that resolving the degeneracy is crucial to precision \nmeasurement of the lepton mixing parameters on which we make \nfurther comments at appropriate points in the subsequent discussions. \nWe also emphasize that it is highly nontrivial that one can formulate \nsuch a global strategy for resolving all the known degeneracies \n(though in a limited range of the mixing parameters) \nonly with the experimental apparatus using conventional muon neutrino \nsuperbeam.\\footnote{\nIt may be contrasted to the method for resolving the degeneracy \nbased on neutrino factory examined in~\\cite{donini}; \nIt uses a 40 kton magnetized iron calorimeter and a 4 kton emulsion chamber, \nand conventional $\\nu_{\\mu}$ beam watched by \na 400 kton water Cherenkov detector. \n}\n\n\nIn some of the previous analyses including ours~\\cite{T2KK,resolve23}, \npeople often tried to resolve the degeneracy of a particular type \nwithout knowing (or addressing) the solutions of the other types of \ndegeneracies. \nBut, then, the question of consistency of the procedure immediately arises; \nCan one solve the degeneracy of type A without knowing the solutions \nof the other degeneracies B and C? \nDoes the obtained solution remain unchanged when the assumed \nsolutions for the other type of degeneracies are changed to the \nalternative ones?, etc. \nWe answer to these questions in the positive in experimental settings \nwhere the earth matter effect can be treated as perturbation. \nWe do so by showing that the resolution of the degeneracy of a \nparticular type decouples from the remaining degeneracies, \nthe property called the ``decoupling between the degeneracies'' \nin this paper. \n\n\nIn Sec.~\\ref{how}, we present a pedagogical discussion of how the \neight-fold degeneracy can be lifted by measurement with \nthe Kamioka-Korea two detector setting. \nIn Sec.~\\ref{decoupling}, we prove the ``decoupling'' and make \na brief comment on its significance.\nIn Sec.~\\ref{23degeneracy}, we discuss some characteristic features \nof the $\\nu_{e}$ and $\\bar{\\nu}_{e}$ appearance probabilities \nthat allow the Kamioka-Korea identical two detector setting to \nresolve the $\\theta_{23}$ octant degeneracy. \nIn Sec.~\\ref{sensitivity}, the actual analysis procedure \nand the obtained sensitivities for solving the $\\theta_{23}$ degeneracy \nare described in detail. \nIn Sec.~\\ref{revisit}, we reexamine the sensitivities to \nthe mass hierarchy and CP violating phase by using our new code \nwith disappearance channels and additional systematic errors.\nIn Sec.~\\ref{summary}, we give a summary and discussions. \n\n\n\\section{How the identical two detector system solves the eight-fold degeneracy?}\n\\label{how}\n\n\nWe describe in this section how the eight-fold parameter degeneracy\ncan be resolved by using two identical detectors, one placed at a medium \nbaseline distance of a few times 100 km, and the other at $\\sim$1000 km or so. \nWe denote them as the intermediate and the far detectors, respectively, \nin this paper. \nWhenever necessary we refer the particular setting of Kamioka-Korea \ntwo detector system, but most of the discussions in this and the next \nsections are valid without the specific setting. \n\n\nTo give the readers a level-one understanding we quote here, \nignoring complications, \nwhich effect is most important for solving which degeneracy: \n\n\\begin{itemize}\n\n\\item\n\nThe intrinsic degeneracy; \nSpectrum information solves the intrinsic degeneracy.\n\n\\item\n\nThe sign-$\\Delta m^2$ degeneracy; \nDifference in the earth matter effect between the intermediate and the far detectors \nsolves the sign-$\\Delta m^2$ degeneracy. \n\n\\item\n\nThe $\\theta_{23}$ octant degeneracy; \nDifference in solar $\\Delta m^2$ oscillation effect \n(which is proportional to $c^2_{23}$) between the intermediate \nand the far detectors solves the $\\theta_{23}$ octant degeneracy. \n\n\\end{itemize}\n\n\nTo show how the eight-fold parameter degeneracy can be resolved, \nwe present in Fig~\\ref{intrinsicKamKorea} a comparison between \nthe sensitivities achieved by the Kamioka only setting and the \nKamioka-Korea setting by taking \na particular set of true values of the mixing parameters which are \nquoted in caption of Fig~\\ref{intrinsicKamKorea}. \nThe left four panels of Fig~\\ref{intrinsicKamKorea} show the \nexpected allowed regions of oscillation parameters \nin the Tokai-to-Kamioka phase-II (T2K II) setting, \nwhile the right four panels show the allowed regions by the \nTokai-to-Kamioka-Korea setting.\nFor both settings we assume \n4 years of neutrino plus 4 years of anti-neutrino \nrunning\\footnote{\nIt was shown in the previous study~\\cite{T2KK} that the sensitivity \nobtained with 2 years of neutrino and 6 years of anti-neutrino \nrunning in the T2K II setting~\\cite{T2K} is very similar to that of \n4 years of neutrino and 4 years of anti-neutrino running. \n}\nand the total fiducial volume is kept to be the same, 0.54 Mton.\nSome more information of the experimental setting and the \ndetails of the analysis procedure are described in the caption of \nFig~\\ref{intrinsicKamKorea} and in Sec.~\\ref{sensitivity}. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{sens.kam.1.eps}\n\\includegraphics[width=0.49\\textwidth]{sens.kam_kor.1.eps}\n\\end{center}\n\\caption{The region allowed in $\\delta-\\sin^2 2\\theta_{13}$ and \n$\\sin^2 \\theta_{23}-\\sin^2 2\\theta_{13}$ spaces \nby T2K II (left four panels) and by the \nKamioka-Korea two detector setting (right four panels) in both of which \n4 years of neutrino plus 4 years of anti-neutrino running are assumed. \nThe upper (lower) four panels show the allowed region for\nthe positive (negative) sign of $\\Delta m^2_{31}$. \nThe detector fiducial volumes of T2K II and Kamioka-Korea settings \nare assumed to be 0.54 Mton and each 0.27 Mton, respectively, \nand the beam power of J-PARC is assumed to be 4 MW. \nThe baseline to the Kamioka and Korea detectors are, \n295 km and 1050 km, respectively. \nThe true solution is assumed to be located at \n$\\sin^2{2\\theta_{13}}$=0.01, $\\sin ^2 \\theta_{23}$=0.60\nand $\\delta$=$\\pi\/4$ \nwith positive sign of $\\Delta m_{31}^2 (=+2.5\\times 10^{-3}$ eV$^2$), which is \nindicated by the green star. \nThe solar mixing parameters are fixed as\n$\\Delta m_{21}^2 = 8\\times 10^{-5}$ eV$^2$ and $\\sin^2{\\theta_{12}}$=0.31.\nThree contours in each figure correspond to\nthe 68\\% (blue line), 90\\% (black line) and 99\\% \n(red line) C.L. sensitivities, which are defined\nas the difference of the $\\chi^2$ being 2.30, 4.61 and 9.21, respectively.\n}\n\\label{intrinsicKamKorea}\n\\end{figure}\n\n\nLet us first focus on the left four panels of Fig~\\ref{intrinsicKamKorea}. \nIn the left-most two panels labeled as (aN) and (aI), one observes \nsome left-over degeneracies of the total eight-fold degeneracy; \nIf we plot the result of a rate only \nanalysis without spectrum information we would have seen \n8 separate (or overlapped) allowed parameter regions. \nThe $\\theta_{23}$ octant degeneracy remains unresolved \nas seen in panels (bN) and (bI). \nNote that the overlapping two regions in (aN) and (aI) are \nnothing but the consequence of unresolved $\\theta_{23}$ degeneracy. \nThe intrinsic degeneracy, horizontal pair seen in (aN), is almost \nresolved apart from 99\\% CL region at the particular set of values \nof the mixing parameters indicated above. \nThe corresponding pair in (aI) is missing because the \nintrinsic degeneracy is completely lifted.\nSince the matter effect plays minor role in the T2K II setting it is likely \nthat the spectral information is mainly responsible for lifting the intrinsic \ndegeneracy. See Sec.~\\ref{intrinsic} for more about it. \n\n\nHere is a brief comment on the property of the intrinsic and the \nsign-$\\Delta m^2$ degeneracies. \nBecause the degenerate solutions of CP phase $\\delta$ satisfy \napproximately the same relationship $\\delta_2 = \\pi - \\delta_1$ \nin both the intrinsic and the sign-$\\Delta m^2$ degeneracies \n\\cite{intrinsic,MNjhep01} \n(see Eqs.~(\\ref{intrinsicDS}) and (\\ref{signdm2DS}) in Sec.~\\ref{decoupling}), \nthe would-be four (one missing) regions in the panels (aN) and (aI) in \nFig.~\\ref{intrinsicKamKorea} forms a cross (or X) shape, \nwith crossing connection between a pair of solutions of \nthe sign-$\\Delta m^2$ degeneracy. \n\n\nIn the right four panels of Fig.~\\ref{intrinsicKamKorea} it is exhibited that \nthe intrinsic degeneracy as well as $\\theta_{23}$ octant degeneracy \nare completely resolved by the Kamioka-Korea \ntwo-detector setting at the same values of the mixing parameters, \nindicating power of the two detector method~\\cite{MNplb97}. \nNamely, the comparison between the spectral shapes in Kamioka \nand in Korea located at the first and nearly the second \noscillation maxima, respectively, supersedes a single detector \nmeasurement in Kamioka with the same total volume \ndespite much less statistics in the Korean detector. \nWe will give a detailed discussion on how $\\theta_{23}$ octant \ndegeneracy can be resolved by the Kamioka-Korea setting in \nSec.~\\ref{23degeneracy}, and present the details of the analysis \nin Sec.~\\ref{sensitivity}.\n\n\nIt should be noted that the sign-$\\Delta m^2$ degeneracy is also \nlifted though incompletely at the particular set of values of the \nmixing parameters as indicated in the panels (cI) \nof Fig.~\\ref{intrinsicKamKorea} where only the 99\\% CL regions remain. \nIn fact, we have shown in our previous paper that the \nKamioka-Korea identical two detector setting is powerful in \nresolving the sign-$\\Delta m^2$ degeneracy in a wide range of the \nmixing parameters~\\cite{T2KK}. \nWe note that resolution of the degeneracy in turn leads to an \nenhanced sensitivity to CP violation than that of T2K II setting in \na region of relatively large $\\theta_{13}$. \nSee~\\cite{T2KK} for comparison \nwith T2K II sensitivity. \nAltogether, we verify that the identical two detector setting in \nKamioka-Korea with neutrino beam from J-PARC solves all the \neight-fold parameter degeneracy {\\em in situ} if $\\theta_{13}$ \nis within reach by the next generation superbeam experiments \nsuch as T2K~\\cite{T2K} and NO$\\nu$A~\\cite{NOvA}.\n\n\n\\section{Decoupling between degeneracies}\n\\label{decoupling}\n\n\nIn this section we discuss the property called the \n``decoupling between degeneracies'' which arises due to the \nspecial setting of baselines shorter than $\\sim$1000 km. \nThe content of this section is somewhat independent of the main line \nof the discussion in this paper, and the readers can skip it \nto go directly to the analysis of $\\theta_{23}$ octant degeneracy \nin Secs.~\\ref{23degeneracy} and \\ref{sensitivity}. \nNonetheless, the property makes the structure of analysis for resolving the \neight-fold degeneracy transparent, and therefore it may worth to report. \n\n\nThe problem of decoupling came to our attention via the following path. \nIn most part of the previous paper~\\cite{T2KK}, we have discussed \nhow to solve the sign-$\\Delta m^2$ degeneracy without worrying \nabout the $\\theta_{23}$ octant degeneracy. \nConversely, the authors of ~\\cite{resolve23} analyzed the latter \ndegeneracy without resolving the former one. \nAre these correct procedure? \nThe answer is yes if the analysis procedure and the results for \nthe $\\theta_{23}$ degeneracy is independent of which solutions \nwe take for the sign-$\\Delta m^2$ degeneracy, and vice versa. \nWe call this property the ``decoupling between the degeneracies''.\\footnote{\nHere is a concrete example for which the decoupling does not work; \nIn the method of comparison between \n$\\nu_{\\mu}$ and $\\bar{\\nu}_{\\mu}$ disappearance measurement for \nlifting $\\theta_{23}$ octant degeneracy \\cite{choubey} one in fact \ndetermines the combined sign of \n$\\cos 2\\theta_{23} \\times \\Delta m^2_{31}$ as noticed in \\cite{resolve23}, \nand hence no decoupling.\n}\nThough discussion on this point was partially given in \\cite{resolve23}, \nwe present here a complete discussion of the decoupling.\n\n\nUnder the approximation of lowest nontrivial order in matter effect, \nwe prove that the decoupling holds between the above two degeneracies, \nand furthermore that it can be generalized, \nthough approximately, to the relation between any two pair of \ndegeneracies among the three types of degeneracies.\nTo our knowledge, leading order in matter perturbation theory appears to \nbe the only known circumstance that the argument goes through. \nFortunately, the approximation is valid for the setting used in this paper \nwith baseline up to $\\sim$1000 km, in particular in the Kamioka-Korea \nsetting. \nIn the following treatment we make a further approximation that the \ndegenerate solutions are determined primarily by the measurement at \nthe intermediate detector. \nIt is a sensible approximation because the statistics is about 10 times higher \nat the intermediate detector, and its validity is explicitly verified \nin the analysis performed in \\cite{T2KK}. \n\n\\subsection{Approximate analytic treatment of the parameter degeneracy}\n\n\nTo make the discussion self-contained, we start from the derivation of \nthe degenerate solutions by using the matter perturbation theory \\cite{AKS}, \nin which the matter effect is kept to its lowest nontrivial order.\nNamely, the matter effect can be ignored in leading order in \nthe disappearance channel whose oscillation probability is order of unity. \nThen, the disappearance probability \n$P(\\nu_{\\mu} \\rightarrow \\nu_{\\mu})$ can be given by \nthe vacuum oscillation approximation with leading order in $s^2_{13}$ \nand the solar $\\Delta m^2_{21}$ corrections as \n\\begin{eqnarray}\n1 - P(\\nu_{\\mu} \\rightarrow \\nu_{\\mu}) &=& \n\\left[ \n\\sin^2 2\\theta_{23}\n+ 4 s^2_{13} s^2_{23} \n\\left(2 s^2_{23} - 1 \\right)\n\\right]\n\\sin^2 \\left(\\frac{\\Delta m^2_{31}L}{4E}\\right) \n\\nonumber \\\\\n& - & c_{12}^2\\sin^22\\theta_{23}\n\\left(\\frac{\\Delta m_{21}^2 L}{4E}\\right) \n\\sin \\left(\\frac{\\Delta m_{31}^2 L}{2E}\\right).\n\\label{Pvac_mumu2}\n\\end{eqnarray}\nThe probability for anti-neutrino channel is the same as that for neutrino \none in this approximation. \nOne can show \\cite{AKS} that the other solar $\\Delta m^2_{21}$ \ncorrections are suppressed further by either a small \n$\\sin^2{2\\theta_{13}} \\lsim 0.1$, \nor the Jarlskog factor \n$J \\equiv c_{12} s_{12} c_{13}^2 s_{13} c_{23} s_{23} \\sin{\\delta} \\lsim 0.04$. \nIn fact, the validity of the approximation is explicitly verified in \\cite{resolve23} \nwhere the matter effect terms are shown to be of the order of $10^{-3}$ \neven at $L=1000$ km. \nA disappearance measurement, therefore, determines $s^2_{23}$ \nto first order in $s^2_{13}$ as \n\\begin{eqnarray}\n(s^2_{23})^{(1)}= (s^2_{23})^{(0)} (1+ s^2_{13}),\n\\label{degene-sol}\n\\end{eqnarray}\nwhere \n$(s^2_{23})^{(0)}$ is the solution obtained by ignoring $s^2_{13}$. \nFrom the first term in Eq.~(\\ref{Pvac_mumu2}), \nthe two solutions of $(s^2_{23})^{(0)}$ is determined as \n$(s^2_{23})^{(0)}= \\frac{1}{2} \\left[ 1 \\pm \\sqrt{1- \\sin^2{2\\theta_{23}}} \\right]$, \nthe simplest form of the $\\theta_{23}$ octant degeneracy. \nFor example, $(s^2_{23})^{(0)} = 0.4$ or 0.6 (0.45 or 0.55) for \n$\\sin^2{2\\theta_{23}}=0.96\\,(0.99)$. \n\n\nFor the appearance channel, we use the \n$\\nu_{\\mu} (\\bar{\\nu}_\\mu) \\to \\nu_{e} (\\bar\\nu_e$) \noscillation probability with first-order matter effect \\cite{AKS}\n\\begin{eqnarray}\nP[\\nu_{\\mu}(\\bar{\\nu}_{\\mu}) &\\rightarrow& \n\\nu_{\\rm e}(\\bar{\\nu}_e)] = \nc^2_{23} \\sin^2{2\\theta_{12}} \n\\left(\\frac{\\Delta m^2_{21} L}{4 E}\\right)^2 \n\\nonumber \\\\\n&+& \\sin^2{2\\theta_{13}} s^2_{23}\n\\left[\n\\sin^2 \\left(\\frac{\\Delta m^2_{31} L}{4 E}\\right)\n-\\frac {1}{2}\ns^2_{12}\n\\left(\\frac{\\Delta m^2_{21} L}{2 E}\\right)\n\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right) \n\\right.\n\\nonumber \\\\\n&&\\hspace*{22mm} {}\\pm\n\\left.\n\\left(\\frac {4 Ea}{\\Delta m^2_{31}}\\right)\n\\sin^2 {\\left(\\frac{\\Delta m^2_{31} L}{4 E}\\right)}\n\\mp \n\\frac{aL}{2}\n\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right) \n\\right]\n\\nonumber \\\\\n&+& \n2J_{r} \\left(\\frac{\\Delta m^2_{21} L}{2 E} \\right)\n\\left[\n\\cos{\\delta}\n\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right) \\mp \n2 \\sin{\\delta}\n\\sin^2 \\left(\\frac{\\Delta m^2_{31} L}{4 E}\\right) \n\\right], \n\\label{Pmue}\n\\end{eqnarray}\nwhere the terms of order \n$s_{13} \\left( \\frac{\\Delta m^2_{21}}{\\Delta m^2_{31}} \\right)^2$ \nand \n$aL s_{13} \\left( \\frac{\\Delta m^2_{21}}{\\Delta m^2_{31}} \\right)$ \nare neglected. \nIn Eq.~(\\ref{Pmue}), $a\\equiv \\sqrt 2 G_F N_e$ \\cite{MSW}\nwhere $G_F$ is the Fermi constant, $N_e$ denotes the averaged \nelectron number density along the neutrino trajectory in \nthe earth, \n$J_r$ $(\\equiv c_{12} s_{12} c_{13}^2 s_{13} c_{23} s_{23} )$ \ndenotes the reduced Jarlskog factor, and the upper and the \nlower sign $\\pm$ refer to the neutrino and \nanti-neutrino channels, respectively. \nWe take constant matter density approximation in this paper.\nThe first term of Eq.~(\\ref{Pmue}) is due to the oscillation driven by the \nsolar $\\Delta m^2_{21}$, which is essentially negligible in the intermediate \ndetector but not at the far detector and is of key importance to resolve the \n$\\theta_{23}$ octant degeneracy. \n\n\nWe make an approximation of ignoring terms of order \n$(\\Delta m^2_{21}\/\\Delta m^2_{31}) J_r \\cos{2\\theta_{23}}$ in \nEq.~(\\ref{Pmue}). \nNote that keeping only the leading order in this quantity \nis reasonable because $J_r \\lsim 0.04$, \n$|\\Delta m^2_{21}\/\\Delta m^2_{31}| \\simeq 1\/30$, and \n$\\cos{2\\theta_{23}}= \\pm 0.2$ for $\\sin^2{2\\theta_{23}} = 0.96$. \nThen, \nthe two degenerate solutions obey an approximate relationship \n\\begin{eqnarray}\n\\left( \\sin^2 {2\\theta_{13}} s^2_{23} \\right)^{\\text{1st}} =\n\\left( \\sin^2 {2\\theta_{13}} s^2_{23} \\right)^{\\text{2nd}}, \n\\label{1st_order}\n\\end{eqnarray}\nor, \n$s_{13}^{\\text{1st}}s_{23}^{\\text{1st}} \n= s_{13}^{\\text{2nd}}s_{23}^{\\text{2nd}}$\nignoring higher order terms in $s_{13}$. \nWe can neglect the leading order correction in $s^2_{13}$ to \n$s^2_{23}$ in these relations because it gives $O(s^4_{13})$ terms. \n\n\n\nAnalytic treatment of the intrinsic and the sign-$\\Delta m^2$ degeneracies \nis given in \\cite{MNP2}. \nIn an environment where the vacuum oscillation approximation applies the \nsolutions corresponding to the intrinsic degeneracy are given by \\cite{intrinsic} \n\\begin{eqnarray}\n\\theta_{13}^{(2)} = \\theta_{13}^{(1)}, \n\\hspace{1cm}\n\\delta^{(2)} = \\pi - \\delta^{(1)},\n\\label{intrinsicDS}\n\\end{eqnarray}\nwhere the superscripts (1) and (2) label the solutions due to the \nintrinsic degeneracy. \nUnder the same approximation the solutions corresponding to the \nsign-$\\Delta m^2$ degeneracy are given by \\cite {MNjhep01}\n\\begin{eqnarray}\n\\theta_{13}^{\\,\\text{norm}} = \\theta_{13}^{\\,\\text{inv}}, \n\\hspace{1cm}\n\\delta^{\\,\\text{norm}} = \\pi - \\delta^{\\,\\text{inv}}, \n\\hspace{1cm}\n(\\Delta m^2_{31})^{\\,\\text{norm}} = - (\\Delta m^2_{31})^{\\,\\text{inv}}, \n\\label{signdm2DS}\n\\end{eqnarray}\nwhere the superscripts ``norm'' and ``inv'' label the solutions with \nthe positive and the negative sign of $\\Delta m^2_{31}$. \nThe degeneracy stems from the approximate symmetry under the \nexchange of these two solutions through which the degeneracy is \nuncovered \\cite {MNjhep01}.\nThe validity of these approximate relationships in the actual experimental \nsetup in the T2K II measurement is explicitly verified in \\cite{T2KK}. \nIt should be noticed that even if sizable matter effect is present \nthe relation (\\ref{intrinsicDS}) holds \nat the energy corresponding to the vacuum oscillation maximum, \nor more precisely, the shrunk ellipse limit \\cite{KMN02}. \n\n\n\n\\subsection{Decoupling between degeneracies}\n\n\nResolution of the degeneracy can be done when a measurement distinguishes \nbetween the values of the oscillation probabilities with the two different \nsolutions corresponding to a degeneracy. \nTherefore, we define the probability difference \n\\begin{eqnarray}\n&&\\Delta P^{ab}(\\nu_{\\alpha} \\rightarrow \\nu_{\\beta}) \n\\nonumber \\\\\n&\\equiv& \nP \\left( \\nu_{\\alpha} \\rightarrow \\nu_{\\beta}; \\theta_{23}^{(a)}, \n\\theta_{13}^{(a)}, \\delta^{(a)}, (\\Delta m^2_{31})^{(a)} \\right) - \nP \\left( \\nu_{\\alpha} \\rightarrow \\nu_{\\beta}; \\theta_{23}^{(b)}, \n\\theta_{13}^{(b)}, \\delta^{(b)}, (\\Delta m^2_{31})^{(b)} \\right)\n\\label{DeltaPdef}\n\\end{eqnarray}\nwhere the superscripts $a$ and $b$ label the degenerate solutions. \nSuppose that we are discussing the degeneracy A. \nThe decoupling between the degeneracies A and B \nholds if $\\Delta P^{ab}$ defined in (\\ref{DeltaPdef}) for the degeneracy A \nis invariant under the replacement of the mixing parameters \ncorresponding to the degeneracy B, and vice versa. \n\n\nThe best example of the decoupling is given by the one between \nthe $\\theta_{23}$ octant and the sign-$\\Delta m^2$ degeneracies. \nBy noting that \n$J_r^{\\,\\text{1st}} - J_r^{\\,\\text{2nd}} \n= \\cos 2\\theta_{23}^{\\,\\text{1st}} J_r^{\\,\\text{1st}}$ \nin leading order in $\\cos{2\\theta_{23}}$, \nthe difference between probabilities with the first and the second octant \nsolutions can be given by \n\\begin{eqnarray}\n&&\\Delta P^{\\,\\text{1st 2nd}}(\\nu_{\\mu} \\rightarrow \\nu_{e}) = \n\\cos{2\\theta_{23}^{\\,\\text{1st}}} \n\\sin^2{2\\theta_{12}} \n\\left(\\frac{\\Delta m^2_{21} L}{4 E}\\right)^2 \n\\nonumber \\\\\n&+&\n2J_{r}^{\\,\\text{1st}} \\cos 2\\theta_{23}^{\\,\\text{1st}} \n\\left(\\frac{\\Delta m^2_{21} L}{2 E} \\right)\n\\left[\n\\cos{\\delta}\n\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right) \\mp \n2 \\sin{\\delta}\n\\sin^2 \\left(\\frac{\\Delta m^2_{31} L}{4 E}\\right) \n\\right]. \n\\label{DeltaP_23}\n\\end{eqnarray}\nThe remarkable feature of (\\ref{DeltaP_23}) is that the leading-order \nmatter effect terms drops out completely. \nTherefore, our approximated treatment remains valid until the \nsecond order matter effect starts to become sizable in the \nappearance oscillation probability. \nMore importantly, $\\Delta P^{\\,\\text{1st~2nd}}$, being composed only of \nthe vacuum oscillation terms, \nis obviously invariant under the replacement \n$normal \\leftrightarrow inverted$ solutions with different signs of \n$\\Delta m^2_{31}$ given in Eq.~(\\ref{signdm2DS}). \nTherefore, the resolution of the $\\theta_{23}$ octant degeneracy \ncan be carried out without worrying about the presence of \nthe sign-$\\Delta m^2_{31}$ degeneracy. \n\n\n\nNext, we examine the inverse problem; \nDoes the determination of mass hierarchy decouple with \nthe resolution of the $\\theta_{23}$ degeneracy?\nOne can show, by using Eq.~(\\ref{Pmue}), that \nthe similar probability difference between the solutions in \nEq.~(\\ref{signdm2DS}) with the normal and the \ninverted hierarchies is given by \n\\begin{eqnarray}\n&&\\Delta P^{\\,\\text{norm~inv}} (\\nu_{\\mu} \\rightarrow \\nu_{e}) \n\\nonumber \\\\\n&=&\n\\sin^2{2\\theta_{13}^{\\,\\text{norm}}} (s_{23}^{\\,\\text{norm}})^2\n\\left[ \n- s^2_{12}\n\\left(\\frac{\\Delta m^2_{21} L}{2 E}\\right)\n\\sin \\left(\\frac{(\\Delta m^2_{31})^{\\,\\text{norm}} L}{2 E}\\right) \n\\right.\n\\nonumber \\\\\n\\hspace*{0mm} {}&\\pm&\n\\left.\n2(aL) \n\\left\\{\n \\left(\\frac{4E}{(\\Delta m^2_{31})^{\\,\\text{norm}} L}\\right) \n\\sin^2 \\left(\\frac{(\\Delta m^2_{31})^{\\,\\text{norm}} L}{4 E}\\right) -\n\\frac{1}{2} \\sin \\left(\\frac{(\\Delta m^2_{31})^{\\,\\text{norm}} L}{2 E}\\right) \n\\right\\}\n\\right]\n\\label{DeltaP_sign}\n\\end{eqnarray}\nwhere the superscripts ``norm'' and ``inv'' can be exchanged \nif one want to start from the inverted hierarchy. \nWe notice that most of the vacuum oscillation terms, including the solar term, \ndrop out because of the invariance under \n$\\delta \\rightarrow \\pi - \\delta$ and \n$\\Delta m^2_{31} \\rightarrow - \\Delta m^2_{31}$. \nNow, we observe that $\\Delta P^{\\,\\,\\text{norm~inv}}$\nis invariant under the transformation \n$\\theta_{23}^{\\,\\text{1st}} \\leftrightarrow \\theta_{23}^{\\,\\text{2nd}}$ and \n$\\theta_{13}^{\\,\\text{1st}} \\leftrightarrow \\theta_{13}^{\\,\\text{2nd}}$, \nbecause $\\Delta P^{\\,\\,\\text{norm~inv}}$ depends upon \n$\\theta_{13}$ and $\\theta_{23}$ only through the combination \n$\\sin^2{2\\theta_{13}} s_{23}^{2}$ within our approximation. \nTherefore, the sign-$\\Delta m^2_{31}$ and the $\\theta_{23}$ \ndegeneracies decouple with each other. \n\n\nIn our previous paper, we have shown that the sign-$\\Delta m^2_{31}$ \ndegeneracy can be lifted by the Kamioka-Korea two detector setting. \nThe above argument for decoupling guarantees that our treatment \nis valid irrespective of the solutions assumed for $\\theta_{23}$ \ndegeneracy.\\footnote{\nWe remark that in most part of \\cite{T2KK}, we have assumed that \n$\\theta_{23}=\\pi\/4$ so that this problem itself does not exist. \nThe above discussion implies that even in the case of \nnon-maximal value of $\\theta_{23}$ the similar analysis \nas in \\cite{T2KK} must go through without knowing in which octant \n$\\theta_{23}$ lives. \nThe resultant sensitivity for resolving the mass hierarchy will change \nas $\\theta_{23}$ goes away from $\\pi\/4$ but only slightly \nas will be shown in Sec.~\\ref{sensitivity}. \n}\n\n\n\\subsection{Including the intrinsic degeneracy}\n\\label{intrinsic}\n\n\nNow we turn to the intrinsic degeneracy for which the situation \nis somewhat different. First of all, in our setting, \nthe intrinsic degeneracy is already \nresolved by spectrum informations at the intermediate detector \nif $\\theta_{13}$ is relatively large, $\\sin^2 2\\theta_{13} \\gsim 0.02$ \nas illustrated in Fig.~\\ref{intrinsicKamKorea} before the information \nfrom the far detector is utilized. \nIt means that there is no intrinsic degeneracy from the beginning \nin the analysis with spectrum informations. \nBecause of this feature, the intrinsic degeneracy decouple from \nthe beginning from the task of resolving the eight-fold degeneracy in \nour setting for the relatively large values of $\\theta_{13}$.\nBecause of a further enhanced sensitivity, the intrinsic degeneracy \nmay be resolved to a smaller values of $\\theta_{13}$ in the Kamioka-Korea setting. \n\n\nIn fact, powerfulness of the spectral information for resolving \nthe intrinsic degeneracy can be understood easily by noting that \n$\\Delta P^{ab}$ defined in (\\ref{DeltaPdef}) is given by using the \nintrinsic degeneracy solution (\\ref{intrinsicDS}) as \n\\begin{eqnarray}\n&&\n\\Delta P^{12}(\\nu_{\\mu} \\rightarrow \\nu_{e}) =\n4 J_{r} \\left(\\frac{\\Delta m^2_{21} L}{2 E} \\right) \n\\cos{\\delta^{(1)}}\n\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right),\n\\label{DeltaP_intrinsic}\n\\end{eqnarray}\nwith notable feature that the matter effect cancels out. \nNotice that even if the matter effect cannot be negligible the solution \n(\\ref{intrinsicDS}) holds for measurement at energies around the \nvacuum oscillation maximum. \nThe right-hand side of Eq.~(\\ref{DeltaP_intrinsic}) is proportional to $E^{-2}$ \nat energies near the vacuum oscillation maximum, \n$\\frac{\\Delta m^2_{21} L}{2 E} = \\pi$, and the steep energy \ndependence can be used to lift the degeneracy. \nHence, the spectrum analysis is a powerful tool for resolving the \nintrinsic degeneracy. \n\n\n\\subsection{The case that the intrinsic degeneracy is not solved} \n\n\nEven if $\\theta_{13}$ is too small, or if the energy resolution is too \npoor for the spectrum analysis to resolve the intrinsic degeneracy, \nwe can show that the intrinsic degeneracy approximately \ndecouples from the other degeneracies. \n$\\Delta P^{12}$ in Eq.~(\\ref{DeltaP_intrinsic}) is not exactly but approximately \ninvariant under the transformation first-octant $\\leftrightarrow$ \nsecond-octant solutions. \nThe difference between $\\Delta P^{12} (\\text{1st})$ and \n$\\Delta P^{12} (\\text{2nd})$\nis of order $\\cos 2\\theta_{23} J_r \\Delta m^2_{21} \/ \\Delta m^2_{31} \n\\simeq 3 \\times 10^{-4}$ for $s^2_{23}=0.4$ and $\\sin^2 2\\theta_{13}= 0.1$\napart from the further suppression by \n$\\sin \\left(\\frac{\\Delta m^2_{31} L}{2 E}\\right)$ at around the \noscillation maximum. \nBeing the vacuum oscillation term $\\Delta P^{12}$ is obviously invariant \nunder the replacement $normal \\leftrightarrow inverted$ solutions \nwith different signs of $\\Delta m^2_{31}$. \nTherefore, resolution of the intrinsic degeneracy can be done, \nto a good approximation, independent of the presence of the \nsign-$\\Delta m^2_{31}$ and the $\\theta_{23}$ octant degeneracies. \n\n\nThe remaining problem we need to address is the inverse problem, \nwhether the resolution of the sign-$\\Delta m^2_{31}$ and the $\\theta_{23}$ \noctant degeneracies can be carried out without knowing solutions \nof the intrinsic degeneracy. \nThe sign-$\\Delta m^2_{31}$ degeneracy decouples from the intrinsic one \nbecause $\\Delta P^{\\,\\text{norm~inv}}$ in (\\ref{DeltaP_sign}) is invariant under \nthe exchange of two intrinsic degeneracy solutions. \nThe $\\theta_{23}$ octant degeneracy also approximately decouples \nfrom the intrinsic one. \n$\\Delta P^{\\,\\text{1st~2nd}}$ in (\\ref{DeltaP_23}) changes under the \ninterchange of two intrinsic degeneracy solutions only by the \nsame amount as the difference between $\\Delta P^{12}$ of \nthe first and the octant $\\theta_{23}$ solutions, \n$\\cos 2\\theta_{23} J_r \\Delta m^2_{21} \/ \\Delta m^2_{31} \n\\simeq 3 \\times 10^{-4}$ (for $s^2_{23}=0.4$ and $\\sin^2 2\\theta_{13}= 0.1$). \n\n\nHere is a clarifying comment on what the decoupling really means; \nBecause of the cross-shaped structure of the degenerate solutions \nof the intrinsic and the sign-$\\Delta m^2_{31}$ degeneracies \n(as was shown in Sec.~\\ref{how}) \nthe decoupling of the former from the latter does {\\em not} imply that \nthe correct value of $\\delta$ can be extracted from the measurement \nwithout knowing the correct sign of $\\Delta m^2_{31}$.\\footnote{\nNotice, however, that it does {\\em not} obscure the CP violation, \nbecause the ambiguity is only two-fold; $\\delta \\leftrightarrow \\pi-\\delta$. \n}\nIt means that the elimination of one of the ``intrinsic'' degenerate pair \nsolutions related by $\\delta \\leftrightarrow \\pi-\\delta$ \nfor a given sign of $\\Delta m^2_{31}$ can be done without knowing \nthe mass hierarchy, the true sign of $\\Delta m^2_{31}$. \nTherefore, the situation that the intrinsic degeneracy is always \nresolved by the spectrum analysis in region of not too small \n$\\theta_{13}$, as is the case in our setting, is particularly transparent \none from this viewpoint. \n\n\nTo sum up, we have shown that to leading order in the matter effect \nthe intrinsic, the sign-$\\Delta m^2_{31}$, and the $\\theta_{23}$ \noctant degeneracies decouples with each other. \nThey do so exactly except for between the intrinsic and the $\\theta_{23}$ \noctant degeneracies for which the decoupling is approximate but \nsufficiently good to allow one-by-one resolution of all the three types \nof degeneracies. \nThe decoupling implies that in analysis for lifting the eight-fold \ndegeneracy the structure of the $\\chi^2$ minimum is very simple in \nmulti-dimensional parameter space, and it may be of use in \ndiscussions of how to solve the degeneracy \nin much wider context than that discussed in this paper.\n\n\n\\section{How identical two detector setting solves $\\theta_{23}$ octant degeneracy?}\n\\label{23degeneracy}\n\n\nNow, we turn to the problem of how the identical two detector \nsetting can resolve the $\\theta_{23}$ degeneracy, \nthe unique missing link in a program of resolving eight-fold \nparameter degeneracy in the Kamioka-Korea two detector setting. \nThe solar $\\Delta m^2$ oscillation term, \nthe first term in Eq.~(\\ref{Pmue}) with the coefficient of $c^2_{23}$, \nmay be of key importance to do the job. \nWhile it was argued on very general ground \\cite{resolve23} that\nthe $\\theta_{23}$ degeneracy is hard to resolve only by accelerator \nexperiments with baseline of $\\lsim 1000$ km or so, \nthe argument can be circumvented if the solar term can be isolated. \nWe emphasize that the accuracy of the determination of $\\theta_{23}$ is \nseverely limited by the octant degeneracy, as discussed in detail \nin \\cite{MSS04}.\n\n\nTherefore, the question we must address first is the relative importance \nof the solar term to the remaining terms in $\\Delta P^{\\,\\text{1st~2nd}}$ \nin Eq.~(\\ref{DeltaP_23}). \nWe note that the ratio of the solar term to the $\\delta$-dependent \nsolar-atmospheric interference term in $\\Delta P^{\\,\\text{1st~2nd}}$ is \ngiven by \n$\\sin^2{2\\theta_{12}} (\\Delta m^2_{21} L \/ 4 E) \/ 4 J_{r}$, \nassuming the square parenthesis in (\\ref{DeltaP_23}) is of order unity. \nThe ratio is roughly given by \n$\\simeq 3 (1 \/ 30) (\\pi \/ 2) 0.86 (1\/ 4 J_{r} ) \\simeq 0.9~(0.16 \/ s_{13}) $ \nwith beam energy having the first oscillation maximum in Kamioka. \nTherefore, the solar term is indeed comparable or lager for\nsmaller $\\theta_{13}$ in size with the interference terms \nin $\\Delta P^{\\,\\text{1st~2nd}}$ at the far detector. \nObviously, the solar term is independent of $\\theta_{13}$,\nwhich suggests that the sensitivity to resolve the $\\theta_{23}$\ndegeneracy is almost independent of $\\theta_{13}$, \nas will be demonstrated in Sec.~\\ref{sensitivity}. \nWe note that while the solar term is the key to resolve \nthe $\\theta_{23}$ degeneracy, the interference terms \nalso contributes to lift the degeneracy. In particular, \nas shown in (\\ref{DeltaP_23}), \nthe $\\sin\\delta$ term has opposite sign when the polarity of the \nbeam is switched from the neutrino to the anti-neutrino runs. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{Patm_Psol_normal.eps}\n\\end{center}\n\\caption{\nThe energy dependence of the solar term (red solid line) is \ncontrasted with the ones of atmospheric plus interference \nterms in the $\\nu_{e}$ \nappearance oscillation probabilities with various values of CP phase \n$\\delta$; \n$\\delta=0$ (dotted line), \n$\\delta=\\pi\/2$ (dashed line), \n$\\delta=\\pi$ (dash-dotted line), and \n$\\delta=3\\pi\/2$ (double-dash-dotted line). \nFor this plot, we used analytic expression in Eq.~(\\ref{Pmue}); \n$P_{\\text{solar}}$ is defined to be the first term in \nEq.~(\\ref{Pmue}) whereas $P_{\\text{atm}}$ is defined to be \nthe rest in Eq.~(\\ref{Pmue}). \n$\\bar{P}_{\\text{solar}}$ and $\\bar{P}_{\\text{atm}}$ refer to the \ncorresponding terms for anti-neutrinos. \n}\n\\label{solar-vs-atm}\n\\end{figure}\n\n\nThe next question we must address is how the solar term can be \nseparated from the other terms to have enhanced sensitivity to \nthe $\\theta_{23}$ degeneracy. \nTo understand the behavior of the solar term and its difference from \nthat of the atmospheric terms in the oscillation probability, \nwe plot in Fig.~\\ref{solar-vs-atm} a comparison between them \nin Kamioka (left panels) and in Korea (right panels) for various \nvalues of $\\delta$.\nAs one can observe in the right panels, the energy dependence \nof the solar oscillation term, a monotonically decreasing \n(approximately $1\/E^2$) behavior with increasing energy, \nis quite different from the oscillating behavior of the atmospheric ones. \nIt is also notable that the ratio of the solar term to the atmospheric-solar \ninterference term is quite different between the intermediate and \nthe far detectors.\nDue to the differing relative importance of the solar term in the two \ndetectors and the clear difference in the energy dependences \nbetween the solar and the atmospheric terms, \nthe spectrum analysis, the powerful method for resolving the \nintrinsic degeneracy, must be able to isolate the solar term \nfrom the remaining ones. \nThis will be demonstrated in the quantitative analysis in the next section.\n\n\nWe note that several alternative methods are proposed to resolve \nthe $\\theta_{23}$ degeneracy. \nThey include: \nthe atmospheric neutrino method \\cite{concha-smi_23,atm23,choubey2}, and \nthe reactor accelerator combined method \\cite{MSYIS,resolve23}, \nthe atmospheric accelerator combined method \\cite{atm-lbl}. \nThe atmospheric neutrino method discussed in \n\\cite{concha-smi_23,atm23} is closest to ours in physics \nprinciple of utilizing the solar mass scale oscillation effect. \nPossible advantage of the present method may be in a clean detection \nof the solar term by the intermediate-versus-far two detector comparison.\n\n\n\\section{Sensitivity for resolving $\\theta_{23}$ octant degeneracy}\n\\label{sensitivity}\n\nIn this section, we describe details of our analysis for resolving \nthe $\\theta_{23}$ octant degeneracy. \nThey include treatment of experimental errors, treatment of \nbackground, and the statistical procedure which is used to \ninvestigate the sensitivity of the experiment. \nThen, the results of our analysis are presented. \n\n\n\\subsection{Assumptions and the definition of $\\chi^2$}\n\n\nIn order to understand the sensitivity of the experiment\nwith the two detector system at \n295~km (Kamioka) and 1050~km (Korea), we carry out \na detailed $\\chi^2$ analysis. \nTo address the $\\theta_{23}$ octant degeneracy, \nit is of course necessary to include $\\nu_{\\mu}$ and $\\bar{\\nu}_{\\mu}$ \ndisappearance channels in addition to the appearance ones in our treatment. \nIn short, the definition of the statistical procedure is similar to the \none used in Ref.~\\cite{T2KK} with necessary extension for including \nmuon events. \nThe assumption on the experimental setting is also identical\nto that of the best performance setting identified in Ref.~\\cite{T2KK}.\nNamely, 0.27~Mton fiducial masses for the intermediate \nsite (Kamioka, 295~km) and the far site (Korea, 1050~km).\nThe neutrino beam is assumed to be 2.5 degree off-axis one produced\nby the upgraded J-PARC 4~MW proton beam. \nIt is assumed that the experiment will continue for 8 years\nwith 4 years of neutrino and 4 years of anti-neutrino runs. \n\nWe use various numbers and distributions available from\nreferences related to T2K \\cite{JPARC-detail}, in which many of \nthe numbers are updated after the original proposal \\cite{T2K}.\nHere, we summarize the main assumptions and the methods \nused in the $\\chi^2$ analysis.\nWe use the reconstructed neutrino energy for single-Cherenkov-ring \nelectron and muon events.\nThe resolution in the reconstructed neutrino energy is 80~MeV \nfor quasi-elastic events. \nWe assume that \n$|\\Delta m_{31}^2|$ should be known precisely by the time when \nthe experiment we consider in this report will be carried out. \nWe take \n$\\Delta m_{31}^2 = \\pm 2.5 \\times 10^{-3} \\text{eV}^2$. \nHence, we assume that the \nenergy spectrum of the beam is the one expected by the\n2.5~degree off-axis-beam in T2K. The shape of the\nenergy spectrum for the anti-neutrino beam is assumed to be\nidentical to that of the neutrino beam. The event rate\nfor the anti-neutrino beam \nin the absence of neutrino oscillations\nis smaller by a factor of 3.4 due mostly to the lower neutrino \ninteraction cross sections and partly to the slightly lower \nflux. The signal to noise ratio is\nworse for the anti-neutrino beam than that for the neutrino beam\nby a factor of about 2. \n\n28 background electron events are expected \nfor the reconstructed neutrino energies between 350 and 850~MeV for\n$(0.75 \\times 0.0225 \\times 5)\\, \\text{MW} \\cdot \\text{Mton} \\cdot \\text{yr}$ \nmeasurement \nwith the neutrino beam. The energy dependence \nof the background rate and the rate itself are taken \nfrom \\cite{JPARC-detail}.\nThe background rate is expected to be higher in the lower \nneutrino energies. The expected number of electron events\nis assumed to be 122 for $\\sin^2 2\\theta_{13} =$0.1 with the \nsame detector exposure\nand beam, assuming the normal mass hierarchy and $\\delta = 0$.\n\n\nWe assume that the experiment is equipped with a near detector \nwhich measures the rate and the energy dependence of the background\nfor electron events,\nun-oscillated muon spectrum, \nand the signal detection efficiency.\nThese measurements are assumed to be carried out\nwithin the uncertainty of 5\\%. \nWe already demonstrated that the dependence on the \nassumed value of the experimental systematic errors is \nrather weak \\cite{T2KK}.\nWe stress that in the present setting \nthe detectors located in Kamioka and in Korea are not only \nidentical but also receive neutrino beams with essentially\nthe same energy distribution (due to the same off-axis angle of 2.5 degree) \nin the absence of oscillations. \nHowever, it was realized recently that, due to a non-circular shape \nof the decay pipe of the J-PARC neutrino beam line, \nthe flux energy spectra viewed at detectors in Kamioka and in \nKorea are expected to be slightly different even at the same off-axis angle, \nespecially in the high-energy tail of the spectrum \n\\cite{Rubbia-Meregaglia-Seoul2006}. \nThe possible difference between fluxes in the intermediate and the \nfar detectors is newly taken into account as a systematic error in \nthe present analysis.\n\n\nWe compute neutrino oscillation probabilities \nby numerically integrating neutrino evolution equation \nunder the constant density approximation. \nThe average density is assumed to be 2.3 and 2.8 g\/cm$^3$ for the \nmatter along the beam line between the production target and \nKamioka and between the target and Korea, respectively \\cite{T2KK}.\nWe assume that the number of electron with respect to that of nucleons \nto be 0.5 to convert the matter density to the electron number density. \nIn our $\\chi^2$ analysis, \nwe fix the absolute value of $|\\Delta m_{31}^2|$ to be $2.5\\times 10^{-3}$ eV$^2$, \nand fix solar parameters as $\\Delta m_{21}^2 = 8\\times 10^{-5}$ eV$^2$ and \n$\\sin^2{\\theta_{12}}$=0.31.\n\n\n\\begin{figure}[htbp]\n\\vglue 0.3cm\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{strategy.1.eps}\n\\end{center}\n\\vglue -0.3cm\n\\caption{Examples of electron and muon events to be observed in \nKamioka and Korea for 4 years of neutrino plus \n4 years of anti-neutrino running \nare presented as a function of reconstructed neutrino energy. \nThe fiducial masses are taken to be 0.27~Mton for both the detectors in \nKamioka and Korea.\nThe dashed histograms for electron events show the background events. \nThe open circles show the expected\nenergy spectrum of signal events with \n$\\sin^2 \\theta_{23} =$0.40 and $\\sin^2 2 \\theta_{13} =$0.01.\nThe solid circles show the expected\nenergy spectrum of signal events with \n$\\sin^2 \\theta_{23} =$0.60 and $\\sin^2 2 \\theta_{13} =$0.0067.\nIn both cases, $\\delta = 3\\pi\/4$ and normal mass\nhierarchy are assumed in simulating the events. \n}\n\\label{fig:energy-spectrum-examples}\n\\end{figure}\n\n\nFig.~\\ref{fig:energy-spectrum-examples} shows an\nexample of the energy spectrum of electron and muon events\nto be observed in Kamioka and Korea for 4 years of\nneutrino beam plus 4 years of anti-neutrino beam.\nThe two sets of parameters give very similar spectrum \nfor both the electron and muon events at the Kamioka detector \nand the muon events at the Korean detector. \nHowever, due to the long baseline distance, the\nsolar term plays some role in the \n$^( \\overline{\\nu}^)_\\mu \\rightarrow\\, ^(\\overline{\\nu}^)_e$ \noscillation \nprobability at the Korean detector. Therefore, \nthe two sets of parameters give slightly different\noscillation probabilities in Korea. \nSince the solar term is proportional to $c^2_{23}$ \nwe use this feature to obtain information on\n$\\sin^2 \\theta_{23}$.\n\n\nThe statistical significance of the measurement considered in this\npaper was estimated by using the following \ndefinition of $\\chi^2$:\n\\begin{equation}\n\\chi^2 = \\sum_{k=1}^{4} \\left( \n\\sum_{i=1}^{5}\n\\frac{\\left(N(e)_{i}^{\\rm obs} - N(e)_{i}^{\\rm exp}\\right)^2}\n{ \\sigma^2_{i} } +\n\\sum_{i=1}^{20}\n\\frac{\\left(N(\\mu)_{i}^{\\rm obs} - N(\\mu)_{i}^{\\rm exp}\\right)^2}\n{ \\sigma^2_{i} }\n\\right)\n+ \\sum_{j=1}^{7} \\left(\\frac{\\epsilon_j}\n{\\tilde{\\sigma}_{j}}\\right)^2, \n\\label{equation:chi2def}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n N(e)_{i}^{\\rm exp} = N_{i}^{\\rm BG} \\cdot \n (1+\\sum_{j=1,2,7} f(e)_{j}^{i}\\cdot\\epsilon_{j}) \n + N_i^{\\rm signal} \\cdot \n (1+\\sum_{j=3,7} f(e)_{j}^{i}\\cdot\\epsilon_{j}) ~,\n\\label{equation:e-number}\n\\\\\n N(\\mu)_{i}^{\\rm exp} = N_{i}^{\\rm non-QE} \\cdot \n (1+\\sum_{j=4,6,7} f(\\mu)_{j}^{i}\\cdot\\epsilon_{j}) \n + N_i^{\\rm QE} \\cdot \n (1+\\sum_{j=4,5,7} f(\\mu)_{j}^{i}\\cdot\\epsilon_{j}) ~.\n\\label{equation:mu-number}\n\\end{eqnarray} \n\n\\noindent\nThe first and second terms in Eq.~(\\ref{equation:chi2def}) are\n for the number of observed \nsingle-ring electron and muon events, respectively.\n$N(e~{\\rm or}~\\mu)^{\\rm obs}_i$ is the number of events to be \nobserved for the given oscillation parameter set,\n and $N(e~{\\rm or}~\\mu )^{\\rm exp}_i$ is the expected number of\nevents for the assumed oscillation parameters \nin the $\\chi^2$ analysis. \n$k=1,2,3$ and $4$ correspond to the four combinations \nof the detectors in Kamioka and in Korea with the \nneutrino and anti-neutrino beams,\nrespectively.\nThe index $i$ represents the reconstructed neutrino energy bin \nfor both electrons and muons. \nFor electron events, both $N(e)^{\\rm obs}_i$ and $N(e)^{\\rm exp}_i$ \ninclude background events.\nThe energy ranges of the five energy bins for electron events \nare respectively\n400-500~MeV, \n500-600~MeV, 600-700~MeV, 700-800~MeV and \n800-1200~MeV. \nThe energy range for the muon events covers \nfrom 200 to 1200~MeV. Each energy bin has 50~MeV width.\n$\\sigma_i$ denotes the\nstatistical uncertainties in the expected data. \nThe third term \nin the $\\chi^2$ definition collects the\ncontributions from variables which parameterize the systematic\nuncertainties in the expected number of signal and background events. \n\n\n$N^{\\rm BG}_i$ is the number of background events\nfor the $i^{\\rm th}$ bin for electrons.\n$N^{\\rm signal}_i$ is the number of electron appearance events\nthat are observed, and depends on \nneutrino oscillation parameters.\nThe uncertainties in $N^{\\rm BG}_i$ and $N^{\\rm signal}_i$\nare represented by 4 parameters $\\epsilon_j$ ($j=1$ to 3 and 7).\nSimilarly, $N^{\\rm non-QE}_i$ are the number of non-quasi-elastic events\nfor the $i^{\\rm th}$ bin for muons. \n$N^{\\rm QE}_i$ are the number of quasi-elastic muon events.\nWe treat the non-quasi-elastic and quasi-elastic muon events separately,\nsince the neutrino energy cannot be properly reconstructed for\n non-quasi-elastic events.\nBoth $N^{\\rm non-QE}_i$ and $N^{\\rm QE}_i$ depend on \nneutrino oscillation parameters.\nThe uncertainties in $N^{\\rm non-QE}_i$ and $N^{\\rm QE}_i$ \nare represented by 4 parameters $\\epsilon_j$ ($j=4$ to 7). \n\n\nDuring the fit, the values of $N(e~{\\rm or}~\\mu)^{\\rm exp}_i$ are\nrecalculated \nfor each choice of the oscillation parameters which are varied freely \nto minimize $\\chi^2$, and so are the systematic error parameters \n$\\epsilon_j$. \nThe parameter $f(e~{\\rm or}~\\mu)^i_j$ represents \nthe fractional change in the predicted\nevent rate in the $i^{\\rm th}$ bin due to a variation of the parameter\n$\\epsilon_j$. \nThe overall background normalization\nfor electron events \nis assumed to be uncertain by $\\pm$5\\% ($\\tilde{\\sigma}_{1}$=0.05). \n It is also assumed that the background events for electron\n events have an energy dependent uncertainty with the functional\n form of $f(e)_2^i=((E_\\nu(rec)-800~\\text{MeV}) \/ 400~\\text{MeV})$. \n 5\\% is assumed to be the\n uncertainty in $\\epsilon_2$ ($\\tilde{\\sigma}_{2}$=0.05).\n The functional form of\n $f(\\mu)_4^i = (E_\\nu(rec)-800~\\text{MeV}) \/ 800~\\text{MeV}$\n is used to define the uncertainty in the spectrum shape for\n muon events ($\\tilde{\\sigma}_{4}$=0.05).\nThe uncertainties in the signal detection efficiency \nare assumed to be 5\\% for both electron and muon events\n($\\tilde{\\sigma}_{3}=\\tilde{\\sigma}_{5}$=0.05).\nThe uncertainty in the separation of quasi-elastic and \nnon-quasi-elastic interactions in the muon events is\nassumed to be 20\\% ($\\tilde{\\sigma}_{6}$=0.20).\nThese systematic errors are assumed to be not correlated between\nthe electron and muon events.\nIn addition, for the number of events in Korea, the possible \nflux difference between Kamioka and Korea is taken into account\nin $f(e~{\\rm or}~\\mu)_7^i$. The predicted flux \ndifference \\cite{Rubbia-Meregaglia-Seoul2006} \nis simply assumed to be the 1~$\\sigma$ uncertainty in the flux \ndifference ($\\tilde{\\sigma}_{7}$).\n\n\n\\subsection{Sensitivity with two-detector complex}\n\n\nNow we present the results of the sensitivity analysis for \nthe $\\theta_{23}$ octant degeneracy. \nThe results for the mass hierarchy as well as CP violation sensitivities \nwill be discussed in the next section.\nFig.~\\ref{sensitivity-theta23-octant} shows\nthe sensitivity to the $\\theta_{23}$ octant determination as a function of \n$\\sin^2 2 \\theta_{13}$ and $\\sin^2 \\theta_{23}$. \nThe areas shaded with light (dark) gray of this figure indicate \nthe regions of parameters where the octant of $\\theta_{23}$ \ncan be determined at 2 (3) standard deviation confidence level, \nwhich is determined by the condition \n$\\chi^2_{\\text{min}} \\text{(wrong~octant)}\n-\\chi^2_{\\text{min}}\\text{(true~octant)}>$ 4 (9). \nThe upper (lower) panels correspond to the case where\nthe true hierarchy is normal (inverted). \nNote, however, that the fit was performed without assuming \nthe mass hierarchy. \nSince the sensitivity mildly depends on the CP phase $\\delta$, \nwe define the sensitivity to resolving the octant degeneracy \nin two ways: \nthe left (right) panels correspond to the case where \nthe sensitivity is defined such that the octant is determined \nfor any value of delta (half of the $\\delta$ space). \nFrom this figure, we conclude that the experiment \nwe consider here is able to solve the octant ambiguity,\nif $\\sin^2 \\theta_{23} < 0.38\\,(0.42) $ or $>0.62\\,(0.58)$ at 3\n(2) standard deviation confidence level. This conclusion depends\nweakly on the value of $\\sin^2 2 \\theta_{13}$, as well \nas the value of the CP phase $\\delta$ and the mass hierarchy.\n\n\n\\begin{figure}[htbp]\n\\vglue 0.3cm\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{region2.1.eps}\n\\end{center}\n\\vglue -0.3cm\n\\caption{ 2 (light gray area) and 3 (dark gray area) \nstandard deviation sensitivities to the $\\theta_{23}$\noctant degeneracy for 0.27~Mton detectors both in Kamioka\nand Korea.\n4 years running with neutrino beam and another 4 years with \nanti-neutrino beam are assumed.\nIn (a), the sensitivity is defined so that the experiment \nis able to identify the octant of $\\theta_{23}$ for any \nvalues of the CP phase $\\delta$. In (b), it is defined so that the \nexperiment is able to identify the octant of $\\theta_{23}$ \nfor half of the CP $\\delta$ phase space.\n}\n\\label{sensitivity-theta23-octant}\n\\end{figure}\n\n\nThe sensitivity of lifting the octant degeneracy by this setting \nis quite high even for rather small values of $\\theta_{13}$ to \n$\\sin^2 2 \\theta_{13} \\sim 10^{-3}$ where the mass hierarchy is \nnot determined, a possible consequence of the decoupling. \nSee Figs.~\\ref{sensitivity-mass-hierarchy} in the next section. \nThe sensitivity depends very weakly on $\\theta_{13}$ \nin relatively small values of \n$\\sin^2 2\\theta_{13}$ where the dominant atmospheric terms \nare small.\nThe feature of almost independence of the sensitivity to $\\theta_{13}$ \nshould be contrasted with that of the accelerator-reactor combined \nmethod in which a strong dependence on $\\theta_{13}$ \nis expected \\cite{resolve23}. \nVery roughly speaking the sensitivity by the present method is \nbetter than the latter method in a region \n$\\sin^2 2 \\theta_{13} \\lsim 0.05-0.06$ \naccording to the result given in Fig.~8 of \\cite{resolve23}. \nThe sensitivity of our method is also at least comparable to that could \nbe achieved by the high statistics observation of atmospheric neutrinos \n\\cite{concha-smi_23,atm23,choubey2}. \n\n\n\\section{Reexamination of sensitivities to neutrino mass hierarchy and CP violation} \n\\label{revisit}\n\n\nIn this section we reexamine the problem of sensitivities \nto the neutrino mass hierarchy and CP violation \nachievable by the Kamioka-Korea identical two detector complex. \nWe want to verify that the sensitivities do not depend on \nwhich octant $\\theta_{23}$ lives, as indicated by our \ndiscussion of the decoupling given in Sec.~\\ref{decoupling}. \nIt is also interesting to examine how the \nsensitivities depend upon $\\sin ^2 2\\theta_{23}$. \nFurthermore, the inclusion of the new systematic error which accounts \nfor difference in the spectral shapes of the neutrino beam between \nthe intermediate and the far detectors makes the reexamination \nworth to do. \n\n\n\\begin{figure}[htbp]\n\\vglue 0.3cm\n\\begin{center}\n\\includegraphics[width=0.73\\textwidth]{region.1.eps}\n\\end{center}\n\\vglue -0.3cm\n\\caption{ 2(thin lines) and 3(thick lines) \nstandard deviation sensitivities to the mass hierarchy \ndetermination for\nseveral values of $\\sin ^2 2 \\theta_{23}$ (red, yellow, \nblack, green and blue lines show the results for \n$\\sin^2 \\theta_{23} =$ 0.40, 0.45, 0.50, 0.55 and 0.60, \nrespectively). The sensitivity is defined in the plane\nof $\\sin ^2 2 \\theta_{13}$ versus CP phase $\\delta$.\nThe top and bottom panels show the cases for positive and \nnegative mass hierarchies, respectively. The experimental\nsetting is identical to that in \nFig.\\ref{sensitivity-theta23-octant}. \n}\n\\label{sensitivity-mass-hierarchy}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\vglue 0.3cm\n\\begin{center}\n\\includegraphics[width=0.73\\textwidth]{region.2.eps}\n\\end{center}\n\\vglue -0.3cm\n\\caption{ Sensitivities to the CP violation, \n$\\sin \\delta \\ne0$. \nThe meaning of the lines \nand colors are identical to that in \nFig.~\\ref{sensitivity-mass-hierarchy}.\n}\n\\label{sensitivity-CP}\n\\end{figure}\n\n\nIn Figs.~\\ref{sensitivity-mass-hierarchy} and \\ref{sensitivity-CP} \nthe regions sensitive to the mass hierarchy and CP violation, \nrespectively, are presented. \nIn both figures, the thin-lines and the thick-lines \nindicate the sensitivity region at 2 and 3 standard deviations, \nrespectively.\nAs in the previous work~\\cite{T2KK}, 2 (3) standard deviation \nsensitivity regions are defined by the conditions, \n$\\chi^2_{\\text{min}} \\text{(wrong~hierarchy)}-\\chi^2_{\\text{min}}\\text{(true~hierarchy)}>$\n4 (9) and \n$\\chi^2_{\\text{min}} (\\delta = 0\\ \\text{or}\\ \\pi)\n-\\chi^2_{\\text{min}}(\\text{true value of}\\, \\delta)>$\n4 (9) for the mass hierarchy and CP violation, respectively. \n\n\nThe sensitivities to the mass hierarchy and CP violation \nat $\\sin ^2 \\theta_{23} = 0.5$ are almost identical to those obtained in \n\\cite{T2KK}. \nIt is evident that the sensitivities do not depend strongly on \n$\\sin ^2 \\theta_{23}$ as far as the value is between 0.40 and 0.60.\nIn fact, the mass hierarchy can be determined even if \nthe $\\theta_{23}$ octant degeneracy is not resolved. \nBut, the sensitivity to mass hierarchy resolution gradually improves \nas $\\sin ^2 \\theta_{23}$ becomes larger, as seen in \nFig.~\\ref{sensitivity-mass-hierarchy}. \nIt is natural because $\\Delta P^{\\,\\text{norm~inv}}$ in (\\ref{DeltaP_sign}), \nor the appearance probability itself is proportional to $\\sin ^2 \\theta_{23}$. \nAn alternative way of presenting the same result is to use \n$s^2_{23} \\sin ^2 2\\theta_{13}$ for the ordinate. \nAn approximate scaling behavior is observed as expected \nby $\\Delta P^{\\,\\text{norm~inv}}$ in (\\ref{DeltaP_sign}).\n\n\n\n\\section{Summary and Discussion}\n\\label{summary}\n\n\nIn this paper, we have shown that a setting with two identical water \nCherenkov detectors of 0.27 Mton fiducial mass, one in Kamioka \nand the other in Korea, which receive almost the same neutrino beam \nfrom J-PARC has capability of resolving the $\\theta_{23}$ \noctant degeneracy {\\em in situ} by observing difference of the \nsolar oscillation term between both detectors. \nThe feature of the sensitivity region indicates that \nthe present method is quite complementary to the reactor-accelerator \ncombined method explored in \\cite{resolve23}. \nTogether with the potential for resolution of the intrinsic and the \nsign-$\\Delta m^2_{31}$ degeneracies previously reported in \\cite{T2KK} \n(with confirmation in Sec.~\\ref{revisit} by an improved treatment), \nwe have demonstrated that the Kamioka-Korea two detector complex \ncan resolve all the eight-fold neutrino parameter degeneracy \nunder the assumption that $\\theta_{13}$ is within reach by the next \ngeneration accelerator experiments and $\\theta_{23}$ \nis not too close to $\\pi\/4$. \n\n\nAs an outcome of these studies, the strategy toward determination \nof the remaining unknowns in the lepton flavor mixing can be\ndiscussed. \nIt is nice to see that such program can be defined only with the single \nexperiment based on the conventional superbeam \ntechnology which does not require long-term R$\\&$D efforts, \nand the well established detector technology.\nIt opens the possibility of accurate determination of the neutrino mixing \nparameters, $\\theta_{23}$, $\\theta_{13}$, $\\delta$, \nas well as the neutrino mass hierarchy, \nby lifting all the eight-fold degeneracy \nwhich should merit our understanding of physics of lepton sector. \n\n\nOur treatment in this paper includes a new systematic error which \naccounts for possible difference in spectral shape of the neutrino beam \nreceived by the two detectors in Kamioka and in Korea. \nWe have shown that, despite the existence of such new uncertainty \nwhich might hurt the principle of near-far cancellation of the \nsystematic errors, the capability of determining neutrino mass hierarchy \nand sensitivity to CP violation are kept intact. \n\n\nWe have also reported a progress in understanding the theoretical \naspect of the problem of how to solve the parameter degeneracy. \nBecause of the property phrased as ``decoupling between degeneracies'' \nwhich is shown to hold in a setting that allows perturbative treatment of \nmatter effect, \none can try to solve a particular degeneracy without worrying \nabout the presence of other degeneracies. \nThis feature may be contrasted to those \nof the very long baseline approaches, such as the neutrino factory, \nin which one would not expect the discussion in this paper to \nhold.\n\n\nAn alternative but closely related approach toward determination \nof the global structure of lepton flavor mixing in a single experiment \nis to utilize an on-axis wide band neutrino beam to explore the \nmultiple oscillation maxima, which may be called the ``BNL strategy'' \n\\cite{BNL,FNALversion}. \nThis strategy can be applied to the far detector in Korea, as examined \nby several authors \\cite{Hagiwara,Dufour-Seoul2006,Rubbia-Seoul2006}.\\footnote{\nVery roughly speaking ignoring the issue of backgrounds\nand assuming the same baseline length, \none would expect that wide band beam option is better in sensitivity \nto the neutrino mass hierarchy, \nwhile the same off-axis angle option studied in this paper is \nadvantageous to resolve the $\\theta_{23}$ octant degeneracy \nfor which low energy bins are essential.\n}\nIn this case, however, one needs to understand the energy \ndependence of the background and the signal efficiency \nas well as the neutrino interaction cross section precisely \nfor both the intermediate and the far detectors.\nIn particular, since the low energy bins are enriched with \nneutral current background contamination that comes from \nevents with higher neutrino energies \n\\cite{Dufour-Seoul2006} \nthe cancellation of the systematic errors between \nthe two detectors, which is the key ingredient in our analysis, \ndoes not hold. \nNonetheless, we emphasize that the potentially powerful method is \nworth to examine further with realistic estimate of the \ndetector performance.\n\n\nFinally, we remark that the J-PARC 2.5 degree off-axis beam\nwith the baseline length of 1,000 to 1,250~km should be \navailable in the Korean Peninsula. Therefore, it may be possible to\nfurther enhance the sensitivity to the $\\theta_{23}$ octant\nby taking a longer baseline length for the Korean detector.\nThe best baseline length and the detector location \nshould be decided so that the experiment has the best \nsensitivities to the oscillation parameters, especially\nto the CP phase $\\delta$, mass hierarchy and the octant of \n$\\theta_{23}$. \n\n\n\\begin{acknowledgments}\nWe would like to thank M.~Ishitsuka and K.~Okumura for \nthe assistance in the analysis code. \nH.N. thanks Stephen Parke and Olga Mena for useful discussion. \nThis work was supported in part by the Grant-in-Aid for Scientific Research, \nNos. 15204016 and 16340078, Japan Society for the Promotion of Science, \nFunda\\c{c}\\~ao de Amparo \\`a Pesquisa do Estado de Rio de Janeiro (FAPERJ) \nand by Conselho Nacional de Ci\\^encia e Tecnologia (CNPq). \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe hierarchical model of structure formation predicts that clusters\nof galaxies form by subsequent merging of smaller structures. A\nconsistent amount of energy ($\\sim 10^{63} - 10^{64}$ ergs) is\nreleased in the Intra Cluster Medium (ICM) as result of such merger\nevents. Cosmological numerical simulations have shown that shocks and\nturbulence associated with these processes do not only heat the ICM\nbut also play an important role in non-thermal phenomena occurring in\nthe ICM (see e. g. the review by Dolag et al. 2008). Although these\nprocesses are not completely understood yet, this energy input is\nsupposed to accelerate and inject relativistic particles on cluster\nscale and amplify the magnetic field in the ICM.\\\\ Extended radio\nsources on cluster scale not associated with any optical counterpart\nbut arising from the ICM have been detected in an increasing number of\ngalaxy clusters. They are called radio halos and radio relics\ndepending on their morphology, location and radio properties. \\\\ Radio\nrelics are extended radio sources located at the outskirts of galaxy\nclusters, and are strongly polarized, with linear fractional\npolarization at 20 cm above 10 $\\%$, reaching values up to 50 $\\%$ in\nsome regions (see e.g. Govoni \\& Feretti 2004; Ferrari et\nal. 2008). Their origin is debated and not well known. There is a\ngeneral consensus that it is related to phenomena occurring in the ICM\nduring merging events. So far, there are $\\sim$ 20 clusters of\ngalaxies where at least one radio relic is present. Their radio\nmorphology and location are quite varied, and could reflect different\nphysical origin or ICM conditions (Kempner et al. 2004; Giovannini \\&\nFeretti 2004). Due to low X-ray brightness at the cluster periphery, a\ncomparison of relic properties with the surrounding medium\n(i. e. temperature and brightness gradient induced by shock waves) is\nnot obvious. Feretti \\& Neumann (2006) did not find any evidence of a\ntemperature jump nearby the Coma cluster relic, and only recently a\ntemperature gradient has been found nearby the relic in Abell 548b by\nSolovyeva et al. (2008); here a precise orientation of the cluster\nmerger with respect to the line of sight is inferred, and the\nprojected displacement of the relic from the shock is thus\nexplained.\\\\ Of particular interest is to explore the connection between\nmerger shock waves and clusters with double relics,\ni.e. clusters hosting two relic radio sources located in the\nperipheral region and symmetric with respect to the cluster center. So\nfar a very small number of clusters with two double relics has been\nfound. One of them is Abell 3667 (R\\\"ottgering et al. 1997;\nJohnston-Hollitt et al. 2002). Here the cluster X-ray emission shows\nan elongated shape, interpreted as the merger axis of two\nsub-clusters, and relics are displaced symmetrically and perpendicular\nto the main axis. X-ray, optical and radio properties have been\nreproduced by a numerical simulation of a merger between clusters with\nmass ratio of 0.2 by Roettiger et al. (1999). We note however that not\nall of the predictions made by such simulations could be tested with\navailable data. Apart from Abell 3667, double relics have been\nobserved in Abell 3376 (Bagchi et al. 2006), and interpreted as\n``Outgoing merger shock waves''. Double relics have also been observed\nin RXCJ 1314.4-2515 (Feretti et al. 2005; Venturi et al. 2007), but no\ndetailed study on the relics formation has been performed on this\ncluster so far. Two more candidates for hosting double relics are\nAbell 2345 (Giovannini et al. 1999) and Abell 1240 (Kempner \\& Sarazin\n2001).\\\\\n\\begin{table*} [t!]\n\\caption{VLA observations} \n\\label{tab:radioobs} \n\\centering \n\\begin{tabular}{|c c c c c c c c|} \n\\hline\\hline \nSource & RA & DEC &$\\nu$&Bandwidth&Config.& Date & Duration \\\\\n & (J2000) & (J2000) & (MHZ) & (MHz) & & & (Hours) \\\\\\hline \n\\hline\nAbell 2345 & 21 27 12.0 & -12 10 30.0 & 325 & 3.125 & B &16-AUG-2006 & 2.0 \\\\% 147779\\\\ \n & & & 325 & 3.125 & C &08-DEC-2006 & 5.4 \\\\%263514\\\\\nAbell 1240 & 11 23 37.0 & 43 05 15.0 & 325 & 3.125 & B &05-AUG-2006 & 2.6\\\\% 214384\\\\\n & & & 325 & 3.125 & C &08-DEC-2006 & 4.7 \\\\%203675\\\\\n\\hline\nAbell 2345-1 & 21 26 43.0 & -12 07 50.0 & 1425 & 50 & C &08-DEC-2006 & 1.9 \\\\%185755 \\\\%(24)\n & 21 26 43.0 & -12 07 50.0 & 1425 & 50 & D &09-APR-2007 & 1.0 \\\\%98376 \\\\%(24)\nAbell 2345-2 & 21 27 36.0 &-12 11 25.0 & 1425 & 50 & C &08-DEC-2006 & 2.0 \\\\%196021\\\\%(24)\n & 21 27 36.0 &-12 11 25.0 & 1425 & 50 & D &09-APR-2007 & 1.0 \\\\%108368\\\\%(24)\nAbell 1240-1 & 11 23 25.0 & 43 10 30.0 & 1425 & 50 & C &08-DEC-2006& 1.8 \\\\% 177879\\\\\n & 11 23 25.0 & 43 10 30.0 & 1425 & 50 & D &12-APR-2007& 1.0 \\\\%80364 \\\\\nAbell 1240-2 & 11 23 50.0 & 43 00 20.0 & 1425 & 50 & C &08-DEC-2006& 1.9 \\\\%189945 \\\\\n & 11 23 50.0 & 43 00 20.0 & 1425 & 50 & D &12-APR-2007& 1.0 \\\\%78099\\\\ \n\\hline\nAbell 2345 & 21 26 57.2 & -12 12 49 & 1490 & 50 & AnB &02-NOV-1991& 0.1\\\\\n\\hline\n\\multicolumn{8}{l}{\\scriptsize Col. 1: Source name; Col. 2, Col. 3: Pointing position (RA, DEC);\nCol. 4: Observing frequency;}\\\\\n\\multicolumn{8}{l}{\\scriptsize Col 5: Observing bandwidth; Col. 6: VLA configuration; \nCol. 7: Dates of observation; Col. 8: Net time on source.}\\\\ \n\\end{tabular}\n\\end{table*}\nWe present here\nnew Very Large Array (VLA) observations of these two clusters at 20\nand 90 cm to confirm and study the double relic emission in the\nframework of relic formation models. Spectral index analysis of both\nradio relics in the same cluster have not been performed so far. In\nAbell 3667 the spectral index image has been obtained for only one of\nthe two relics, and no spectral index information are available for\nrelics in Abell 3376. Only integrated spectral index information are\navailable for the relics in RXCJ 1314.4-2515. Study of the spectral\nindex and of the polarization properties of relics offers a powerful\ntool to investigate the connection between double relics and outgoing\nshock waves originating in a merger event. In fact, theoretical models\nand numerical simulations make clear predictions on the relic spectral\nindex trend and magnetic field properties (see Ensslin et al.\n1998; Roettiger et al. 1999; Hoeft \\& Br{\\\"u}ggen 2007).\\\\\nThe paper is\norganized as follows: in Sec. 2 observations and data reduction are\ndescribed, in Sec. 3 and 4 we present the analysis of the cluster\nAbell 2345 and Abell 1240. Results are discussed in Sec. 5 in\ncomparison with other observed relics and theoretical models, and\nconclusions are presented in Sec. 6. We assume a $\\Lambda$CDM\ncosmological model with $H_0=$71 km s$^{-1}$ Mpc$^{-1}$, $\\Omega_M$=0.27,\n$\\Omega_{\\Lambda}$=0.73.\n\n\\section{VLA radio observations}\n\\subsection{Total intensity data reduction}\n\\label{sec:radioobs}\nObservations have been performed at the Very Large Array (VLA) at 20\ncm in the C and D configuration and at 90 cm in the B and C\nconfiguration, in order to obtain the same spatial frequency coverage\nin the UV plane. Observations details are given in\nTab. \\ref{tab:radioobs}. \\\\ {\\bf Observations at 20 cm (1.4 GHz)} have\nbeen pointed separately on the two relics in both of the clusters\nbecause of the smaller full width at half power of the primary\nbeam. Observations of the cluster Abell 1240 have been calibrated\nusing the source 3C286 as primary flux density calibrator\\footnote{we\nrefer to the flux density scale by Baars \\& Martin (1990)}. The source\n1156+314 has been observed at intervals of about 30 min and used as\nphase calibrator. Observations of Abell 2345 have been calibrated\nusing the sources 3C48 as primary flux density calibrator. Phase\ncalibration has been performed by observing the source 2137-207 at\nintervals of $\\sim$ 30 min.\\\\ We performed standard calibration and\nimaging using the NRAO Astronomical Imaging Processing Systems\n(AIPS). Cycles of phase self-calibration were performed to refine\nantennas phase solutions, followed by a final amplitude and gain\nself-calibration cycle.\\\\ In addition we recovered from the VLA data\narchive a short observation performed with AnB array. The source 3C48\nis used as primary flux density calibrator and the source 2121+053 is\nused as phase calibrator. We reduced and calibrated these data as\nexplained above, details are given in\nTab. {\\ref{tab:radioobs}}. \\\\{ \\bf Observations at 90 cm (325 MHz)}\nhave been performed in the spectral line mode, using 32 channels with\n3.127 MHz bandwidth. This observing method avoids part of the VLA\ninternal electronics interferences and allows us to remove accurately\nRadio Frequency Interferences (RFI). This also reduces bandwidth\nsmearing, that is quite strong at low frequencies. Primary flux\ndensity and phase calibrators were the same sources used in 1.4 GHz\nobservations. 3C48 and 3C286 were also used for bandpass\ncalibration. RFI are particularly strong at low radio frequency, so\nthat an accurate editing has been done channel by channel, resulting\nin a consistent flag of data. This in conjunction with bad data coming\nfrom EVLA antennas results in a loss of $\\sim$ 40 \\% of observing\ntime. Calibration has been performed following the ``Suggestions for\nP band data reduction'' by Owen et al. (2004).\\\\ After the initial\nbandpass calibration channels from 1 to 4 and from 28 to 32 have been\nflagged because of the roll-off of the bandpass. In the imaging\nprocedure data have been averaged to 8 channels. Imaging has been\nperformed using the wide field imaging technique to correct for non\ncomplanarity effects over a wide field of view. 25 facets covering\nthe main lobe of the primary beam have been used in the cleaning and\nphase-self calibration processes. We also searched in\nthe NVSS data archive for sources stronger then 0.5 Jy over a radius\nas large as 10$^{\\circ}$. These sources have been included in the\ninitial cleaning and self calibration steps.\\\\ Each (u,v) data set at\nthe same frequency but observed with different configurations has been\ncalibrated, reduced and imaged separately and then combined to produce\nthe final images. Images resulting from the separate pointed\nobservations at 1.4 GHz have been then linearly combined with the AIPS\ntask LTESS. We combined the data set and produced images at higher and\nlower resolution (herein after HR images and LR images) giving uniform\nand natural weight to the data. For the purposes of the spectral\nanalysis, the final images at 325 MHz and 1.4 GHz, have been restored\nwith the same beam (reported in Tab. \\ref{tab:2345radiofinali} and\n\\ref{tab:1240radiofinali}) and corrected for the primary beam effects.\n\\begin{table}[h!]\n\\caption{ Abell 2345}\n\\label{tab:2345radiofinali}\n\\centering\n\\begin{tabular} {|l c c c c|} \n\\hline\\hline\nSource name & $\\nu$ & $\\theta$ & $\\sigma_{I}$ & $~~~$Fig. \\\\\n & MHz & arcsec & mJy\/beam & \\\\\n\\hline\nAbell 2345-1 HR & 1425 & 37 X 20 & 0.08 & \\\\\nAbell 2345-1 LR & 1425 & 50 X 38 & 0.09 & \\ref{fig:A2345_spixcut}, central panel\\\\\nAbell 2345-2 HR & 1425 & 37 X 20 & 0.09 & \\\\\nAbell 2345-2 LR & 1425 & 50 X 38 & 0.09 & \\ref{fig:A2345_spixcut}, central panel\\\\\nAbell 2345 HR & 325 & 37 X 20 & 1.7 & \\\\\nAbell 2345 LR & 325 & 50 X 38 & 2.0 & \\ref{fig:A2345_spixcut}, right panels\\\\\n\\hline\nAbell 2345 & 1490 &6X6 & 0.13 & \\ref{fig:A2345_otticoradio}, central panel\\\\\n\\hline\n\\multicolumn{5}{l}{\\scriptsize Col. 1: Source name; Col. 2: Observation frequency;\n}\\\\\n\\multicolumn{5}{l}{\\scriptsize Col. 3: Restoring beam; Col. 4: RMS noise of the \nfinal images;}\\\\\n\\multicolumn{5}{l}{\\scriptsize Col 5: Figure of merit.}\n\\end{tabular}\n\\end{table}\n\\begin{table}[h!]\n\\caption{ Abell 1240}\n\\label{tab:1240radiofinali}\n\\centering\n\\begin{tabular} {|l c c c c|} \n\\hline\\hline\nSource name & $\\nu$ & $\\theta$ & $\\sigma_{I}$ & $~~~$Fig. \\\\ \n & MHz & arcsec & mJy\/beam &\\\\\n\\hline\nAbell 1240-1 HR & 1425 & 22 X 18 & 0.04&\\ref{fig:A1240_otticoradio.ps} \\\\\nAbell 1240-1 LR & 1425 & 42 X 33 & 0.04&\\ref{fig:A1240spix},central panel \\\\\nAbell 1240-2 HR & 1425 & 22 X 18 & 0.04&\\ref{fig:A1240_otticoradio.ps} \\\\\nAbell 1240-2 LR & 1425 & 42 X 33 & 0.05&\\ref{fig:A1240spix}, central panel \\\\\nAbell 1240 HR & 325 & 22 X 18 & 0.9 &\\\\\nAbell 1240 LR & 325 & 42 X 33 & 1.0& \\ref{fig:A1240spix}, left panels\\\\\n\\hline\n\\multicolumn{5}{l}{\\scriptsize Col. 1: Source name; Col. 2: Observation frequency;}\\\\\n\\multicolumn{5}{l}{\\scriptsize Col. 3: Restoring beam; \n Col. 4: RMS noise of the final images;}\\\\\n\\multicolumn{5}{l}{\\scriptsize Col. 5: Fig. of merit.}\n\\end{tabular}\n\\end{table}\n\\begin{table}[h!]\n\\caption{Total and polarization intensity radio images at 1425 MHz}\n\\label{tab:radiopol}\n\\centering\n\\begin{tabular}{|c c c c c| } \n\\hline\\hline \nSource name & $\\theta$ & $\\sigma_I$ & $\\sigma_{Q,U}$ & Fig. \\\\ \n & arcsec & \\scriptsize(mJy\/beam) & \\scriptsize(mJy\/beam)&\\\\ \n\\hline \nAbell 2345-1 & 23 X 16 & 0.05 & 0.02 & \\ref{fig:A2345_pol},\nright panel\\\\ Abell 2345-2 & 23 X 16 & 0.07 & 0.02 &\n\\ref{fig:A2345_pol}, left panel\\\\ Abell 1240-1 & 18 X 17 & 0.04 & 0.02\n&\\ref{fig:A1240pol}, top panel\\\\ Abell 1240-2 & 18 X 17 & 0.04 & 0.01\n&\\ref{fig:A1240pol}, bottom panel \\\\ \\hline\n\\multicolumn{5}{l}{\\scriptsize Col. 1: Source name; Col. 2: Restoring\nbeam; }\\\\ \\multicolumn{5}{l}{ \\scriptsize Col. 4: RMS noise of the I\nimage; Col 5: RMS noise of the Q and U images}\\\\\n\\multicolumn{5}{l}{\\scriptsize Col 6: Figure of merit.}\n\\end{tabular}\n\\end{table}\n\\subsection{Polarization intensity data reduction}\nObservations at 20 cm (1.425 GHz) include full polarization\ninformation. Polarization data observed with the D array are unusable\nbecause of bad quality of data of the polarization calibrator. The\nabsolute polarization position angle has been calibrated by observing\n3C286 for both clusters in C configuration. The instrumental\npolarization of the antennas has been corrected using the source\n1156+314 for Abell 1240 and the source 2137-207 for Abell 2345.\\\\\nStokes parameters U and Q images have been obtained. We then derived\nthe Polarization intensity image ($P=\\sqrt{U^2+Q^2}$), the\nPolarization angle image ($\\Psi=\\frac{1}{2}arctan\\frac{U}{Q}$) and the\nFractional Polarization image ($FPOL=\\frac{P}{I}$), with I being the\ntotal intensity image. Further details are given in\nTab. \\ref{tab:radiopol}.\\\\\n\\begin{table*}\n\\caption{Abell 2345 and Abell 1240 properties} \n\\label{tab:x} \n\\centering \n\\begin{tabular}{|c c c c c c c |} \n\\hline\\hline \nSource name & RA & DEC & z & scale & F$_X$ & L$_X$ \\\\\n & (J2000) & (J2000) & & (kpc\/$''$) & $10^{-12}$ erg\/s\/cm$^2$ & $10^{44}$ erg\/s \\\\\n\\hline\nAbell 2345 & 21 27 11.00 & -12 09 33.0 & 0.1765& 2.957 & 5.3 & 4.3\\\\\nAbell 1240 & 11 23 32.10 & 43 06 32 & 0.1590& 2.715 & 1.3 & 1.0\\\\\n\\hline\n\\multicolumn{7}{l}{\\scriptsize Col. 1: Source name; Col. 2, Col. 3: Cluster X-ray centre (RA,\nDEC); Col 4: Cluster redshift; Col 5: arcsec to kpc conversion scale;} \\\\\n\\multicolumn{7}{l}{\\scriptsize Col 6: Flux in the 0.1-\n2.4 keV band (Abell 2345) and in the 0.5-2 keV (Abell 1249); Col 7: X-ray cluster luminosity }\\\\\n\\multicolumn{7}{l}{\\scriptsize in the 0.1-2.4 keV band (Abell 2345) and in the 0.5-2 keV (Abell 1240);}\\\\\n\\multicolumn{7}{l}{\\scriptsize Data from B\\\"ohringer et al. (2004) for Abell 2345 and from\nDavid et al. (1999) for Abell 1240, corrected for the adopted cosmology.}\n\\end{tabular}\n\\end{table*}\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=19.5cm]{comboA2345otticoradio.ps}\n\\caption{The cluster Abell 2345. In the center: DPOSSII optical\nemission (red band) in colors overlaid onto radio contours at 1.490\nGHz. First contours are $\\pm$ 0.4 mJy\/beam and are then spaced by a\nfactor 2. The beam in 6$''$$\\times$6$''$. Top inset shows the zoomed\nimages of the central sources: the central cD galaxy and two radio\ngalaxies are visible. Bottom inset shows the zoomed image of the\nsouthern radio source. Red Boxes mark the region of the relics,\ncompletely resolved in the high resolution image. In the Left and\nRight panels zoomed image of the red boxes is shown. Here colors\nrepresent the optical DPOSSII emission, while contours represents the\nrelic radio emission at the resolution of 23$''\\times$16$''$. The\nrelic A2345-1 is visible in the right panel, while A2345-2 is in the\nleft panel. Contours start at $\\pm$0.15mJy\/beam and are spaced by a\nfactor 2. Red arrows indicate the position of the discrete sources\nembedded in the relic emission.}\n\\label{fig:A2345_otticoradio}\n\\end{figure*}\n\\section{The Cluster Abell 2345}\nOptical information are available for this cluster, while little is\nknown about its X-ray emission. General data are reported in\nTab. \\ref{tab:x}.\\\\\nWeak gravitational lensing analysis has been performed by Dahle et al.\n(2002) and by Cypriano et al. (2004). Optical data cover the inner\npart of the cluster ($\\sim$3$'$$\\times$3$'$). They find that this cluster\nhas a well defined core dominated by a cD galaxy, and both the light\nand galaxy number density distribution have several peaks close to the\ncentral galaxy. The authors derived that the projected mass\ndistribution has the most prominent peak displaced from the central cD\nby $\\sim$1.5$'$, although a secondary peak is closer to the central\ncD. No information about the possible presence of a cooling flow\nassociated with this galaxy is present in the literature. Dahle et\nal. (2002) conclude from their analysis that the cluster may be a\ndynamically young system. Cypriano et al. (2004) report the mass\ndistribution derived from weak lensing analysis and find that the best\nfit to their data is a singular isothermal ellipsoid with the main\naxis oriented in the E-W direction.\\\\\n\\smallskip\\\\ The radio emission of Abell 2345 is characterized by the\npresence of two relics visible in the NVSS (Giovannini et al. 1999).\\\\\nOur new VLA observations confirm the presence of two regions where\nnon-thermal emission is present at the cluster periphery, nearly\nsymmetrical with respect to the cluster center. These new\nobservations together with the archive data, allow the study of the\ncluster radio emission in a wide range of resolutions going from\n$\\sim$6$''$ to $\\sim$50$''$. Therefore, it is possible to separate the\ncontribution of discrete sources whose emission is not related to the\nrelic's physical properties. In Fig. \\ref{fig:A2345_otticoradio} the\nradio emission of Abell 2345 at 6$''$ resolution is shown overlaid\nonto the optical emission (taken from the Digitalized Palomar Sky\nSurvey II, red band). Two central radio-tail sources are associated\nwith optical galaxies in the cluster center. The central cD is visible\nin the optical image. Relics are not visible in this image because of\nthe lack of short baselines. This confirms that the emission detected\nin lower resolution observations is indeed extended and it is not due\nto the blending of discrete sources. In the same figure we also report\nthe radio relic emission as detected by C array observations. The\nwestern relic (Abell 2345-1) is located at $\\sim$ 1 Mpc from the\ncluster X-ray center while the eastern relic (Abell 2345-2) is $\\sim$\n890 kpc far from the cluster center (see\nTab. \\ref{tab:A2345_relics}).\\\\ There are several discrete sources in\nproximity of the western relic, A2345-1, visible in the 1.4 GHz image,\nthey are labeled with letters from A to F in the right panel of\nFig.\\ref{fig:A2345_otticoradio}. The sources A, C, D, E and F could be\nassociated with the optical galaxies visible in the DPOSSII image,\nwhereas B does not have any obvious optical identification. Optical\nemission is present at 35$''$ in NE direction from the radio\npeak. This is larger then the error associated with the beam, that is\nonly 6$''\\times$6$''$ in the highest resolution image. We can then\nconclude that no optical counterpart of the B radio source is detected\nin the DPOSSII image. The sources D E and F are not visible in the 325\nMHz image (see Fig. \\ref{fig:A2345_spixcut}, top left panel). This is\nconsistent with a radio source having a spectral index\n$<1.2$\\footnote{The spectral index $\\alpha$ is derived according to\n$S_{\\nu} \\propto \\nu^{-\\alpha}$.}. There is only one discrete source\nin proximity of the relic A2345-2, labeled with G in the\nFig.\\ref{fig:A2345_otticoradio} without any obvious optical\nidentification. This source is also detected in the higher resolution\nimage. \\\\ The whole extension of the relics is properly revealed by LR\nimages (Fig. \\ref{fig:A2345_spixcut}). The morphology of the relics\nis similar at 1.4 GHz and 325 MHz, although only the brightest regions\ncan be seen at 325 MHz due to the higher rms noise level of these\nobservations with respect to the 1.4 GHz ones. The total flux of the\nrelics at the 2 frequencies, excluding the contribution of the\ndiscrete sources, are reported in Tab. \\ref{tab:A2345_relics}, where\nthe main physical parameters are summarized.\\\\ The relic A2345-1 shows\nan elongated shape at high resolution, while at lower resolution it\nshows a weaker wide emission extending in the West direction\ni.e. toward the cluster outskirts. We note that this circular\nfilamentary morphology is not seen in other double relic sources, as\ndiscussed in the introduction.\\\\\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=19cm]{comboA2345spix.ps}\n\\caption{Center: the cluster Abell 2345 radio emission at 1.4 GHz. The\nbeam is 50$''$$\\times$38$''$. Contours start at 3 $\\sigma$ (0.24\nmJy\/beam) and are then spaced by a factor 2. The cross marks the X-ray\ncluster center. Left: Colors represent the spectral index of the\nrelic A2345-1 (top) and A2345-2 (bottom) superimposed over the radio\nemission at 325 MHz (contours). The beam is 50$''$$\\times$38$''$,\ncontours start at 3$\\sigma$ (6 mJy\/beam) and are then spaced by a\nfactor 2. Right: Spectral index error image (colors) superimposed onto\nthe emission at 325 MHz (contours are as above). }\n\\label{fig:A2345_spixcut}\n\\end{figure*}\n\\begin{figure*}[t!]\n \\label{fig:A23451spixprofile.ps}\n \\centering\n \\includegraphics[width=9cm]{comboA2345-1anuli.ps}\n \\includegraphics[width=9cm]{comboA2345-2anuli.ps}\n \\caption{Spectral index radial trend of A2345-1 (left) and A2345-2\n (right), computed in shells of $\\sim 50''$ in width. It has\n been computed excluding the contribution of the discrete\n sources. Crosses refer to spectral index values computed in\n shells where the mean brightness is $>3\\sigma$ at both 325\n MHz and 1.4 GHz. Arrows are $3\\sigma$ upper limits on the\n spectral index mean value (see text). The red cross refers\n to the cluster X-ray center, the blue cross refers to the\n center of the spherical shells. In the insets: displacement\n of the shells over which the mean spectral index has been\n computed. Circles refer to the discrete sources embedded in\n the relic emission. The red cross refers to the cluster\n X-ray center, the blue cross is the center of the spherical\n shells. }\n \\end{figure*}\n\\subsection{Spectral index analysis}\n\\label{A2345spectralindex}\nWe derived the spectral index image of the cluster's relics comparing\nthe LR images at 1.4 GHz and 325 MHz. The rms noise of the images are\nreported in Tab. \\ref{tab:2345radiofinali}. Spectral index and\nspectral index noise images are shown in\nFig. \\ref{fig:A2345_spixcut}. They have been obtained by considering\nonly pixels whose brightness is $> 3\\sigma$ at both frequencies. We\nnote that relics are more extended at 1.4 GHz than at 325 MHz. This\ncan be due to the different sensitivities at 1.4 GHz and 325\nMHz. Confusion and RFI strongly affect the low frequency image, where\nthe noise level is significantly higher than the thermal noise. A\nconsistent spectral index analysis has to consider the different\nextension at the two frequencies. In fact, as already pointed out by\nOrr\\'u et al. (2007) if we compute spectral index analysis considering\nonly regions that have a signal to noise ratio $>$ 3 at both\nfrequencies, we introduce a bias, since we are excluding a priori low\nspectral index regions, whose emission cannot be detected at 325\nMHz. For instance, the relic A2345-1 radio brightness at 1.4 GHz\ndecreases as the distance from the cluster center increases. The\nfainter region could be detected in the 325 MHz image only if its\nspectral index, $\\alpha$, were steeper than $\\sim$1.8.\\\\ In both of\nthe relics the spectral index is patchy. The spectral index rms is\n$\\sigma_{spix}\\sim$ 0.4 while the mean spectral index noise is $<$Spix\nNoise$>\\sim$ 0.1 for both relics. Thus, by comparing these two\nquantities we can conclude that spectral index features are\nstatistically significant. \\\\ Our aim here is to investigate if there\nis a systematic variation of the relic spectral index with distance\nfrom the cluster center as found in other radio relics (e.g. 1253+275\nby Giovannini et al. 1991; Abell 3667 by R\\\"ottgering et al. 1997;\nAbell 2744 by Orr\\'u et al. 2007; Abell 2255 by Pizzo et al. 2008;\nAbell 521 by Giacintucci et al. 2008; ).\\\\ In order to properly\nobtain the radial trend of the spectral index, we integrated the radio\nbrightness at 325 MHz and 1.4 GHz in radial shells of $\\sim 50''$ in\nwidth wherever the 1.4 GHz brightness is $>3\\sigma$, and then we\ncomputed the value of the spectral index in each shell. We have\nexcluded the regions where discrete radio sources are embedded in the\nrelic emission (see insets in Fig. \\ref{fig:A23451spixprofile.ps}).\nThe shells have been centered in the extrapolated curvature center of\nthe relic A2345-2, that is 2.6$'$ South the cluster X-ray\ncenter. Shells are then parallel to the relics main axis. We computed\nthe integrated brightness in each shell at 20 and 90 cm , and\ncalculated the associated error as $\\sigma\\times\\sqrt{N_{beam}}$,\nwhere $\\sigma$ is the image rms noise, $N_{beam}$ is the number of\nbeams sampled in the shell. In those shells where the brightness is\n$>3\\sigma$ in the 1.4 GHz image but $<3\\sigma$ in the 325 MHz image,\nonly upper limits on the mean spectral index can be derived. The\nspectral index profiles thus obtained are shown in\nFig. \\ref{fig:A23451spixprofile.ps}. These plots shows that the\nspectral index in the relic A2345-1 increases with distance from the\ncluster center, indicating a spectral steepening of the emitting\nparticles. The spectral index, in each shell, is rather high, going\nfrom $\\sim$1.4 in the inner rim to $\\sim$1.7 in the central rim of the\nrelic. The spectral index trend derived for the outer shells is\nconsistent with a further steepening.\\\\ The spectral index of the\nrelic A2345-2 shows instead a different trend, going from $\\sim$1.4 in\nthe inner shell to $\\sim$1.1 in the outer rim\n(Fig. \\ref{fig:A23451spixprofile.ps}).\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=1.2\\columnwidth]{A2345g1.5Xradio.ps}\n\\caption{Abell 2345 X-ray emission (colors) in the energy band 0.1-2.4\nkeV from ROSAT PSPC observations. The image has been smoothed with a\nGaussian of $\\sigma\\sim$60$''$;\ncontours represent the radio image of the cluster at 1.4 GHz. The beam\nis 50$''$$\\times$38$''$. Contours are 0.24 mJy\/beam and are then spaced\nby a factor 2. Arrows mark the position of the X1 X2 and X3 regions. }\n\\label{fig:A2345X}\n\\end{figure*}\n\\subsection{Radio-X-ray comparison}\nNo X-ray studies are present in the literature for this cluster.\nX-ray observations in the energy band $0.1-2.4$ keV have been\nretrieved by the ROSAT data archive. The cluster is $\\sim$ 100$'$\noffset from the ROSAT pointing. Observations have been performed with\nthe ROSAT PSPC detector for a total exposure time of $\\sim$ 14\nksec. After background subtraction the event file has been divided by\nthe exposure map. We smoothed the resulting image with a Gaussian of\n$\\sigma=60''$. The resulting image is shown in\nFig. \\ref{fig:A2345X}.\\\\ The X-ray emission of this cluster is\nelongated in the NW-SE direction. Two bright regions are visible at\n$\\sim$ 10$'$ and 14$'$ in N-W direction from the cluster center (we\nrefer to them as X1 and X2 respectively). The galaxy J21263466-1207214\n(RA =21h26m34.6s, DEC= -12d07m22s, z=0.178221) is close to the first\none. Another bright region is present at $\\sim$ 4$'$ South from the\ncluster center (X3). \\\\ Data presented here allow an interesting\ncomparison among cluster emission at different wavelengths. We note\nthat mass distribution from weak lensing studies (Cypriano et\nal. 2004) is well represented by an ellipsoid with the major axis\ndirected in the E-W direction and relics are found perpendicular to\nthis axis. Consistently with the optical analysis, the X-ray emission\nis elongated in the NW-SE direction, indicating a possible merger\nalong that direction, and relics are displaced perpendicular to that\naxis. In Fig. \\ref{fig:A2345X} the X-ray emission is superimposed onto\nradio contours. A2345-2 is located at the edge of the X-ray emission,\nas found in relics of Abell 3667 and A3376. A2345-1, instead, is\nlocated between eastern edge of the cluster and the X1 region, 10$'$\nfrom Abell 2345 center, and its radio emission extends toward X1.\\\\\nFrom the same figure, in the X3 region a Narrow Angle Tail radio\ngalaxy is visible in radio images at every resolution (see\nFig.\\ref{fig:A2345_otticoradio} and \\ref{fig:A2345_spixcut}). Although\nredshift is not available for this radio source, its structure favors\na connection to the cluster and\/or to the close X-ray peak. One\npossibility is that these X-ray multiple features are galaxy clumps\ninteracting with Abell 2345. \\\\ A self consistent scenario arises from\nthis analysis, indicating that the cluster Abell 2345 could be\nundergoing multiple merger with X3 and X1 groups, and this could\nexplain the peculiar properties of A2345-1. More sensitive and\nresolved X-ray observations in conjunction with optical studies are\nrequired to shed light on the connection between the radio emission of\nA2345, X1 X2 and X3.\\\\\n\\subsection{Equipartition Magnetic field}\n\\label{secs:Equip}\nUnder the assumption that a radio source is in a minimum energy\nconditions, it is possible to derive an average estimate of the\nmagnetic field strength in the emitting volume (see e.g. Pacholczyk\n1970). We assume that the magnetic field and relativistic particles\nfill the whole volume of the relics, and that energy content in\nprotons and electrons is equal. We further assume that the volume of\nthe relics is well represented by an ellipsoid having the major and\nminor axis equal to the largest and smallest linear scale visible in\nour images; we estimated the third axis to be the mean between the\nmajor and minor one. The synchrotron luminosity is calculated from a\nlow frequency cut-off of 10 MHz to a high frequency cut-off of 10\nGHz. The emitting particle energy distribution is assumed to be a\npower law in this frequency range ($N(E)\\propto E^{-p}$), with\n$p=2\\alpha+1$. We used the mean value of $\\alpha=$1.5 and 1.3 for\nA2345-1 and A2345-2 respectively and found $B_{eq}\\sim$1.0 $\\mu$G in\nA2345-1 and 0.8 $\\mu$G in A2345-2. These values are consistent with\nequipartition magnetic field found in other relics.\\\\ It has been\npointed out by Brunetti et al. (1997) that synchrotron luminosity\nshould be calculated in a fixed range of electron energies rather than\nin a fixed range of radio frequencies (see also Beck \\& Krause\n2005). In fact, electron energy corresponding to a fixed frequency\ndepends on the magnetic field value, and thus the integration limits\nare variable in terms of the energy of the radiating particles. Given\nthe power law of the radiating particles and the high value of the\nradio spectral index, the lower limit is particularly relevant\nhere. We adopted a low-energy cut off of $\\gamma_{min}$=100 and\nassumed $\\gamma_{max}>>\\gamma_{min}$, obtaining $B'_{eq}\\sim$ 2.9\n$\\mu$G in A2345-1 and 2.2 $\\mu$G in A2345-2.\\\\ We derived the minimum\nnon thermal energy density in the relic sources from $B'_{eq}$\nobtaining $U_{min}\\sim$8.1 and 4.3 10$^{-13}$erg\/cm$^{-3}$ for A2345-1\nand A2345-2. The corresponding minimum non-thermal pressure is then\n$\\sim$5.0 and 2.7 10$^{-13}$erg\/cm$^{-3}$. \\\\ We are aware that\n the extrapolation to low energies or frequencies could over estimate\n the number of low energy electrons, leading to over estimate the\n equipartition magnetic field if a spectral curvature is present. We\n note that a detailed study of the radio spectrum on a large\n frequency range is available for three peripheral relics: the one in\n Abell 786, in the Coma cluster (see Giovannini \\& Feretti, 2004 and\n references therein) and in Abell 521 (Giacintucci et al. 2008). In\n these relics a straight steep radio spectrum is observed. We also note that a\n low frequency cut-off of 10 MHz and a magnetic field of\n $\\sim$1$\\mu$G imply a low energy cut-off of\n $\\gamma_{min}\\sim$1500. Thus, also if the spectrum of the emitting\n particles is truncated at $\\gamma>$1500, both $B'_{eq}$ and $B_{eq}$\n could over estimate the magnetic field strength. Future\n low-frequency radio interferometers such as SKA and LWA will likely shed\n light on this point. On the other hand, it is possible to derive an\n independent estimate of the magnetic field from X-ray flux due to\n inverse Compton scattering of CMB photons by relativistic electrons\n in the relic source. These studies have been performed so far on a\n scarce number of radio relics and have led to lower limits on the\n magnetic field strength: $B>$0.8$\\mu$G in the relic 1140+203 of\n Abell 1367 (Henriksen \\& Mushotzky 2001); $B>$1.05$\\mu$G in 1253+275\n of the Coma cluster (Feretti \\& Neumann 2006; $B>$0.8$\\mu$G in\n 0917+75 in Rood27 cluster (Chen et al. 2008); and $B>$2.2$\\mu$G in\n the relic 1401-33 in the Abell S753 cluster (Chen et al. 2008). In\n these cases, the lower limits derived from IC arguments are\n consistent with equipartition estimates, thus indicating that the\n equipartition value could be used as a reasonable approximations of\n the magnetic field strength in relics.\n\\begin{table*} [t]\n\\caption{ Abell 2345}\n\\label{tab:A2345_relics}\n\\centering\n\\begin{tabular} {|c c c c c c c |} \n\\hline\\hline\nSource name & Proj. dist & LLS & F$_{20 cm}$ & F$_{90 cm}$ & B$_{eq}$ - B$'_{eq}$& $<\\alpha>$ \\\\ \n & kpc & kpc & mJy & mJy & $\\mu$G & \\\\\n\\hline\nAbell 2345-1 & 340$''$=1000& 390$''$= 1150 &30.0$\\pm$0.5& 291$\\pm$ 4 & 1.0 -2.9 & $1.5\\pm$0.1 \\\\\nAbell 2345-2 & 300$''$=890 & 510$''$= 1500 &29.0$\\pm$0.4& 188$\\pm$ 3 & 0.8 -2.2 & $1.3\\pm$0.1 \\\\\n\n\\hline \\multicolumn{7}{l}{\\scriptsize Col. 1: Source name; Col. 2:\nprojected distance from the X-ray centroid; Col. 3: Largest linear\nscale measured on the 20 cm images.}\\\\ \n\\multicolumn{7}{l}{\\scriptsize Col. 4 and 5: Flux density at 20 and 90 cm; Col. 6: equipartition\nmagnetic field computed at fixed frequency - fixed energy }\\\\\n \\multicolumn{7}{l}{\\scriptsize (see Sec. \\ref{secs:Equip}); Col. 7: mean spectral index in\nregion where both 20 and 90 cm surface brightness is $>$ 3 $\\sigma$}\n\\end{tabular}\n\\end{table*}\n\\begin{figure*}[t!]\n \\centering \\includegraphics[width=18cm]{combopol.ps}\n \\caption{Abell 2345: Polarized emission of Abell 2345 at 1,4\n GHz. In the central panel the polarized radio emission at 1.4 GHz\n is shown. The restoring beam is $23''\\times 16''$. Left and\n right panels: contours refer to the radio image Abell 2345-2 and\n Abell 2345-1 (see Tab. \\ref{tab:radiopol} for further\n details). Contours start from $3 \\sigma$ and are spaced by a\n factor 2. E vectors are superimposed: line orientation indicates\n the direction of the E field, while line length is proportional\n to the polarization intensity (Left panel: 1$''$ corresponds to\n 5.5 $\\mu$Jy\/beam; Right panel: 1$''$ corresponds to 10\n $\\mu$Jy\/beam)}\n \\label{fig:A2345_pol}\n \\end{figure*} \n\\subsection{Polarization analysis}\n\\label{sec:A2345pol}\nAnother important set of information about the magnetic field in the\nrelics can be derived through the study of polarized emission. As\npreviously mentioned, we could calibrate polarization only for\nobservations at 1.4 GHz with the C array.\\\\ In Fig. \\ref{fig:A2345_pol} the P radio image of\nthe cluster is shown. The noise achieved in the P, Q and U images\n(Tab. \\ref{tab:radiopol}) are lower than those obtained in total\nintensity image. In fact total intensity image are affected by\ndynamical range limitation due to the presence of powerful radio\nsources near our target. These sources are not strongly polarized, so\nthat P images are not affected by such limitation, and weaker\npolarized emission can be revealed. We note in fact that polarized\nradio emission of the relic A2345-2 reveals an arc-like structure that\nis more extended than in total intensity emission. The arc-like\nstructure of this relic indicates that the shock wave, possibly\nresponsible of the radio emission, has been originated $\\sim 2.6'$\nsouthern the present X-ray center.\\\\ The mean fractional polarization\nis $\\sim 22 \\%$ in A2345-2, reaching values up to 50\\% in the eastern\nregion. The relic A2345-1, shows a mean fractional polarization of\n$\\sim 14 \\%$ with higher polarized region ( $\\sim 60 $\\%) in the\nnorth-western part of the relic. The amount of fractional\npolarization allows to estimate the level of order of the magnetic\nfield in the source. Following Burn (1966), if we assume that the\nmagnetic field is composed by an ordered component ${\\bf B_o}$ plus a\nrandom isotropic component represented by a Gaussian with variance\nequal to $2\/3 B_{r}^2$, it results\n\\begin{equation}\nP_{oss}=P_{int}\\frac{1}{1+(B_r^2\/B_o^2)}\n\\label{eq:pol}\n\\end{equation}\n where $P_{oss}$ is the observed fractional polarization, while\n$P_{intr}$ is given by $P_{intr}=\\frac{3\\delta+3}{3\\delta+7}$. For\nthe relic A2345-1 we obtain that $B_r^2\/B_o^2 \\sim 4$, meaning that\nthe magnetic energy density in the random component is three times\nlarger than the one in the ordered component. For the relic A2345-2,\ninstead, we obtain that $B_r^2\/B_o^2 \\sim 2$. This indicates that the\nmagnetic field in the region of the relic A2345-2 has a higher degree\nof order. We also have to consider possible beam depolarization,\ninternal depolarization and ICM depolarization, so that what we can\nconclude from this analysis is $B_r^2\/B_o^2 < 4$ and $<2$ in A2345-1\nand A2345-2 respectively.\\\\ In A2345-1 the magnetic field is\nmainly aligned with the sharp edge of the radio emission, i.e. in the\nSW-NE direction. In the northern part of the relic the E vectors rotate\nand in the N-W part they are almost aligned toward the SW-NE\ndirection. In A2345-2 the E vectors are perpendicular to the relic\nmajor axis, following the arc-like structure that is marginally\nvisible in the total intensity image. \\\\\n\\subsection{Results on Abell 2345}\n\\label{Sec:A2345results}\nThe presented analysis confirms that non-thermal emission is\nassociated with the ICM of Abell 2345.\n\\begin{itemize}\n\\item {The properties of the western relic, A2345-1 are quite\n peculiar. We note indeed that its morphology is rather\n circular and filamentary, its brightness distribution is higher in\n the inner region of the relic and its spectral index steepens\n toward the cluster periphery. Although the statistic is really\n poor, these features have not been found in other double relics so\n far. The level of polarization, the magnetic field direction\n mainly aligned with the sharp edge of the radio emission, and the\n value of the equipartition magnetic field are instead in agreement\n with other observed relics.\\\\ Diffusive shock acceleration models\n predict a steepening of the radio spectrum towards the cluster\n center (e.g. Ensslin et al. 1998; Hoeft \\& Br{\\\"u}ggen 2007) as a\n consequence of the electron energy losses after shock\n acceleration. It is worth mentioning here that theoretical\n predictions rely on some assumptions about the\n shock symmetry and the magnetic field structure that could be not\n representative of this specific cluster environment. Moreover, if\n the relic is not seen edge on, projection effects could further\n complicate the observed radio emission. Taking all of these into\n account, the observed spectral index trend of A2345-1 cannot be\n used as an argument to exclude an outgoing shock wave. \\\\ We\n however note that the position of A2345-1 is in between the main\n cluster and the possibly merging group X1. Thus we suggest the\n possibility that its radio properties could be affected by this\n ongoing merger. In particular, if the relic is seen edge-on, and\n if the magnetic field strength is almost uniform in the relic\n region, the observed spectral index trend could be the sign of a\n shock wave moving inward, toward the cluster center. It could\n result from the interaction with X1. Detailed optical and X-ray\n observations would be needed to shed light on this point.}\\\\\n\\item{The relic A2345-2 shows the classical feature of ``elongated\n relic sources'' also found in double relics of Abell 3667 and Abell\n 3376, as well as in single relic sources as 1253+275 (Andernach et\n al. 1984, Giovannini et al. 1991) and A521 (Ferrari 2003,\n Giacintucci et al. 2008). It is located far from the cluster center,\n its spectral index is steep with mean value $\\sim$ 1.3 and steepens\n towards the cluster center, as expected by relic formation theories\n if the relic is observed edge-on. The value of the\n equipartition magnetic field, the direction of the E vectors and the\n detected level of polarization are consistent with previous\n observations of elongated relics and agree with expectations from\n theoretical models as well. The polarized emission image reveals\n the arc-like structure of the relic A2345-2. If we assume that the\n relic is originated by a spherical shock wave, we can infer the\n propagation center of the shock by extrapolating the curvature\n radius of the relic. It results that the propagation center is $\\sim\n 2.6'$, southern the present X-ray center of the cluster Abell 2345\n (see Fig. \\ref{fig:A23451spixprofile.ps}).\\\\ This corresponds to a\n physical distance of 450 kpc at this redshift. From weak lensing\n analysis the galaxy velocity dispersion in this cluster results\n $\\sim$900 km\/s (Dahle et al. 2002; Cypriano et al. 2004). As we will\n see in Sect.\\ref{sec:discussion}, the expected Mach number is of\n about 2.2 for this relic. Since the galaxy velocity dispersion is\n comparable to the sound speed in the ICM (see e.g. Sarazin 1988), a\n Mach number 2.2 corresponds to a velocity of $\\sim$2000 km\/s. The\n relic A2345-2 is $\\sim$800 kpc far from the spherical-shock\n center. A shock wave with $M\\sim$2.2 travels this distance in\n $\\sim$ 0.4 Gyr (if the shock speed remains constant). Thus the\n merging between the two substructures should have occurred at $\\sim$\n 1200 km\/s to explain the shift of the X-ray center in this\n scenario. This is a reasonable value for cluster merger\n velocity.\\\\ Although a precise estimate should consider the amount\n of energy injected in the ICM as the shock wave passes through it,\n and despite the number of assumptions and approximations, we suggest\n that the relic indicates the position of the merger center as it was\n $\\sim$ 0.4 Gy ago. The time that the shock wave has taken to get the\n present relic position is the time that the sub cluster has taken to\n get the current X-ray center position.\\\\}\n\\end{itemize}\n\\begin{figure}[h!]\n\\centering\n \\includegraphics[width=0.9\\columnwidth]{A1240otticoDPSSradio2.ps}\n \\caption{Abell 1240. Colors: Optical emission from DPOSSII (red\n band); Contours: radio emission at 1.4 GHz (HR image). Contours\n start at $\\pm$3$\\sigma$ and are then spaced by 2. Red cross\n signs the X-ray center, labels refer to the discrete sources\n embedded in A1240-1. }\n \\label{fig:A1240_otticoradio.ps}\n \\end{figure}\n\n\\section{The Cluster Abell 1240}\nLittle is known in the literature about this cluster. It is a rich\ncluster classified as Bautz-Morgan type III. In Tab. \\ref{tab:x}\ngeneral data about this cluster are reported.\\\\ Kempner \\& Sarazin\n(2001) have revealed the presence of two roughly symmetric relics from\nthe Westerbork Northern Sky Survey (WENSS). From WENSS images relics\nare visible at 2 and 2.5 $\\sigma$ level. Our VLA observation confirm\nthe presence of two weak radio emitting regions in the cluster's\noutskirts. The radio image of the cluster is shown in\nFig. \\ref{fig:A1240_otticoradio.ps} (contours) overlaid into optical\nemission (from the DPOSSII, red band). The Northern relic (A1240-1) is\nlocated at $\\sim$ 270$''$ from the cluster X-ray center. This distance\ncorresponds to $\\sim$ 700 kpc at the cluster's redshift. This relic is\nmainly elongated in the E-W direction, and its radio brightness\ndecreases going from the western to the eastern part of the relic (see\nFig. \\ref{fig:A1240spix}). At 325 MHz only the eastern brightest part\nis visible. This is likely due to the higher noise in the 325 MHz\nimage. In fact from the mean brightness of the weaker part of the\nrelic, we estimated that it should have a spectral index $>$3 to be\ndetected at 325 MHz. Three radio sources are embedded in the relic\nemission, they are labeled with A B and C in\nFig. \\ref{fig:A1240_otticoradio.ps}. The sources A and B are not\ndetected in the 325 MHz observations. This is consistent with spectral\nindex values $<$1, as commonly found in radiogalaxies. A weak emission\nat 1.4 GHz links the A radio source at the relic (see\nFig. \\ref{fig:A1240spix}). \\\\\nThe southern relic (A1240-2) is located at\n$\\sim$ 400$''$ (1.1 Mpc) from the cluster X ray center. At 1.4 GHz it is\nelongated in the E-W direction extending $\\sim $ 480$''$. No discrete\nsources have been found embedded in the relic emission. Also in this\ncase at 325 MHz the relic's extension is reduced to $\\sim$ 350$''$ along\nthe main axis, and only the brightest regions are visible at 325\nMHz.\\\\ The relic's physical parameters are reported in\nTab. \\ref{tab:A1240_relics}. The quantity are computed excluding the\nregion where discrete sources (A,B and C) are present.\\\\\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=1.99\\columnwidth]{comboA1240spix.ps}\n\\caption{Center: the cluster Abell 1240 radio emission at 1.4 GHz. The\nbeam is 42$''$$\\times$33$''$. Contours start at 3$\\sigma$ (0.13\nmJy\/beam) and are then spaced by a factor 2. The cross marks the\ncluster X-ray center. Left: colors represent the spectral index of the\nrelic A1240-1 (top) and A1240-2 (bottom) superimposed over the radio\nemission at 325 MHz (contours) The beam is 42$''$$\\times$33$''$, first\ncontours are 2 $\\sigma$ (2 mJy\/beam), 3$\\sigma$ and are then spaced by\na factor 2. Right: Spectral index error image (colors) superimposed\nonto the emission at 325 MHz (contours are as above).}\n\\label{fig:A1240spix}\n\\end{figure*}\n\\begin{figure*}[t!]\n \\centering\n \\includegraphics[width=9cm]{comboA1240-1anuli.ps}\n \\includegraphics[width=9cm]{comboA1240-2anuli.ps}\n \\caption{Spectral index radial trend of A1240-1 (left) and\n A1240-2 (right), computed in shells of $\\sim 50''$ in\n width. It has been computed excluding the contribution of\n the discrete sources. Crosses refer to spectral index values\n computed in shells where the mean brightness is $>3\\sigma$\n at both 325 MHz and 1.4 GHz. Arrows are $3\\sigma$ upper\n limits on the spectral index mean value (see text). In the\n inset: displacement of the shells over which the mean\n spectral index has been computed. Circles refer to the\n discrete sources embedded in the relic emission. The red\n cross refers to the cluster X-ray center, the blue cross is\n the center of the spherical shells. }\n \\label{fig:A1240spixprofile.ps}\n \\end{figure*}\n\n\\subsection{Spectral index analysis}\nWe report in Fig.\\ref{fig:A1240spix} the spectral index map and the\nspectral index map error for the relics of Abell 1240. They have been\nobtained considering only those pixels that have a brightness\n$>$2$\\sigma$ at both frequencies.\\\\ Fig. \\ref{fig:A1240spix} shows\nthat the spectral index image is patchy. The spectral index image rms,\n$\\sigma_{spix}$, is $\\sim$ 0.3 and 0.4 for A1240-1 and A1240-2\nrespectively, while the mean of the spectral index error image,\n$<$Spix Noise$>$ is $\\sim$ 0.2 for both of the relics. We can then\nconclude that features in A1240-2 are statistically significant, while\ngiven the small difference between $\\sigma_{spix}$ and $<$Spix\nNoise$>$ in A1240-1, we cannot exclude that local features are a noise\nartifact in this case. In the relic A1240-2 a gradient is visible\nalong the main axis of the relic, as has been found in Abell 2256 by\nClarke \\& Ensslin (2006).\\\\ In Fig. \\ref{fig:A1240spixprofile.ps} the\nradial spectral index trend is shown for A1240-1 and A1240-2\nrespectively. They have been obtained as described in\nSec. \\ref{A2345spectralindex}. Spherical shells are centered close to\nthe X-ray cluster center and they are parallel to the main axis of\nboth relics.\\\\ Despite the small extension of the relics at 325 MHZ,\nit is still possible to derive some important results on the spectral\nindex radial trends in these relics: in the relic A1240-1 the spectral\nindex is steeper in the inner part of the relic and flatter in the\nouter part, as found in A2345-2 and predicted by ``outgoing merger\nshock'' models if relics are seen edge-on (Roettiger et al. 1999;\nBagchi et al. 2006). The same trend is consistent with the spectral\nindex profile derived in A1240-2, although a firm conclusion cannot be\nderived from these data. We note in fact that errors and upper limit\nin the inner shell cannot exclude a constant spectral index or even an\nopposite trend.\\\\\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=10cm]{A1240ROSAT60asecBand2+radio.ps}\n\\caption{Colors: Abell 1240X-ray emission in the energy band 0.5-2 keV\nfrom ROSAT PSPC observations. The image has been smoothed with a\nGaussian of $\\sigma\\sim$60$''$; contours represent the radio image of\nthe cluster at 1.4 GHz. The beam is 42$''$$\\times$33$''$. First contour is\n0.13 mJy\/beam, further contours are then spaced by a factor 2. }\n\\label{fig:A1240X}\n\\end{figure} \n\\subsection{Radio-X-ray comparison}\n We retrieved from the ROSAT data archive X-ray observations in the\nenergy band $0.5-2$ keV. The cluster is $\\sim$ 28$'$ offset from the\ncenter of the ROSAT pointing. Observations have been performed with\nthe ROSAT PSPC detector for a total exposure time of $\\sim$ 12\nksec. After background subtraction the event file has been divided by\nthe exposure map. We smoothed the resulting image with a Gaussian of\n$\\sigma=60''$. The resulting image is shown in\nFig. \\ref{fig:A1240X}.\\\\In Fig. \\ref{fig:A1240X} the X-ray emission of\nthe cluster is superimposed onto radio contours. The X-ray emission of\nthis cluster is elongated in the S-N direction and shows a double\nX-ray morphology. As already stated by Kempner \\& Sarazin (2001) this\nmorphology is consistent with a slightly asymmetric merger. \\\\ Relics\nare located at the edge of the X-ray emission. Their emission shows\nthe characteristic elongated shape, and their main axis is\nperpendicular to the main axis of the X-ray emission, as found in\ndouble relics of Abell 3367 and Abell 3376.\\\\\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=3cm, angle=-90]{A1240-1pol.ps}\n\\centering\n\\includegraphics[width=3cm, angle=-90]{A1240-2pol.ps}\n\\caption{Top panel: A1240-1 radio emission at 1.4 GHz, lines represent\nthe E vectors. The line direction indicates the E vector direction and\nthe line length is proportional to the polarized flux intensity. 1$''$\ncorresponds to 3$\\mu$Jy\/beam. The beam is\n18$''$$\\times$18$''$. Contours start at 0.12 mJy\/beam and are then\nspaced by a factor 2. Bottom panel: A1240-2 radio emission at 1.4\nGHz. The line direction indicates the E vector direction and\nthe line length is proportional to the polarized flux intensity. 1$''$\ncorresponds to 2$\\mu$Jy\/beam. Contours are as above.}\n\\label{fig:A1240pol}\n\\end{figure}\n\\begin{table*} \n\\caption{ Abell 1240}\n\\label{tab:A1240_relics}\n\\centering\n\\begin{tabular} {|c c c c c c c |} \n\\hline\\hline\nSource name & Proj. dist & LLS & F$_{20 cm}$ & F$_{90 cm}$ & B$_{eq}$ - B$'_{eq}$ & $<\\alpha>$ \\\\ \n & kpc & kpc & mJy & mJy & $\\mu$G & \\\\\n\\hline\nAbell 1240-1 & 270$''$=700 & 240$''$= 650 &6.0$\\pm$0.2 & 21.0$\\pm$0.8& 1.0 -2.4 & 1.2 $\\pm$0.1 \\\\\nAbell 1240-2 & 400$''$=1100& 460$''$= 1250 &10.1$\\pm$0.4 & 28.5$\\pm$1.1& 1.0 -2.5 & 1.3 $\\pm$0.2 \\\\\n\\hline \\multicolumn{7}{l}{\\scriptsize Col. 1: Source name; Col. 2:\nprojected distance from the X-ray centroid; Col. 3: Largest linear\nscale measured on the 20 cm images.}\\\\ \\multicolumn{7}{l}{\\scriptsize\nCol. 4 and 5: Flux density at 20 and 90 cm; Col. 6: equipartition\nmagnetic field computed at fixed frequency - fixed energy (see Sec. \\ref{sec:A1240pol})\n}\\\\ \n\\multicolumn{7}{l}{\\scriptsize Col. 7: mean spectral index in\nregion where both 20 cm and 90cm surface brightness is $>$ 3 $\\sigma$.}\n\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Equipartition magnetic field}\n\\label{sec:A1240pol}\nUnder the same assumptions explained in Sec. \\ref{sec:A2345pol} , we\ncalculated the equipartition magnetic field for the relics A1240-1 and\nA1240-2. Values obtained are reported in\nTab. \\ref{tab:A1240_relics}. We note that these values have been\ncomputed considering the brightness of those pixels for which we have\nwell constrained information about the spectral index value,\ni. e. those regions whose emission is detected at both\nfrequencies. Since the emission at 325 MHz is only detected in a small\nregion of the relics, while at 1.4 GHz relics are more extended, the\nequipartition estimates refer to the same small regions, and different\nestimates could be representative of the wider relic emission detected\nat 1.4 GHz.\\\\ We derived the minimum non thermal energy density in the\nrelic sources from $B'_{eq}$ obtaining $U_{min}\\sim$5.5\n10$^{-13}$erg\/cm$^{-3}$ for A1240-1 and A1240-2. The corresponding\nminimum non-thermal pressure in then $\\sim$3.4 and $\\sim$3.5\n10$^{-13}$erg\/cm$^{-3}$. The consistency between magnetic field\n equipartition values and magnetic field lower limits derived by\n X-ray emission in other few clusters (see discussion in\n Sec. \\ref{secs:Equip}) indicates that equipartition magnetic field\n can be used as a reasonable approximation of the magnetic field in relics.\n\\subsection{Polarization analysis}\nWe obtained the polarized intensity images for the relics as described\nin Sec. \\ref{sec:A2345pol}. In Tab. \\ref{tab:radiopol} the parameters\nrelative to the polarization images of the relics A1240-1 and A1240-2\nare reported.\\\\ In Fig. \\ref{fig:A1240pol} the polarized emission of\nthe two relics is shown. Observations performed with C array cannot\nreveal the weak extended emission, and thus only the most compact an\nbright regions are visible in this image. In these regions the\nmagnetic field is mainly aligned along the relic main axis in both of\nthe relics. This is consistent with what has been observed in the\nrelics of Abell 2345 and with what is expected from the models that\nexplain the origin of these sources (e. g. Ensslin et al. 1998;\nRoettiger et al. 1999). The mean fractional polarization of A1240-1 is\n26\\%, reaching values up to 70\\%. In the relic A1240-2 the mean\nfractional polarization is 29\\%, reaching values up to 70\\%. From\nEq. \\ref{eq:pol} we derive that $B_r^2\/B_o^2 \\sim$1.4 and 1.2\nrespectively. Because of possible beam depolarization, internal\ndepolarization and ICM depolarization, we conclude that\n$B_r^2\/B_o^2 <$1.4 and $<$1.2. This means that the magnetic energy\ndensity in the random and ordered component is similar.\n\\subsection{Results on Abell 1240 }\nOur observations confirm the presence of two relics in\nAbell 1240 with as steep spectral index values as $\\sim$1.2 and\n$\\sim$1.3. The spectral index trends derived for these relics indicate\na radial flattening toward the cluster outskirts. This\nis the trend predicted by ``outgoing merger shock'' models.\\\\ The\ndouble relics radio morphology and location are similar to the double\nrelics found in Abell 3667 and Abell 3376.\\\\ The polarization level is\nhigh in both of the relics, although we have to consider that our\npolarization observations lack the weak extended regions, that are\nprobably less polarized. The magnetic field estimate achieved under\nthe minimum total energy assumption reveals magnetic field of the\norder of $\\mu$G at the cluster periphery in the relic regions, ordered\non Mpc scale, indicating a magnetic field amplification and ordering.\n\\section{Discussion}\n\\label{sec:discussion}\nWe confirm the presence of double relics in the cluster Abell\n1240. Their symmetry and properties strongly suggest a common origin\nof A1240-1 and A1240-2.\\\\ In the cluster Abell 2345 we confirm the\nexistence of two relics. However, while A2345-2 is a classic extended\nperipheral relic source similar to 1253+275, in the Coma cluster (see\nGiovannini et al. 1991 and references therein), A2345-1 shows a more\ncomplex structure. We suggest that its properties could be due to its\npeculiar position in between the cluster Abell 2345 and the possibly\nmerging group X1, and thus affected by a more recent merger.\\\\ Several\nmodels have been proposed to explain the origin of radio relics. They\ncan be divided into 2 classes:\n\\begin{enumerate}\n\\item{Diffusive Shock Acceleration by Fermi-I process (Ensslin et\nal. 1998; Roettiger et al. 1999; Hoeft \\& Br{\\\"u}ggen 2007). }\n\\item{Re-acceleration of emitting particles due to adiabatic\ncompression of fossil radio plasma (Ensslin \\& Gopal-Krishna 2001). }\n\\end{enumerate} \nIn both of these models the presence of a shock within the gas is\nrequired. The second one also requires the presence of a nearby radio\nsource to provide the fossil radio plasma which can be re-energized by\nthe shock wave. Simulations of cluster mergers show indeed that the\nmerging of two sub-clusters leads to the formation of shocks in the\ncluster outskirts (Ryu et al. 2003).\\\\ In favor of the second scenario\nthere is the observational evidence that relics resemble individual\nobjects and do not trace the entire shock front (Hoeft et al.\n2004). Moreover, when a radio ghost is passed by a shock wave with\ntypical velocity of 10$^3$ km\/s, it is adiabatically compressed\nbecause of the higher value of the sound speed in the radio ghost\n(Ensslin \\& Br\\\"uggen 2002). We have to remind, however, that the\nequation of state of the radio emitting plasma is still poorly known,\nand that if the radio plasma has a high mass load due to undetectable\ncool gas , it should get shocked (Ensslin \\& Gopal Krishna 2001).\\\\\nThe presence of double relics itself favors the first scenario,\nbecause of the low probability to find two symmetric regions with\nfossil radio plasma. \\\\ Several independent cosmological simulations\nhave identified two main categories of cosmological shocks:\\\\ (i)\n``accretion shocks'' resulting from accretion of cold gas onto already\nformed structure, characterized by high Mach numbers;\\\\ (ii) ``merging''\nor ``internal'' shocks due to merging of substructures such as galaxy\nclusters or groups, with moderate Mach numbers: $2\\le M \\le 4$ (see\nreview by Bykov et al. 2008 and references therein).\n\\subsection{Relics from merging shocks}\nThe presence of double relics is particularly interesting in this\nscenario since the shape, morphology and properties of these extended\nstructures strongly suggest the presence of shock waves propagating\nfrom the cluster center to the peripheral regions. Because of the\nshort radiative lifetime of relativistic electrons, radio emission is\nproduced close to the location of the shock waves. These models\npredict that the magnetic field is aligned with the shock front and\nthat the radio spectrum is flatter at the shock edge, where the radio\nbrightness is expected to decline sharply. \\\\ The shock compression\nratio can be estimated from the radio spectral index $\\alpha$ (assuming an\nequilibrium electron population accelerated and cooled at the same\ntime, and assuming a polytropic index 5\/3, see Drury 1983), as\n\\begin{equation}\nR=\\frac{\\alpha+1}{\\alpha-0.5}.\n\\end{equation}\nThe pressure and temperature jumps across the shock can be estimated\n from the theory of shocks (Landau \\& Lifschitz 1966) as\n\\begin{equation}\n\\frac{P_2}{P_1}=\\frac{4R-1}{4-R}= \\frac{\\alpha+1.5}{\\alpha-1}; \\frac{T_2}{T_1}=\\frac{P_2}{RP_1}\n\\end{equation}\nhere and after the index 2 refers to down stream regions and 1 to\n up-stream regions i.e. regions inside and outside the cluster shock\n front. These parameters are reported in Tab. \\ref{tab:DSA}.\n\\begin{table*}\n\\caption{Predictions from the shock acceleration model}\n\\label{tab:DSA}\n\\centering\n\\begin{tabular}{|c c c c c c c |} \n\\hline\\hline\nRelic &$\\alpha$ & M & R & $P_2\/P_1$ & $T_2\/T_1$ & $(B_2\/B_1)_{isoP}$\\\\\n\\hline \nAbell 2345-1 &1.5$\\pm$0.1 &2.8$\\pm$0.1&2.5$\\pm$0.2&6$\\pm$1 & 2.4$\\pm$0.4 &2.4$\\pm$0.2\\\\\nAbell 2345-2 &1.3$\\pm$0.1 &2.2$\\pm$0.1&2.9$\\pm$0.2&9$\\pm$3 & 3$\\pm$1 &3.0$\\pm$0.5\\\\\nAbell 1240-1 &1.2$\\pm$0.1 &3.3$\\pm$0.2&3.1$\\pm$0.3&14$\\pm$6 & 4$\\pm$2 &3.7$\\pm$0.8\\\\\nAbell 1240-2 &1.3$\\pm$0.2 &2.8$\\pm$0.3&2.9$\\pm$0.4&9$\\pm$3 & 3$\\pm$2 &3.0$\\pm$0.5\\\\\n\\hline\n\\multicolumn{7}{l}{\\scriptsize Col. 1: \nSource name; Col 2: spectral index value; Col. 3: Mach number; Col 4: Shock compression ratio estimated }\\\\\n\\multicolumn{7}{l}{\\scriptsize from the radio spectral index; Col. 5, 6: \nPressure and temperature jump across the shock; Col. 7: Magnetic field }\\\\\n\\multicolumn{7}{l}{\\scriptsize strength in the pre and post shock regions required to\nsupport the relic against the thermal pressure }\\\\\n\\end{tabular}\n\\end{table*}\nThe Mach number of the shock can be estimated from the radio spectral\nindex under some assumptions: if the emitting particles are linearly\naccelerated by shock, the spectral index of the particle energy\nspectrum p ($=2\\alpha+1$) is related to the Mach number $M$ of the\nshock through:\n\\begin{equation}\np=2\\frac{M^2+1}{M^2-1}+1\n\\end{equation}\nincluding the effect of particle aging (continuous injection and\nInverse Compton energy losses, see e.g. Sarazin 1999). Mach number\nvalues we obtained are reported in Tab. \\ref{tab:DSA}. These values\nare lower than Mach number expected for accretion shocks (e. g. Bykov\net al. 2008), and are instead consistent with those expected for\nweaker shocks due to merging of structures. \\\\ The spectral index\ntrend clearly detected in A2345-2 and in both relics of Abell 1240\nagrees with the predictions of this scenario. If relics are seen\nedge-on, the flattest region, in the outer part of the relics, would\ncorresponds to the current shock location, indicating shock waves\nmoving outward from the cluster center. As discussed\n in Sec. \\ref{Sec:A2345results}, A2345-1 shows a more complex radio\n emission. It could be affected by a more recent merger with the X1\n group. It could trace a merger shock moving inward to the cluster\n center as a result of the Abell 2345 - X1 group interaction.\\\\\n\\subsubsection{Magnetic field and merging shocks}\n\\label{sec:Magnetic-Shock}\n The study of the magnetic field associated with the relics offers\n further opportunities to investigate the connection between relics\n and merger shock waves. First of all the presence of relics itself\n indicates the existence of significant magnetic field at the cluster\n periphery on the Mpc scale. Furthermore, the detected level of\n polarization shows that the magnetic field in these regions is rather\n ordered. \\\\ The effect of passage of a shock wave in the ICM\n could be twofold: (i) order and compress a magnetic field that was\n randomly oriented before the shock passage or (ii) compress a\n magnetic field that was already ordered on the relic scale before\n the shock passage . This depends on the turbulence development at\n the cluster periphery, that could either give rise to a random\n field in the cluster outskirts (case i) or not (case ii). Little is\n known about this point from observational point of\n view. Observational evidence from the gas pressure map of the Coma\n cluster (Schuecker et al. 2004) indicates the relevance of chaotic\n motions within the ICM. Cosmological numerical simulations\n (e.g. Bryan \\& Norman 1998; Sunyaev, Bryan \\& Norman 2003) suggest\n that the level of ICM turbulence is larger at increasing radial\n distances from the cluster center. If the simple\n Kolmogorov's picture of incompressible fluid turbulence is assumed, this\n implies a more developed turbulence in the outermost region (since\n the decay time is L\/$\\sigma$, where L is the typical scale where\n the bulk of turbulence is injected, and $\\sigma$ is the rms\n velocity of turbulence). Recently Ryu et al. (2008) argued that\n turbulence is likely well developed in clusters and\n filaments, and not in more rarefied regions such as sheets and\n voids. On the other hand Dolag et al.(2005) suggested that the bulk\n of turbulence is injected in the core of galaxy clusters, thus\n implying a more developed turbulence in the innermost regions,\n compared to the outermost ones. The main limitation of\n cosmological simulations is the lack of resolutions in low density\n environments, that makes it difficult to discriminate if the\n turbulent cascade is developed in these regions. Moreover, details\n of the conversion process of large\u2013scale velocity fields into MHD\n modes is still poorly understood. Thus, from the theoretical point\n of view the overall picture seems still uncertain.\\\\\n\n\n \n\n\n \n\n\n\n \n \n \n \n \n \nIn the case that the magnetic field in the cluster\noutskirts is randomly oriented before the shock passage (i. e. the\nturbulence is developed in the cluster outskirts) and that it has been\namplified and ordered, by the passage of the shock wave (case i\nabove), the observed ratio $B_r\/B_o$ derived by polarization analysis\n(Sec. \\ref{sec:A2345pol} and \\ref{sec:A1240pol}) could be used to\nestimate the magnetic field amplification due to the passage of the\nshock.\\\\\n Following Ensslin et al. (1998), if the relic is seen at some angle\n $\\delta>$0 between the line of sight and the normal of the shock\n front, the projected magnetic field should appear perpendicular to\n the line connecting the cluster center and the relic. This is indeed\n what polarization data presented here show. The magnetic field\n amplification, the observed integral polarization and the\n preferential direction of the field revealed by the E vectors\n orientation could be derived, provided that $\\delta$ and R, the shock\n compression factor, are known. Present data do not allow to infer the\n angle $\\delta$. Future X-ray and optical observations could\n reconstruct the merging geometry for these two clusters, as done,\n e.g. in Abell 521 by Ferrari et al. (2003, 2006). Despite this, if\n relics are supported by magnetic pressure only, the upstream and\n downstream fields are related by $(B_2^2\/B_1^2)_{isoP}=P_2\/P_1$\n (``strong field'' case in Ensslin et al. 1998). This ratio can be\n compared to the ratio derived by the polarization properties of the\n relics, under the assumption that $B_2$ corresponds to the ordered\n component of the field and $B_1$ to the random one. In\n Tab. \\ref{tab:DSA} the $(B_2\/B_1)_{isoP}$ ratio is reported for the\n relics in Abell 2345 and Abell 1240. These values are comparable to\n the observed ratio $B_r\/B_o$ derived by polarization analysis\n (Sec. \\ref{sec:A2345pol} and \\ref{sec:A1240pol}). \\\\ Another\n indication of the magnetic field amplification in the relics may be\n obtained by comparing the magnetic field in the relic with the\n cluster magnetic field intensity expected at the relic\n location. Relics are located at 700-1100 kpc from the cluster\n center in Abell 2345 and Abell 1240. At these distances the cluster\n magnetic field strength is expected to be of the order of\n $\\sim$10$^{-1}\\mu$G (see e. g. Dolag et al. 2008; Ferrari et\n al. 2008 and references therein). Equipartition magnetic field\n values are of the order of $\\mu$G (see Sec. \\ref{secs:Equip} and\n \\ref{sec:A1240pol}), thus about 10 times higher. Despite the number\n of uncertainties and assumptions related with the equipartition\n estimate, this is consistent with the ratio $(B_2^2\/B_1^2)_{isoP}$\n and $B_r\/B_o$.\\\\Even if no firm conclusion can be obtained by this\n analysis, we can conclude at least that the drawn picture has no\n inconsistencies with the presented observations.\n\n\n\n\n\n\\subsection{Relics from Adiabatic compression}\nAnother model to explain the origin of cluster radio relics has been\nproposed by Ensslin \\& Gopal-Krishna (2001). This idea has been\ninvestigated with the help of 3-dimensional Magneto Hydro Dynamical\nsimulations by Ensslin \\& Br\\\"uggen (2002) and in a more realistic\ncosmological environment by Hoeft et al. (2004). In this scenario\ncluster radio relics would originate by the compression of fossil\nradio plasma by shock wave occurring in the process of large scale\nstructure formation. The expected high sound velocity of that still\nrelativistic plasma should forbid the shock to penetrate into the\nradio plasma, so that shock acceleration is not expected in this\nmodel. The plasma gains energy adiabatically from the compression and\nthe magnetic field itself is amplified by such compression. If the\nelectron plasma is not older than 2 Gyr in the outskirts of a cluster,\nthey can emit radio wave again. Simulations performed by Ensslin \\&\nBr\\\"uggen (2002) show that the radio morphology of the resulting radio\nrelic in the early stage after the shock passage is sheet-like. Then\nthe formation of a torus is expected when the post shock gas starts to\nexpand into the volume occupied by the radio plasma. Thus it is\nexpected in this scenario that some correlation should exist between\nthe morphology of the radio relic and its spectral index, that traces\nthe time passed after the shock wave has compressed and re-energized\nthe emitting particles. A2345-1 shows indeed a torus-like radio\nstructure and a spectral index higher than A2345-2, A1240-1 and\nA1240-2, that exhibit a sheet-like structure. The simulations\nperformed by Ensslin \\& Br\\\"uggen (2002) indicate that the compression\nof the radio plasma by the shock can be estimated from a cluster radio\nrelic with a toroidal shape. Assuming the idealized case of a\ninitially spherical and finally toroidal radio cocoon, the compression\nfactor is given by:\n\\begin{equation}\n\\label{eq:plasma}\nR'=\\frac{2r_{max}^2}{3\\pi r_{min}^2}\n\\end{equation}\nwhere $r_{max}$ and $r_{min}$ refer to the outer and inner radius of\nthe torus. In the case of A2345-1 we assume that the observed torus\nlike structure can be described by taking $r_{max}\\sim$ the LLS of the\nrelic and $r_{min}$ the thickness of the filament in the N-E part of\nthe relic, as suggested by the same authors in the case of non perfect\ntoroidal filamentary relics. With $r_{max}\\sim$ 1 Mpc, $r_{min}\\sim$\n200 kpc it results $R'\\sim$ 5. This is higher than the value of\n the maximum compression ratio for mono-atomic gas (that is 4); this\n would indicate that the radio plasma has a different equation of\n state. However no conclusion can be drawn since Eq. \\ref{eq:plasma}\n is based on too simplistic assumptions, in particular a spherical\n model for the compressed relic.\\\\\n\\section{Conclusions}\nWe have presented 1.4 GHz and 325 MHz observations of Abell 2345 and\nAbell 1240. The presence of double relics in these cluster had been\ninferred by Giovannini et al. (1999) for Abell 2345 and by Kempner \\&\nSarazin (2001) for Abell 1240 from NVSS and WENSS. We confirm the presence\nof two relics in each of these clusters.\nBy combining 1.4 GHz and 325 MHz observations we obtained the spectral\nindex image of the diffuse radio emission. The study of the polarized\nemission at 1.4 GHz has been presented as well. The analysis of both\nthe spectral index distribution and the polarization properties of\nrelics allows to test several independent predictions of the relic\nformation models.\\\\ We report the summary of the results from the\npresented analysis:\\\\\n\\begin{enumerate}\n\\item{ {\\bf A2345}: two relics have been detected in the cluster\noutskirts at both 1.4 GHz and 325 MHz. They are not perfectly\nsymmetrical with respect to the cluster center; the normals to the\nrelic main axis form an angle of $\\sim$150$^{\\circ}$. A2345-2 is a\nclassical peripheral relic and A2345-1 is a peculiar relic with a\ntorus-like structure possibly related to a merging region.}\\\\\n\\item{{\\bf A1240}: relics are fainter than relics in A2345. Their\nextended emission is detected at 1.4 Ghz while only their brightest\npart are detected at 325 MHz. They are symmetrical with respect to the\ncluster center and the angle between their normals is\n$\\sim$180$^{\\circ}$ as found in the other known double relics: Abell 3667,\nAbell 3376 and RXCJ1314.4-2515.}\\\\\n\\item{ Relics are located at the edge of the X-ray emission of Abell 2345\nand Abell 1240. The X-ray emission of Abell 2345 shows multiple substructures\nthat could be galaxy groups interacting with A2345. Peculiar features\nof A2345-1 could arise from this multiple interaction, but only\ndetailed X-ray and optical analysis could shed light on this\npoint.}\\\\\n\\item{Relics in Abell 1240 are located perpendicular to the cluster main\naxis revealed by X-ray observations. The double X-ray morphology of\nthe cluster is typical of merging clusters. }\\\\\n\\item {The average spectral indexes are steep. We found 1.5 $\\pm$ 0.1\n and 1.3 $\\pm$0.1 for A2345-1 and A2345-2 and 1.2$\\pm$ 0.1, 1.3$\\pm$\n 0.2 for A1240-1 and A1240-2.}\\\\\n\\item{The spectral index distribution in the relics is rather\nirregular and patchy, although this, a clear radial trend is present\nin the relics of these two clusters. A2345-2 spectral index ranges\nfrom $\\sim$1.5 in the region closer to the cluster center to $\\sim$1.1\nin the outer rim. This trend is consistent with shock models\npredictions. The same trend is observed in both of Abell 1240\nrelics. A1240-1 spectral index ranges from $\\sim$1.1 to $\\sim$1.6\ngoing from the outer to the inner rim, A1240-2 spectral index is also\nconsistent with a similar trend (going from $\\alpha <$ 1.5 in the\ninner rim to $\\alpha\\sim$ 1.1 in the outer one). An opposite trend is\ninstead detected in A2345-1. Spectral index values are lower in the\ninner rim ($\\sim$1.3) and increase toward the outer part of the relic\nreaching values $\\sim$1.7. This trend could be due to its peculiar\nposition between two merging clumps.}\\\\\n\\item{The magnetic field, as revealed by polarized emission is mainly\n aligned with the relic main axis. In Abell 2345 the polarized\n emission reveals the arc-like structure morphology of the relic\n A2345-2. Under equipartition conditions, values of $\\sim$ 2.2 - 2.9 $\\mu$G are\n derived. The field has been likely amplified,\n consistently with shock models predictions.}\\\\\n\\end{enumerate}\n These results have been discussed in the framework of relic\n formation models. The Mach numbers derived from the value of radio\n spectral index disfavour the ``accretion shock'' scenario, since\n they are too small. Outgoing merger shock waves, proposed to explain\n double relic emission in Abell 3667 and A3376, could also work in\n Abell 1240 and Abell 2345. For the latter cluster we suggest that\n the peculiar emission of A2345-1 could be explained by a shock wave\n moving inward, due to the interaction of the main cluster with the\n X1 group.\\\\ The toroidal shape of A2345-1 could be produced by\n adiabatic compression, however the available data and models do not\n allow a conclusive comparison.\n \n\\begin{acknowledgements}\n The authors grateful to Franco Vazza, Marica Branchesi and\n Elisabetta Liuzzo for helpful discussions and to Klaus Dolag for\n elucidating discussion on cluster magnetic fields. The authors\n thank the anonymous referee for useful suggestions and\n comments. NRAO is a facility of the National Science Foundation,\n operated under cooperative agreement by Associated Universities,\n Inc. This work was partly supported by the Italian Space Agency\n (ASI), contract I\/088\/06\/0, by the Italian Ministry for University\n and Research (MIUR) and by the Italian National Institute for\n Astrophysics (INAF). This research has made use of the NASA\/IPAC\n Extragalactic Data Base (NED) which is operated by the JPL,\n California Institute of Technology, under contract with the\n National Aeronautics and Space Administration.\n\\end{acknowledgements}\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{Conjectures}\nThe semantics of RDF makes it impossible to express contradictory points of view that act on the same data.\n\nEvery approach applied in the past has not solved the main issue, that is, being able to express different statements whose truth value is not known, or even in contrast with each other, without fully asserting them, therefore making them undoubtedly true.\n\nIn this paper we introduce and examine the characteristics of \\textit{conjectures}, a proposed extension to RDF 1.1 that by design allows the expressions of graphs whose truth value is unknown.\n\nThe approach revolves around two main concepts:\n\\begin{itemize}\n \\item \\textit{conjecture}: a \\textit{concept whose truth value is not available} and its representation;\n \\item \\textit{collapse to reality}: a mechanism to fully assert, when the time comes, the truth value of conjectures in their RDF-esque form.\n\\end{itemize}\n\nConjectures encapsulate plain RDF statements or other conjectures in specially marked named graphs.\nThey can be expressed in a strong form:\n\n\\begin{verbatim}\nCONJECTURE :deVereWroteHamlet {\n :Hamlet dc:creator :EdwardDeVere .\n}\n\\end{verbatim}\n\nor, allowing us to be able to include them in plain RDF 1.1 datasets, in a weak form, where the predicate is expressed as a \\textit{conjectural predicate}:\n\n\n\\begin{verbatim}\n@prefix conj0001: \n\nGRAPH :deVereWroteHamlet {\n :Hamlet conj0001:creator :EdwardDeVere .\n conj0001:creator conj:isAConjecturalFormOf dc:creator . \n}\n:deVereWroteHamlet prov:wasAttributedTo :JThomasLooney .\n\n\\end{verbatim}\n\n\nAdditional statements adding information to the conjecture, but external to it, are the \\textit{grounds} or \\textit{ground statements}. One of them is the last triple in the last example. \n\nThis paper is organized as follows: \nin section 2 we show a number of fundamental concepts and results of the conceptual model of conjectures. \n\nIn section 3 we introduce the impact of conjectures onto the standard RDF Simple interpretation according to \\cite{HAYES04}. \n\nSections 4 through 7 describe the simple interpretation of the main concepts of Conjectures: conjectures of individual triples, collapse to reality of individual triples, conjectures and collapses of graphs, and conjectures involving blank nodes. \n\nIn section 8 we discuss the simple interpretation of the additional features of conjectures: nested conjectures, conjectural collapses, ad cascading collapses. \n\nIn section 9 we draw some conclusions about the formal model presented here.\n\n\\section{Set-theoretical formal model}\n\nAssume the disjoint sets $\\mathcal{I}$ (all IRIs), $\\mathcal{B}$ (blank nodes), and $\\mathcal{L}$ (literals). \nAn RDF triple is a tuple $(s, p, o) \\in \\mathcal{T} = (\\mathcal{I} \\cup \\mathcal{B}) \\times \\mathcal{I} \\times (\\mathcal{I} \\cup \\mathcal{B} \\cup \\mathcal{L})$. \n\nFor every RDF predicate $p$, let $\\mathcal{S}_{p} \\subseteq (\\mathcal{I} \\cup \\mathcal{B})$ be its domain and\n$\\mathcal{O}_{p} \\subseteq (\\mathcal{I} \\cup \\mathcal{B} \\cup \\mathcal{L})$ its range. \n\nWe denote with \n\\begin{verbatim}\n x { s p o } \n\\end{verbatim}\n\na triple $(s, p, o)$ that is referred to in the examples by\nthe name $x$.\n\n\\textbf{Conjecturing}: Conjecturing is the function $conj : \\mathcal{T} \\rightarrow \\mathcal{T}$ such that, for every RDF triple $t_{1} = (s_{1} , p, o_{1}), conj (t_{1}) = (s, p_{s,o} , o)$ where:\n\\begin{enumerate}\n \\item Identity of subject: $s_{1} = s$.\n \\item Identity of object: $o_{1} = o$.\n \\item Disjointness: $\\forall s_{j}, o_{k}$ such that $(s_{j}, p_{s,o} , o_{k}) \\in \\mathcal{T}$, we have that $s = s_{j}$ and $o = o_{k}$ . \n\\end{enumerate}\n\n\n\\textbf{Conjectures, conjectural predicates, conjecturing triple}: given the triple $(s, p, o) \\in \\mathcal{T}$, its conjecture is the triple\n\n\\[conj((s, p, o)) = (s, p_{s,o}, o) .\\] \n\nConjectural predicates (or \\textit{weak predicates}) of predicate $p$ are all the predicates that are\nmembers of the set $\\mathcal{C}onj_{p}$ , such that:\n\n\\[ \\mathcal{C}onj_{p} = \\{cp \\in \\mathcal{I} | \\exists s \\in S_{p}, o \\in O_{p}, conj((s, p, o)) = (s, cp, o) \\}\\].\n\n\\textbf{Theorem 1}: every conjectural predicate is used in one triple only.\n\n\\textbf{Proof}: derives from item 3 of definition 1 (Disjointness):\nGiven two triples $(s_{1}, p_{s,o}, o_{1}) , (s_{2}, p_{s,o}, o_{2}) \\in \\mathcal{T}$.\n\nFor item 3 of definition 1 (Disjointness), we have that $s_{1} = s_{2}$ and $o_{1} = o_{2}. \\forall (s, p, o) \\in \\mathcal{T}, \\exists! p_{s,o}$ such that $conj((s, p, o)) = (s, p_{s,o} , o)$.\n\n\\textbf{Corollary 1}: the function $conj$ is unique (barring predicate name changes).\n\n\\textbf{Proof}: derives immediately from Theorem 1.\n\n\\textbf{Conjectural Form}: predicate $q$ is said to be a \\textit{conjectural form} of $p$ if there exists a pair of subjects and objects $s, o$ such that $conj ((s, p, o)) = (s, q, o)$.\n\n\n\\textbf{Collapsing}: Collapsing is a function $collapses: \\mathcal{T} \\rightarrow \\mathcal{T}$ such that, for every RDF triple $t = (s, p, o)$,\n$collapses(s, p, o) = (s, p_{s,o} , o)$ iff $conj((s, p, o)) = (s, p_{s,o} , o)$, and is undefined otherwise.\n\n\\textbf{Collapsed predicate; collapse}: let $(s, p_{s,o}, o)$ be a conjecture. \nThe collapsed predicate of $p_{s,o}$ is the predicate $p$ such that \n\\[collapses (s, p, o) = (s, p_{s,o} , o)\\]. \nThe collapse is the triple \n\\[( (s, p, o), collapse, (s, p_{s,o}, o))\\].\n\n\\section{RDF Simple interpretation and Conjectures}\n\nIn RDF, a simple interpretation $I$ of a vocabulary $V$ consists of:\n\\begin{enumerate}\n\\item A non-empty set $IR$ of resources, called the domain or universe of $I$.\n\\item A set $IP$, called the set of properties of $I$.\n\\item A mapping $IEXT$ from $IP$ into the powerset of $IR \\times IR$ i.e. the set of sets\nof pairs $< x, y >$ with $x$ and $y$ in $IR$ .\n\\item A mapping $IS$ from IRIs into $IR$ - in order to map resources and properties\n\\item A partial mapping $IL$ from literals into $IR$ - in order to map literals\n\\end{enumerate}\n\n$IEXT(p)$ is the extension of $p$, that is the set of pairs that are the arguments for which the property $p$ is true.\n\nAccording to \\cite{HAYPSC14}, a semantic extension is a set of additional semantic assumptions that gives IRIs additional meanings on the basis of other specifications or conventions.\nWhen this happens, the semantic extension must conform to the minimal truth conditions already enunciated, extending from them to accommodate the additional meanings.\n\nTherefore, we extend the RDF Simple Interpretation adding a new set of conjectural properties, $IPC$, disjoint from the set of properties $IP$, where the conjectural predicates are created on the fly.\n\nWe add a new mapping $IEXTC$ from $IPC$ to the Cartesian product $IR \\times IR$. $IEXTC(cp)$ identifies the pair for which the property $cp$ is true.\n\nBecause of the Disjointness property of the conjecturing function $conj$ seen in section 1, the pair satisfying the property $cp$ will always be unique.\n\nWe need to specify an additional mapping $CONJFORM$ from $IP$ into $IPC$ to map the conjectural forms of the properties. \n\nOur full simple interpretation of $I$ in RDF with Conjectures is:\n\n\\begin{enumerate}\n\\item A non-empty set $IR$ of resources, called the domain or universe of $I$.\n\\item A set $IP$, called the set of properties of $I$. \n\\item { \\color{blue} A set $IPC$, called the set of conjectural properties of $I$. $IPC \\cap IP = \\emptyset$ }\n\\item A mapping $IEXT$ from $IP$ into the powerset of $IR \\times IR$ i.e. the set of sets of pairs $< x, y >$ with $x, y \\in IR$ .\n\\item { \\color{blue} An injective mapping $IEXTC$ from $IPC$ into $IR \\times IR$, in other words the set of pairs $< x, y >$ with $x, y \\in IR$. By definition of injective mapping, if $IEXTC(a)=IEXTC(b)$, then $a=b$, that is, any $cp \\in IPC$ uniquely applies to a specific pair $$.}\n\\item { \\color{blue} A mapping $CONJFORM$ from $IP$ into $IPC$ in order to map the conjectural forms of the properties }\n\\item A mapping $IS$ from IRIs into $IR$ - in order to map resources, properties and conjectural properties\n\\item A partial mapping $IL$ from literals into $IR$ - in order to map literals\n\\item {\\color{blue} A conjecture might be represented as a set of individual statements composing as a whole the conjecture. This set is a \\textit{conjectural graph}}\n\n\\end{enumerate}\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=\\textwidth]{Interpretation1}\n \\caption{Schematics of the interpretation}\n \\label{fig:Interpretation}\n\\end{figure}\n\n\n\\section{Conjectures}\n\\begin{itemize}\n\\item if $E$ is a literal then $I(E) = IL(E)$\n\\item if $E$ is an IRI then $I(E) = IS(E)$\n\\item if $E$ is a ground triple $s \\: p \\: o \\: .$ then\n \\begin{itemize}\n \\item $I(E) = true$ if $I(p) \\in IP$ and\n \\item the pair $ \\in IEXT(I(p))$\n \\item otherwise $I(E) = false$.\n \\end{itemize}\n\\item { \\color{blue} if $E$ is a Conjecture triple $s \\: cp \\: o \\:.$ then }\n \\begin{itemize}\n \\color{blue}\n \\item $I(E) = true$ if $I(cp) \\in IPC$ and\n \\item the pair $ \\in IEXTC(I(cp))$\n \\item $I(cp) \\in CONJFORM(I(p))$ for some $I(p) \\in IP$\n \\item otherwise $I(E) = false$.\n \\end{itemize}\n\\item if $E$ is a ground RDF graph then $I(E) = false$ if $I(E') = false$ for some triple $E' \\in E$, otherwise $I(E) =true$.\n\\item { \\color{blue} if $E$ is a ground conjectural graph then $I(E) = false$ if $I(E') = false$ for some triple $E' \\in E$, otherwise $I(E) =true$.}\n\\end{itemize}\n\n\nThe last clause captures the definition of conjectural graph seen before.\n\n\\subsection{Model}\nGiven a RDF graph $G$, we say that the interpretation $I$ is a model of the graph $G$ if all the triples of graph $G$ are satisfied, that is, if they are true according to rules of $I$.\nIn this case, we can say that Interpretation $I$ satisfies $G$.\n\nThe notion of model is at the basis of the (simple) entailment: \nfollowing standard terminology, we say that $I$ (simply) satisfies $G$ when $I(G)=true$, that $G$ is (simply) satisfiable when a simple interpretation exists which satisfies it, otherwise (simply) unsatisfiable, and that a graph $E$ simply entails a graph $G$ when every interpretation which satisfies $E$ also satisfies $G$. If two graphs $G$ and F each entail the other then they are logically equivalent. \\cite{HAYPSC14} \n\n\nIs our interpretation $I$ a model of this conjecture graph?\n\n\\begin{verbatim}\n:deVereWroteHamlet { \n :Hamlet conj001:creator :EdwardDeVere .\n conj001:creator conj:isAConjecturalFormOf dc:creator .\n}\n\\end{verbatim}\n\n\n\\begin{itemize}\n \\item $IR=\\{dVWH, h, c, cc1, e, iacf\\}$\n \\item $IP=\\{c, iacf\\}$\n \\item $IPC=\\{cc1\\}$\n\\end{itemize}\n\nThe functions:\n\n\\begin{itemize}\n \\item $IS(:deVereWroteHamlet) \\rightarrow dVWH$\n \\item $IS(:Hamlet)\\rightarrow h$\n \\item $IS(conj001:creator)\\rightarrow cc1$\n \\item $IS(:EdwardDeVere)\\rightarrow e$\n \\item $IS(conj:isAConjecturalFormOf)\\rightarrow iacf$\n \\item $IS(dc:creator)\\rightarrow c$\n \\item $IEXT(c)=\\emptyset$\n \\item $IEXT(iacf)=\\{\\}$\n \\item $IEXTC(cc1)=\\{\\}$\n \\item $CONJFORM(c) = cc1$\n\\end{itemize}\n\nIEXT(c) is the empty set. We are still dealing with a conjecture, there is no assertion regarding any \"real\" property (in this case $dc:creator$) yet.\n\nAll the clauses seem to hold. We can say that simple interpretation (s-interpretation) $I$ is a model of our graph.\n\n\\section{Collapse to reality} \\label{collapseToReality}\nAt a certain point, someone may want to consider our conjectures as true. \n\nIn this case we would need to specify a mechanism in order to express the statements in the conjecture in full force. \n\nThis mechanism is called \"collapse to reality\".\n\nIt consists of exactly two triples added to the base, where:\n\n\\begin{itemize}\n \\item in the first new triple, the \"effective\" property is used instead of the corresponding conjectural property.\n \\item the first new triple is then declared to \"collapse\" the conjecture .\n\\end {itemize}\n\nTwo new triples are added, nothing gets replaced or deleted. \nSo we can keep track of what's happening and all the relationships between the graphs.\n\nLet's reason on an example of a collapse:\n\n\\begin{verbatim}\n:attr1 { \n :Hamlet conj003:creator :Shakespeare .\n conj003:creator conj:isAConjecturalFormOf dc:creator.\n}\n\n:attr1Cot {\n :Hamlet dc:creator :Shakespeare .\n}\n\n:attr1Cot conj:collapses :attr1 .\n\n\\end{verbatim}\n\n$:attr1$ is collapsed by adding a triple $:attr1Cot$ where the conjectural predicate is replaced by the corresponding \"real\" predicate.\n\nThe final additional triple functions as the explication of the collapse. \n$:attr1Cot$ is declarad to collapse the conjecture $:attr1$ by means of the property \"conj:collapses\".\n\nFor the sake of clarity:\n\\begin{itemize}\n \\item the conjecture triple to be collapsed will be the \"conjecture triple\";\n \\item the new triple collapsing the conjecture will be the \"collapsing triple\";\n \\item the last triple will be the \"collapse triple\".\n\\end{itemize} \n\n\\subsection{Interpretation $[I + COLLAPSE]$} \\label{I+COLLAPSE}\n\nWe need to extend once more our Interpretation with Conjectures $I$ by adding the Collapse to Reality rules.\n\nWe define a new mapping $collapses$ from triples to triples that maps the relationship between the conjecture triple and its collapsing triple:\n\n$collapses(s,p,o)=(s,cp,o)$ iff $conj(s,p,o)=(s,cp,o)$.\n\nThe collapse collapses the conjecture triple if, and only if, the latter is the (unique) conjecture of the collapsing triple. \n\nWe can clearly see that the \"collapses triple\" is just the translation into RDF of the definition of collapse we have introduced in section 1, that is, the triple:\n\n\\[((s,p,o),\\quad collapses,\\quad (s,cp,o) )\\] \n\nFrom a more formal point of view, given a conjectural triple $E$, we add the triple $E_{cot}$ as a collapsing triple $\\{s \\quad p \\quad o .\\}$.\n\nThe semantic conditions of the collapse to reality should be the following:\n\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E$ be a conjecture triple\n$\\\\ \\{s \\quad cp \\quad o \\quad .\\}$\n\nlet $E_{cot}$ be a collapsing triple \n$\\\\ \\{ s \\quad p \\quad o \\quad . \\}$\n\nand finally let $E_{collapse}$ be the collapse triple\n$\\\\ \\{ E_{cot} \\quad \\mathit{collapses} \\quad E \\}$\n\nthen } \n \\begin{itemize}\n \\color{blue}\n \\item $I(E) = true$ if $I(cp) \\in IPC$ and\n \\item $I(E_{cot}) = true$ if $I(p) \\in IP$ and\n \\item $CONJFORM(I(p))=I(cp)$ and\n \\item the pair $ \\in IEXTC(I(cp))$ and\n \\item the pair $ \\in IEXT(I(p))$ and\n \\item $I(E_{collapse}) = true$ if $collapses(I(E_{cot}))=I(E)$, that is:\n $collapses(I(s),I(p),I(o))=(I(s),I(cp),I(o))$ \n \\item otherwise $I(E) = false$ and $I(E_{cot}) = false$ and $I(E_{collapse})=false$.\n \\end{itemize}\n\\end{itemize}\n\n\\subsection{Model}\n\nNow, let's check if our Interpretation $[I + COLLAPSE]$ can be a model of our example graph.\n\\begin{itemize}\n \\item $IR=\\{a1, h, cc3, s, iacf, c, a1cot, cl \\}$\n \\item $IP=\\{iacf, c, cl\\}$\n \\item $IPC=\\{cc3\\}$\n\\end{itemize}\nThe functions:\n\\begin{itemize}\n \\item $IS(:attr1) \\rightarrow a1$\n \\item $IS(:Hamlet)\\rightarrow h$\n \\item $IS(conj003:creator)\\rightarrow cc3$\n \\item $IS(:Shakespeare) \\rightarrow s$\n \\item $IS(conj:isAConjecturalFormOf)\\rightarrow iacf$\n \\item $IS(dc:creator)\\rightarrow c$\n \\item $IS(:attr1Cot)\\rightarrow a1cot$\n \\item $IS(conj:collapses)\\rightarrow cl$\n \\item $IEXT(iacf)=\\{\\}$\n \\item $IEXT(c)=\\{\\}$\n \\item $IEXT(cl)=\\{\\}$\n \\item $IEXTC(cc3)=\\{\\}$\n \\item $CONJFORM(c) = cc3$\n\\end{itemize}\n\nLet's check our semantic conditions for the collapse to reality:\n\\begin{itemize}\n \\item $cc3 \\in IPC$? Yes; \n \\item $c \\in IP$? Yes;\n \\item $CONJFORM(c)=cc3$? Yes;\n \\item The pair $ \\in IEXTC(cc3)$? Yes;\n \\item The pair $ \\in IEXTC(c)$? Yes;\n \\item $collapses(h,c,s)=(h,cc3,s)$? Yes, because $CONJFORM(c)=cc3$, hence $conj(h,c,s)=(h,cc3,s)$\n\\end{itemize}\n\nAll of them seem to hold.\n\nHence, our interpretation $[I + COLLAPSE]$ is a model of our graph.\n\nAs we will see in section \\ref{CollapseGraphs}, the \\textit{collapsing triple} and the \\textit{collapse} triple can be safely merged into a \\textit{collapse graph}:\n\n\\begin{verbatim}\n:attr1 { \n :Hamlet conj003:creator :Shakespeare .\n conj003:creator conj:isAConjecturalFormOf dc:creator.\n}\n\n:collapseOfattr1 {\n :Hamlet dc:creator :Shakespeare .\n :collapseOfattr1 conj:collapses :attr1 .\n}\n\n\n\\end{verbatim}\n\n\n\\section{Conjectural Graphs and Collapse Graphs}\n\\subsection{Conjectural Graphs}\nA conjectural graph is a representation of a conjecture by means of a set of individual statements composing as a whole the said conjecture.\n\nA ground Conjectural Graph is a graph with no blank nodes.\n\nAll triples inside a Conjectural Graph must be either with a conjectural predicate used exactly once, or a \"conj:isAConjecturalFormOf\" triple connecting a conjectural predicate to an original one.\n\nThe semantic conditions for Conjectural Graphs are:\n\\begin{itemize}\n\\item if $E$ is a literal then $I(E) = IL(E)$\n\\item if $E$ is an IRI then $I(E) = IS(E)$\n\\item { \\color{blue} if $E$ is a Conjecture triple $s \\: cp \\: o \\:.$ then }\n \\begin{itemize}\n \\color{blue}\n \\item $I(E) = true$ if $I(cp) \\in IPC$ and\n \\item the pair $ \\in IEXTC(I(cp))$\n \\item $I(cp) \\in CONJFORM(I(p))$ for some $I(p) \\in IP$ and\n \\item $E$ must be unique in the graph\n \\item otherwise $I(E) = false$.\n \\end{itemize}\n\\item { \\color{blue} if $E$ is a triple $cp conj:isAConjecturalFormOf p$ then }\n \\begin{itemize}\n \\color{blue}\n \\item $I(E) = true$ if $I(cp) \\in IPC$ and\n \\item $I(p) \\in IP$ and\n \\item $I(cp) \\in CONJFORM(I(p))$ and\n \\item $I(conj:isAConjecturalFormOf) \\in IP$\n \\item the pair $ \\in IEXT(I(conj:isAConjecturalFormOf))$\n \\item otherwise $I(E) = false$\n \\end{itemize}\n\\item { \\color{blue} if $E$ is a ground conjectural graph then $I(E) = false$ if $I(E') = false$ for some triple $E' \\in E$, otherwise $I(E) =true$.}\n\\end{itemize}\n\n\n\\subsection{Collapse Graphs} \\label{CollapseGraphs}\n\nCollapse graphs are graphs that connect explicitly with a conjecture and make it possible to track which conjecture was collapsed.\n\nThey must be composed at least of:\n\\begin{itemize}\n \\item the \"effective form\" of the conjecture to be collapsed \n \\item a \"conj:collapse\" triple connecting the conjecture to the collapse\n\\end{itemize}\n\nThe semantic conditions are:\n\n\\begin{itemize}\n\\item let $E_{conj}$ be a conjecture triple $s \\: cp \\: o \\: .$\n\\item let $E_{collapse}$ be a collapse graph\n\\item if $E$ is a literal then $I(E) = IL(E)$\n\\item if $E$ is an IRI then $I(E) = IS(E)$\n\\item { \\color{blue} if $E$ is a triple $s \\: p \\: o \\:.$ then }\n \\begin{itemize}\n \\color{blue}\n \\item $I(E) = true$ if $I(p) \\in IP$ and\n \\item the pair $ \\in IEXT(I(p))$\n \\item $CONJFORM(I(p))=I(cp)$ where $cp$ is the conjectural predicate of the conjecture to be collapsed\n \\item otherwise $I(E) = false$.\n \\end{itemize}\n\\item { \\color{blue} if $E$ is a triple $E_{collapse} \\quad conj:collapses \\quad E_{conj}$ then }\n \\begin{itemize}\n \\color{blue}\n \\item $I(E) = true$ if $I(conj:collapses) \\in IP$ and\n \\item the pair $ \\in IEXT(I(conj:collapses))$\n \\item otherwise $I(E) = false$\n \\end{itemize}\n\\item { \\color{blue} if $E$ is a ground conjectural graph then $I(E) = false$ if $I(E') = false$ for some triple $E' \\in E$, otherwise $I(E) =true$.}\n\\end{itemize}\n\n\n\n\n\n\\section{Blank Nodes}\nWith conjectures, we would want to be able to express even more \"uncertain\" concepts involving unnamed entities or unspecified values.\n\nIn RDF this is done through \\textit{blank nodes}, which indicate the existence of an entity without using a IRI to identify any particular one.\n\nIn this section we will use a simple sentence implying the reliance on a blank node, namely \"Muammar al-Qaddafi claimed that the author of Othello was someone who was an Arab\" \\footnote{He really did it in December 1988 - see \"William Shakespeare's Othello\" by Jibesh Bhattacharyya, ISBN 9788126904747}. \n\nIn this case we don't know who someone is, and we can't identify it with any IRIs.\n\nNevertheless, the information we are conveying with our conjecture maintains some degree of meaningfulness, and it definitely is something we could reason upon.\n\nSticking to our examples' style, we could say it like this: \n\"it is conjectured that Othello was written by somebody and this somebody was an Arab. And this claim has been attributed to Muammar al-Qaddafi\".\n\nIn RDF:\n\\begin{verbatim}\n:ArabWroteOthello { \n :Othello conj002:creator _:z .\n conj002:creator conj:isAConjecturalFormOf dc:creator .\n _:z dbpedia-owl:nationality :Arab .}\n\n:ArabWroteOthello prov:wasAttributedTo :MalQaddafi .\n\\end{verbatim}\n\nIt comes pretty natural to conceive the term \"somebody\" as a blank node.\n\n\\subsection{Interpretation $[I + A]$}\n\nIn order to deal with blank nodes, we need to add a new mapping $A$ from blank nodes into IR.\n\nTherefore, we extend our interpretation $I$:\n\\begin{itemize}\n\\item $[I + A](x)=I(x)$ for names\n\\item $[I + A](x)=A(x)$ if x is a blank node.\n\\end{itemize}\n\nWe add a couple of semantic conditions to our interpretation for blank nodes, one is the \"standard\" one for RDF graphs, the other one is for conjectures:\n\n\\begin{itemize}\n \\item if $E$ is a RDF graph then $I(E) = true$ if $[I+A](E) = true$ for some mapping $A$ from the set of blank nodes in $E$ to $IR$, otherwise $I(E) =false$.\n \\item { \\color{blue} if $E$ is a conjectural graph then $I(E) = true$ if $[I+A](E) = true$ for some mapping $A$ from the set of blank nodes in $E$ to $IR$, otherwise $I(E) =false$.}\n\\end{itemize}\n\n\\subsection{Model}\n\nIs our Interpretation $[I+A]$ with blank nodes a model of our example graph? \n\nLet's reason on $:ArabWroteOthello$ part only.\n\nBe $A$ our blank nodes mapping into $IR$: $\\_:z \\rightarrow zz$. \n\nOur interpretation $[I + A]$ will be: \n\n\\begin{itemize}\n \\item $IR=\\{awo, o, c, cc2, iacf, n, a, zz \\}$\n \\item $IP=\\{c, iacf, n\\}$\n \\item $IPC=\\{cc2\\}$\n\\end{itemize}\n\nThe functions:\n\n\\begin{itemize}\n \\item $IS(:ArabWroteOthello) \\rightarrow awo$\n \\item $IS(:Othello)\\rightarrow o$\n \\item $IS(conj002:creator)\\rightarrow cc2$\n \\item $IS(dc:creator)\\rightarrow c$\n \\item $IS(dbpedia-owl:nationality) \\rightarrow n$\n \\item $IS(:Arab)\\rightarrow a$\n \\item $IS(conj:isAConjecturalFormOf)\\rightarrow iacf$\n \\item $IEXT(c)=\\emptyset$\n \\item $IEXT(iacf)=\\{\\}$\n \\item $CONJFORM(c) = cc2$\n \\item $IEXT(n)= \\{\\}$\n \\item $IEXTC(cc2)=\\{\\}$\n\\end{itemize}\n\nEven in this case $IEXT(c)$ is empty because there is no assertion regarding the property $dc:creator$ yet, so our conjecture is, well, still a conjecture.\n\nThe last two functions hold because of the mapping $A$ of our blank node.\n\nAll the clauses are true. Our interpretation $I + A$ is a model of our graph.\n\nThis approach allows us to reason with the blank nodes in what-if scenarios, where we could define the $A$ mapping from blank nodes to specific resources of our choice, therefore exploring new relationships arising between the triples.\n\n\n\n\n\\section{Further applications of conjectures}\n\n\\subsection{Conjecture of a conjecture - nested conjectures}\n\nLet's imagine we want to express a conjecture on another conjecture.\nOf course, the process can involve as many conjectures over conjectures we want.\n\nSuch as:\n\n\\begin{verbatim}\n:conjecture01 {\n :Hamlet conj004:creator :EdwardDeVere .\n conj004:creator conj:isAConjecturalFormOf dc:creator .\n}\n\n:conjecture02 {\n :conjecture01 conj004:wasAttributedTo :JThomasLooney .\n conj004:wasAttributedTo conj:isAConjecturalFormOf prov:wasAttributedTo .\n}\n\n:conjecture03 {\n:conjecture02 conj004:wasInformedBy .\nconj004:wasInformedBy conj:isAConjecturalFormOf prov:wasInformedBy .\n}\n\n:conjecture03 prov:wasAttributedTo :FabioVitali . \n \n\\end{verbatim}\nThey are three different conjectures, one becoming the subject of the other one:\n\\begin{itemize}\n \\item the first one says that Hamlet is conjectured to have been written by Edward De Vere;\n \\item the second one says that the previous conjecture could possibly have been made by J. Thomas Looney;\n \\item The third one says that it might be that the second conjecture could have been brought to light by the resource with URL https:\/\/bit.ly\/3wSH76A.\n\\end{itemize}\nThe fourth triple says that the last conjecture has been attributed to Fabio Vitali.\n\nReading it (more or less) backwards:\nFabio Vitali has stated that the resource at https:\/\/bit.ly\/3wSH76A might have brought to light that J. Thomas Looney could have possibly said that Hamlet is conjectured to have been written by Edward De Vere.\n\nWe could imagine it as a stack, or a stair, where at level 0 we have the first conjecture, and then as we get on the higher steps, the conjectures we find are built with the conjecture at the level below as a subject (or object, or both):\n\\begin{verbatim}\nLevel 2: E{3}={E{2}, cp3, o3}\nLevel 1: E{2}=(E{1}, cp2, o2}\nLevel 0: E{1}=(s1, cp1, o1)\n\\end{verbatim}\n \nDeveloping the Level 2 we see they are nested in each other:\n$E{3}=(E{1}, cp2, o2),cp3, o3)=(((s1, cp1, o1), cp2, o2), cp3, o3)$\n\n\\subsubsection{Interpretation $[I + NESTEDCONJ]$}\n\nIn order to delineate the rules for the conjectures of conjectures (or nested conjectures), we can reason with pairs of conjectures.\n\nAs stated before, the conjectures at levels $> 0$ could be based on lower-level conjectures as their subject or object, or both.\n\nFor the sake of conciseness, we momentarily limit ourselves to reason with the case of the lower-level conjectures as their subject only.\n\n\\begin{enumerate}\n \\item if $E_{0}$ is a ground conjecture $(s_{0}, cp_{0}, o_{0})$, $E_{1}= (E_{0}, cp_{1}, o_{1})$\n \\item if $E_{1}$ is a ground conjecture $(E_{0}, cp_{1}, o_{1})$, $E_{2}= (E{1}, cp_{2}, o_{2})$\n [...]\n \\item if $E_{k-1}$ is a \"higher level\" ground conjecture $(E_{k-2}, cp_{k-1}, o_{k-1})$, $E_{k}=(E_{k-1}, cp_{k}, o_{k})$ \n \\item if $E_{k}$ is a \"higher level\" ground conjecture $(E_{k-1}, cp_{k}, o_{k})$, $E_{k+1}= (E_{k}, cp_{k+1}, o_{k+1})$ \n\\end{enumerate}\n\nWe should have the following cases:\n\\begin{itemize}\n \\item base case:\n \\begin{itemize}\n \\item $E_{0}=(s_{0}, p_{0}, o_{0})$\n \\end{itemize}\n \\item 1st cases:\n \\begin{itemize}\n \\item $E_{1}=(E_{0}, cp_{1}, o_{1})$\n \\item $E_{1}=(s_{1}, cp_{1}, E_{0})$\n \\end{itemize}\n \\item $k^{th}$ cases:\n \\begin{itemize}\n \\item $E_{k}=(E_{k-1)}, cp_{k}, o_{k})$\n \\item $E_{k}=(s_{k}, cp_{k}, E_{k-1})$\n \\end{itemize}\n\\end{itemize}\n\n\nLet's extend our interpretation $I$ adding new rules.\n\nThe extension will be subdivided into cases, depending on the type of conjectures to be evaluated.\n\nWe must also enforce an order on the operations: we start from the conjecture(s) at the \"lowest level\", that is, the one(s) not involving other conjectures, evaluate them and \"climb\" up the \"stair\".\n\nTherefore, the extension to the interpretation $I$ will be:\n\\newline\n\\textbf{Base case} - for the lowest level conjecture at level 0:\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E_{0}$ be a conjecture triple\n$\\\\ \\{s_{0} \\quad cp_{0} \\quad o_{0} \\quad .\\}$\nthen } \n \\begin{itemize}\n \\color{blue}\n \\item $I(E_{0}) = true$ if $I(cp_{0}) \\in IPC$ and\n \\item the pair $ \\in IEXTC(I(cp_{0}))$\n \\item otherwise $I(E_{0}) = false$ \n \\end{itemize}\n\\end{itemize}\n\nAs we \"climb\" up the \"stair\" and get to the conjecture $E_{k}$, we can assume that the conjecture $E_{k-1}$ has already been proved by the previous steps.\n\\newline\n\n\\textbf{$K^{th}$ Case S} - for conjectures at the first step and above having another conjecture as the subject:\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E_{k-1}$ be a conjecture triple\n\nlet $E_{k}$ be a conjecture triple\n$\\\\ \\{ E_{k-1} \\quad cp_{k} \\quad o_{k}\\quad . \\}$\n\nthen } \n \\begin{itemize}\n \\color{blue}\n \\item $I(E_{k}) = true$ if $I(E_{k-1}) = true$ and \n \\item $I(cp_{k}) \\in IPC$ and\n \\item the pair $ \\in IEXTC(I(cp_{k}))$\n \\item otherwise $I(E_{k}) = false$\n \\end{itemize}\n\\end{itemize}\n\n\n\\textbf{$K^{th}$ Case O} - for conjectures at the first step and above having another conjecture as the object:\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E_{k-1}$ be a conjecture triple\n\n\nlet $E_{k}$ be a conjecture triple\n$\\\\ \\{ s_{k} \\quad cp_{k} \\quad E_{k-1} \\quad . \\}$\n\nthen } \n \\begin{itemize}\n \\color{blue}\n \\item $I(E_{k}) = true$ if $I(E_{k-1}) = true$ and\n \\item $I(cp_{k}) \\in IPC$ and\n \\item the pair $ \\in IEXTC(I(cp_{k}))$\n \\item otherwise $I(E_{k}) = false$\n \\end{itemize}\n\\end{itemize}\n\n\n\\textbf{$K^{th}$ Case SO} - for conjectures at the first step and above having another conjecture as the subject and yet another one as the object:\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E_{(k-1)a}$ be a conjecture triple\n\nlet $E_{(k-1)b}$ be a conjecture triple\n\nlet $E_{k}$ be a conjecture triple\n$\\\\ \\{ E_{(k-1)a} \\quad cp_{k} \\quad E_{(k-1)b} \\quad . \\}$\n\nthen } \n \\begin{itemize}\n \\color{blue}\n \\item $I(E_{k}) = true$ if $I(E_{(k-1)a}) = true$ and\n \\item $I(E_{(k-1)b}) = true$ and\n \\item $I(cp_{k}) \\in IPC$ and\n \\item the pair $ \\in IEXTC(I(cp_{k}))$\n \\item otherwise $I(E_{k}) = false$\n \\end{itemize}\n\\end{itemize}\n\n\n\\subsubsection{Model}\nIs our Interpretation $[I + NESTEDCONJ]$ a model of the nested conjectures example seen before?\n\nWe define the sets:\n\n\\begin{itemize}\n \\item $IR=\\{c1, h, cc4, edv, iacf, c, c2, cwa4, jtl, pwa, c3, cwib4, http, pwib, fv\\}$\n \\item $IP=\\{c, iacf, pwa, pwib\\}$\n \\item $IPC=\\{cc4, cwa4, cwib4\\}$\n\\end{itemize}\n\nThe functions:\n\n\\begin{itemize}\n \\item $IS(:conjecture01) \\rightarrow c1$\n \\item $IS(:Hamlet)\\rightarrow h$\n \\item $IS(conj004:creator)\\rightarrow cc4$\n \\item $IS(:EdwardDeVere)\\rightarrow edv$\n \\item $IS(conj:isAConjecturalFormOf)\\rightarrow iacf$\n \\item $IS(dc:creator)\\rightarrow c$\n \\item $IS(:conjecture02) \\rightarrow c2$\n \\item $IS(conj004:wasAttributedTo) \\rightarrow cwa4$\n \\item $IS(:JThomasLooney)\\rightarrow jtl$\n \\item $IS(prov:wasAttributedTo)\\rightarrow pwa$\n \\item $IS(:conjecture03) \\rightarrow c3$\n \\item $IS(conj004:wasInformedBy) \\rightarrow cwib4$\n \\item $IL() \\rightarrow http$\n \\item $IS(prov:wasInformedBy)\\rightarrow pwib$\n \\item $IS(:FabioVitali) \\rightarrow fv$\n \\item $IEXT(iacf)=\\{,,\\}$\n \\item $IEXT(c)=\\emptyset$\n \\item $IEXT(pwa)=\\{\\}$\n \\item $IEXT(pwib)=\\emptyset$\n \\item $IEXTC(cc4)=\\{\\}$\n \\item $IEXTC(cwa4)=\\{\\}$\n \\item $IEXTC(cwib4)=\\{\\}$\n \\item $CONJFORM(c) = cc4$\n \\item $CONJFORM(cwa4) = pwa$\n \\item $CONJFORM(cwib4) = pwib$\n\\end{itemize}\n\nLet's check the validity of the rules of the new semantic extension.\n\nWe must start from the conjecture not depending on other conjecture, that is $:conjecture01$, and we use the Base Case:\n\n\\begin{itemize}\n \\item Is $cc4 \\in IPC$? Yes\n \\item Is the pair $ \\in IEXTC(cc4)$ Yes\n \\end{itemize}\n\nThe \"base\" conjecture seems to hold.\nWe can say that $c1 = true$.\n\\newline\n\nNow we \"climb the stair\" to $:conjecture02$. Since it is based on another conjecture ($:conjecture01$, already proved true) as its subject, we use \"$k^{th}$ Case S\" \n\n \\begin{itemize}\n \\item is $c1 = true$? Yes\n \\item is $cwa4 \\in IPC$? Yes\n \\item is the pair $ \\in IEXTC(cwa4)$? Yes\n \\end{itemize}\n\nThen we can say that $c2 = true$.\n\\newline\n\nLet's climb one step higher. $:conjecture03$ is based on $:conjecture02$ as its subject. We still use \"$k^{th}$ Case S\"\n\n \\begin{itemize}\n \\item is $c2 = true$? Yes\n \\item is $cwib4 \\in IPC$? Yes\n \\item is the pair $ \\in IEXTC(cwib4)$? Yes\n \\end{itemize}\n\n$\\rightarrow c3 = true$.\n\\newline\nThe last triple $:conjecture03 prov:wasAttributedTo :FabioVitali$ is satisfied by the rules of the simple intepretation $I$\n\\newline\nEverything seems to hold.\n\nWe can say that our Interpretation $[I + NESTEDCONJ]$ satisfies all the triples of the graph.\n\n\n\n\\subsection{Conjecture of a collapse} \\label{ConjectureOfACollapse}\nLet us consider the following sentence: \n\n\\begin{quote}According to Encyclopaedia Britannica (\\url{https:\/\/bit.ly\/3qgakFT}), Samuel Johnson attributed Hamlet to William Shakespeare, and he was right in saying so.\\end{quote}\n\nThis sentence is more than a mere collapse: it is a conjecture by an article in Encyclopedia Britannica a) attributing to Samuel Johnson a conjecture (about the authorship of Hamlet), and b) expressing total confidence in such conjecture (i.e., collapsing it). \n\nIts representation is:\n\n\\begin{verbatim}\n:attribution01 {\n :Hamlet conj005:creator :WilliamShakespeare .\n conj005:creator conj:isAConjecturalFormOf dc:creator .\n}\n\n:collapseOfAttribution01 {\n :attribution01 conj005:wasAttributedTo :SamuelJohnson .\n conj005:wasAttributedTo conj:isAConjecturalFormOf prov:wasAttributedTo .\n\n :collapseOfAttribution01 conj005:collapses :attribution01 .\n conj005:collapses conj:isAConjecturalFormOf conj:collapses .\n}\n\n:collapseOfattribution01 prov:wasInformedBy .\n\\end{verbatim}\n\nIn this case we have a conjecture of a collapse that, if collapsing, as a side effect, triggers the collapse of another conjecture, therefore asserting it in full force.\nPlease note that, in its current and \"uncollapsed\" status, everything is conjectured.\n\nWe can see the familiar predicate $collapses$ in its conjectural form, not yet in force in this case.\nThis predicate, once met in its \"effective\" form, will trigger the collapse of its subject. \nWe will see it in the next section.\n\n\n\\subsection{Cascading collapses}\n\nCascading collapses are all the collapses that take place recursively when multiple and nested conjectures of collapses collapse.\n\nFor example, what if we wanted to state that the $collapseOfAttribution01$ is true? We would collapse it by adding the \"collapsing triple\" and the \"collapse triple\" as per the rules of Interpretation [I + COLLAPSE] seen before in \\ref{I+COLLAPSE}.\n\nFor the sake of simplicity, we resort to the notion of collapse graph (\\ref{CollapseGraphs}). According to that, we can safely group the added triples into the collapse graph $:collapseOfcollapseOfAttribution01$.\n\nNow that our conjecture of a collapse is declared true, therefore the collapse is true and effective, we have to enforce it, collapsing $:attribution01$\n\nGeneralising, once a conjecture of a collapse is declared true, we must enforce a new special rule for its collapse to reality:\n\nWhenever $collapses$ is met in its effective form inside a collapse graph, its object, if it is a conjecture, must be collapsed.\n\nIf the object is a conjecture triple $s \\:cp \\:o$, we need to add the \\textit{collapsing triple} and the \\textit{collapse triple}.\n\nGetting back to our example, we can see that the \\textit{collapse triple} is already defined in its effective form, in fact we have that:\n\\begin{verbatim}\n :collapseOfAttribution01 conj:collapses :attribution01 .\n\\end{verbatim}\n\nSo we just need to add the collapsing triple, stating the conjecture in its full force, in the graph where the $conj:collapses$ property is met in full force.\n\nThe result will be as follows:\n\n\\begin{verbatim}\n:attribution01 {\n :Hamlet conj005:creator :WilliamShakespeare .\n conj005:creator conj:isAConjecturalFormOf dc:creator .\n}\n\n:collapseOfAttribution01 {\n :attribution01 conj005:wasAttributedTo :SamuelJohnson .\n conj005:wasAttributedTo conj:isAConjecturalFormOf prov:wasAttributedTo .\n\n :collapseOfAttribution01 conj005:collapses :attribution01 .\n conj005:collapses conj:isAConjecturalFormOf conj:collapses .\n}\n:collapseOfattribution01 prov:wasInformedBy .\n\n:collapseOfcollapseOfAttribution01 {\n :attribution01 prov:wasAttributedTo :SamuelJohnson .\n \n :collapseOfAttribution01 conj:collapses :attribution01 .\n\n :collapseOfcollapseOfAttribution01 conj:collapses :collapseOfAttribution01 . \n .\n\n :Hamlet dc:creator :WilliamShakespeare.\n}\n\n\\end{verbatim}\n\nA collapse graph is built for the first collapse of the conjecture of the collapse, and then, at the end, the final \"effective\" triple.\n\nWe need to extend the Interpretation [I + COLLAPSE] seen before in \\ref{I+COLLAPSE}.\n\nAfter we collapse the conjecture of the collapse following the rules in [I+COLLAPSE], we will find the property $collapses$ in its effective form inside the collapse graph.\n\nWe need to extend the notion of $collapses$ defining it from collapse graphs to conjectural graphs:\n$collapses(G)=(E)$ iff $\\forall (s,cp,o) \\in E, conj(s,p,o)=(s,cp,o) with (s,p,o) \\in G$.\n\nIf the object of the $collapses$ property is a triple, which can be considered as a special case of a graph, we will add its effective form to the conjectural graph $collapses$ is in. \n\nIf the object of $collapses$ property is a conjectural graph, we need to add the effective forms of all its conjectures.\n\nOne important thing to be aware of is that we disregard the subject of the $collapses$ property: the effective form of the conjecture(s) in the object(s) is added to the \\textit{current} collapse graph no matter their subject(s).\n\nIn the semantic conditions below we will simplify for clarity by adopting the notion of a generic graph as the theoretical subject of a $collapses$ property. The graphs will in fact be more than one, if the subjects of the conjectures which are objects of $collapses$ are distinct (as in the example above). \nAlso The conjecture graphs\/triples can be more than one.\n\nThe semantic conditions are:\n\n\\textbf{conjecture triple:}\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E$ be a conjecture triple\n$\\\\ \\{s \\quad cp \\quad o \\quad .\\}$\n\nlet $E_{g}$ be a generic graph\n\nlet $E_{cg}$ be a collapse graph \n$\\\\ \\{ E_{g} \\quad \\mathit{conj:collapses} \\quad E \\\\\n s \\quad p \\quad o . \\}$\n\nthen } \n \\begin{itemize}\n \\color{blue}\n \\item $I(E) = true$ if $I(cp) \\in IPC$ and\n \\item $I(E_{cg}) = true$ if $I(p) \\in IP$ and\n \\item $CONJFORM(I(p))=I(cp)$ and\n \\item the pair $ \\in IEXTC(I(cp))$ and\n \\item the pair $ \\in IEXT(I(p))$ and\n \\item $I(conj:collapses) \\in IP$ and\n \\item the pair $ \\in IEXT(I(conj:collapses))$\n \\item otherwise $I(E) = false$ and $I(E_{cg})$ is false.\n \\end{itemize}\n\\end{itemize}\n\n\\textbf{conjectural graph:}\nIf we are dealing with a conjectural graph as the object of $conj:collapses$:\n\\begin{itemize}\n\\item { \\color{blue} \nlet $E$ be a conjectural graph\n$\\\\ \\{s_{0} \\quad cp_{0} \\quad o_{0} \\quad . \\\\\n {[...]} \\\\\n s_{k} \\quad cp_{k} \\quad o_{k} \\quad . \\}$\n\nlet $E_{g}$ be a generic graph\n\nlet $E_{cg}$ be a collapse graph \n$\\\\ \\{ s_{0} \\quad p_{0} \\quad o_{0} . \\\\\n {[...]} \\\\\n s_{k} \\quad p_{k} \\quad o_{k} \\quad . \\\\\n E_{g} \\quad \\mathit{conj:collapses} \\quad E\\}$\n\nthen } \n \\begin{itemize}\n \\color{blue}\n \\item $I(E) = true$ if $I(cp_{k}) \\in IPC \\forall cp_{k} in E$ and\n \\item $I(E_{cg}) = true$ if $I(p_{k}) \\in IP \\forall p_{k} in E_{cg}$ and\n \\item $CONJFORM(I(p_{k}))=I(cp_{k}) \\forall p_{k} in E$ and\n \\item any pair $ \\in IEXTC(I(cp_{k}))$ and\n \\item any pair $ \\in IEXT(I(p_{k}))$ and\n \\item $I(conj:collapses) \\in IP$ and\n \\item the pair $ \\in IEXT(I(conj:collapses))$\n \\item otherwise $I(E) = false$ and $I(E_{cg})$ is false.\n \\end{itemize}\n\\end{itemize}\n\n\\subsubsection{Model}\n\nLet's check once again if the interpretation can be a model for our last example.\n\nWe define the sets:\n\n\\begin{itemize}\n \\item $IR=\\{a1, h, cc5, ws, iacf, c, coa1, c5wat, sj, pwat, c5cl, cl, pwib, url, ccoa1 \\}$\n \\item $IP=\\{c, iacf, pwat, pwib, cl\\}$\n \\item $IPC=\\{cc5, c5wat, c5cl\\}$\n\\end{itemize}\n\nThe functions:\n\n\\begin{itemize}\n \\item $IS(:attribution01) \\rightarrow a1$\n \\item $IS(:Hamlet)\\rightarrow h$\n \\item $IS(conj005:creator)\\rightarrow cc5$\n \\item $IS(:WilliamShakespeare)\\rightarrow ws$\n \\item $IS(conj:isAConjecturalFormOf)\\rightarrow iacf$\n \\item $IS(dc:creator)\\rightarrow c$\n \\item $IS(:collapseOfAttribution01) \\rightarrow coa1$\n \\item $IS(conj005:wasAttributedTo) \\rightarrow c5wat$\n \\item $IS(:SamuelJohnson)\\rightarrow sj$\n \\item $IS(prov:wasAttributedTo)\\rightarrow pwat$\n \\item $IS(conj005:collapses)\\rightarrow c5cl$\n \\item $IS(conj:collapses)\\rightarrow cl$\n \\item $IS(prov:wasInformedBy)\\rightarrow pwib$\n \\item $IL()\\rightarrow url$\n \\item $IS(:collapseOfcollapseOfAttribution01) \\rightarrow ccoa1$\n \\item $IEXT(c)=\\{\\}$\n \\item $IEXT(iacf)=\\{, , \\}$\n \\item $IEXT(pwat)=\\{\\}$\n \\item $IEXT(cl)=\\{, \\}$\n \\item $IEXT(pwib)=\\{\\}$\n \\item $IEXTC(cc5)=\\{\\}$\n \\item $IEXTC(c5wat)=\\{\\}$\n \\item $IEXTC(c5cl)=\\{\\}$\n \\item $CONJFORM(c) = cc5$\n \\item $CONJFORM(pwat) = c5wat$\n \\item $CONJFORM(cl) = c5cl$\n\\end{itemize}\n\n\nLet's check now the rules. \nWe focus the check on the cascading collapses only. \n\nWe need to proceed in steps. The first $conj:collapses$ in our final collapse graph has $:attribution01$ as its object.\nWe can consider $:attribution01$ as a single triple, as it contains only one conjecture:\n\n\\begin{itemize}\n\\color{blue}\n \\item is $cc5 \\in IPC$? Yes;\n \\item is $c \\in IP$? Yes;\n \\item is $CONJFORM(c)=cc5$? Yes;\n \\item is the pair $ \\in IEXTC(cc5)$? Yes;\n \\item is the pair $ \\in IEXT(c)$? Yes;\n \\item is $cl \\in IP$? Yes;\n \\item the pair $ \\in IEXT(cl)$?Yes.\n\\end{itemize}\n\nIt seems to hold.\n\nLet's get to the second $con:collapses$ of our final collapse graph. It has $:collapseOfAttribution01$ as its object.\nIt is a graph. Let's apply rules for the graphs:\n\n\n \\begin{itemize}\n \\color{blue}\n \\item $c5wat, c5cl \\in IPC$? Yes;\n \\item $pwat, cl \\in IP$? Yes;\n \\item $CONJFORM(c5wat)=pwat; CONJFORM(c5cl)=cl?$ Yes;\n \\item $ \\in IEXTC(c5wat)$? Yes; $ \\in IEXTC(c5cl)$? Yes;\n \\item $ \\in IEXT(pwat)$? Yes; $ \\in IEXT(cl)$? Yes;\n \\item $cl \\in IP$? Yes;\n \\item the pair $ \\in IEXT(cl)$? Yes\n \\end{itemize}\n\n\nIt can be easily verified that all the other triples of the example can be satisfied by the interpretation [I + COLLAPSE].\n\nOur interpretation is a model of the graph.\n\n\\section{Conclusions}\nIn this paper we have attempted to build a formal representation of the semantics of the \\textit{conjectures}.\n\nWe have started with the formal model and extended the simple interpretation and model of RDF 1.1.\n\nThen we have explored the key features of the \\textit{conjectures} and verified that they fit in the semantics, expanding it when needed.\n\nThe \\textit{conjectures} allow to express concepts whose truth value is unknown, without asserting it, something that is currently missing in RDF.\n\nWith this work we have demonstrated that, with a somewhat limited extension to its model, it is possibile to add this powerful feature to RDF 1.1.\n\n\\printbibliography[\ntitle={Bibliography}\n]\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{{INTRODUCTION}}\n\\label{introduction}\n\nProcessing the enormous volume of data sets generated by the social networks such as Facebook\n\\cite{facebook}\nand Twitter\n\\cite{twitter}\n, financial institutions, health-care industry, etc. has become a major motivations for data-parallel processing. In data-parallel processing applications, an incoming task needs a specific piece of data which is physically replicated on different servers. The task receives the service from a server which is close to the physical location of the stored data faster than another server far from the location where the data is stored. For instance, for a task, the speed of receiving the service from a server which has the data stored on its memory or local disk is much faster than receiving the service from a server which does not have the data. That forces the server to fetch the data from another server from the same rack, or even other remote racks. Unless the speed of data center networks have been increased, the differences between service time is still obviously large \\cite{intro1}, \\cite{intro2}, \\cite{intro3}, \\cite{nd1}. While assigning tasks, it is a critical consideration to schedule a task on a local server \\cite{nd1}, \\cite{nd2}, \\cite{nd3}, and \\cite{nd4}. Scheduling in this setting is called the near-data scheduling problem, or scheduling with data locality. As a result, scheduling and assigning tasks to more suitable servers makes the system stable in the capacity region of the system, and also reduces the mean delay experienced by all tasks in average sense. Therefore, to evaluate different algorithms for scheduling tasks to different servers, two optimality criteria are defined as follows:\n\\begin{itemize}\n\\item Throughput Optimality: A throughput optimal algorithm stabilizes the system in the whole capacity region. That is, the algorithm is robust to arrival rate changes, as long as the arrival rate is strictly in the capacity region of the system.\n\\item Heavy-traffic Optimality: A heavy-traffic optimal algorithm minimizes the mean task completion time as the arrival rate vector approaches the boundary of the capacity region (note that task completion time consists of both waiting time of the task to be assigned to a server and the service time). As a result, a heavy-traffic optimal algorithm assigns tasks to servers efficiently when the system is in the peak loads close to the capacity region boundary. This not only stabilizes the system, but also minimizes the mean delay in stressed load conditions.\\\\\n\\end{itemize}\n\nAlthough there are various heuristic algorithms taking the near-data scheduling into consideration for data centers with multiple levels of data locality \\cite{heuristic1}, \\cite{nd1}, \\cite{heuristic2}, their throughput and heavy-traffic optimality properties are not studied. In this paper, we discuss different real-time scheduling algorithms for systems with multiple levels of data locality with theoretical guaranties for throughput and heavy-traffic optimality. We also compare those algorithms against each other and evaluate them for mean task completion time in the capacity region, including both in high load and low load.\n\nIn the following, we first explain the common data center structures and problem definition. In Section \\ref{affinity}, we discuss prime algorithms such as Fluid Model Planning and Generalized c$\\mu$-Rule. We then discuss the problem for two and then three levels of data locality, in Section \\ref{twolevel} and Section \\ref{threelevel}, respectively. The paper is concluded in Section \\ref{conclusion}.\n\n\\section{{SYSTEM MODEL}}\n\\label{systemmodel}\nConsider the system consists of $M$ servers indexed by $m \\in \\{1, 2, \\dots, M \\} \\overset{\\Delta}{=} \\mathcal{M}$. Each server belongs to a specific rack denoted by $k \\in \\{1, 2, \\dots, K \\} \\overset{\\Delta}{=} \\mathcal{K}$. Without loss of generality, let's assume that the servers in a rack are indexed sequentially. That is, servers $\\{ 1, 2, \\dots, i \\}$ are in the first rack, servers $\\{ i + 1, i + 2, \\dots, j \\}$ are in the second rack, and so on. Let the rack for the server $m$ be $K(m)$. Each data chunk is stored on a set of servers denoted by $\\bar{L}$. In real world applications, $\\bar{L}$ consists of three servers. The reason to store the data in different servers is to allow the data to be accessed through other servers if a server disconnects from the network or fails to continue working. The larger the set $\\bar{L}$ is, the more secure the data would be. However, as the storage of servers is limited, the data is usually stored on no more than three servers. So, $\\bar{L} \\in \\{ (m_1, m_2, m_3) \\in \\mathcal{M}^3, m_1 < m_2 < m_3 \\}$, and the set of all task types is denoted by $\\mathcal{L}$. Therefore, different tasks can be labeled by the location of their associated data chunk. For example, all tasks with their stored data in servers $1, 2,$ and $7$ are denoted by type $\\bar{L} = \\{ 1, 2, 7 \\}$ task. All servers in the set $\\bar{L}$ are named local servers to the task type $\\bar{L}$ since they keep the data needed for the task to be processed. The set of servers $\\{ m \\notin \\bar{L}: \\exists n \\in \\bar{L} \\ \\text{s.t.} \\ K(m) = K(n) \\}$ are rack-local servers, and all other servers are named remote servers to type $\\bar{L}$ task. If server $m$ is local, rack-local, or remote to task of type $\\bar{L}$, it is denoted by $m \\in \\bar{L}$, $m \\in \\bar{L}_k$, or $m \\in \\bar{L}_r$, respectively. The system model is assumed to be discrete-time. If task $\\bar{L}$ is scheduled to server $m \\in \\bar{L}$ ($\\bar{L}_k$, or $\\bar{L}_r$), the probability that the task receives service in a time slot and departs the system at the end of the time slot is $\\alpha$ ($\\beta$, or $\\gamma$), where $\\alpha > \\beta > \\gamma$. In other words, the local, rack-local, and remote services follow $Geo(\\alpha)$, $Geo(\\beta)$, and $Geo(\\gamma)$, respectively. As a result, on average it takes less time for a task to receive service from a local server ($\\frac{1}{\\alpha}$), other than a rack local-local server ($\\frac{1}{\\beta}$), or a remote server ($\\frac{1}{\\gamma}$). On the other hand, the arrival of type $\\bar{L}$ task at the beginning of time slot $t$ is denoted by $A_{\\bar{L}}(t)$ which is bounded with average arrival $\\lambda_{\\bar{L}}$. The arrival of type $\\bar{L}$ tasks is assumed to be i.i.d. \\newline\nWith the service time distribution illustrated above, when a new task arrives, there might not be no local, rack-local, or remote available servers to serve the task immediately. Therefore, multiple queues exist in the data centers where the tasks are kept, waiting to receive service. Based on the structure of the data center and the scheduling algorithm, the number of queues can be less than, equal, or larger than the number of servers. For example, FIFO algorithm just needs one single queue for any number of servers to be implemented. In the rest of the paper, we will point out the number of queues needed for different algorithms. \\newline\nA question that might be raised here is \"At which queue should a new arriving task wait at so to finally receive service from a server?\". This part is handled by a \\textit{routing} algorithm which takes care of routing new tasks to appropriate queue. On the other hand, when a server is done with processing of a task and becomes idle, it needs to decide which task to pick among all tasks queued at all servers. The act of assigning tasks to idle servers is called \\textit{scheduling} with a little bit abuse of terminology. Therefore, an algorithm is fully defined by both its routing and scheduling policies. \\newline\nAs a new terminology, three levels of data locality refers to the case of having all kinds of local, rack-local, and remote services in the system. The number of data locality levels depends directly on the structure of the system. For example, if tasks receive their services just locally or remotely (no rack structure exists), then just two levels of data locality exists.\n\n\\subsection{{Capacity Region Realization}}\nLet $\\lambda_{\\bar{L}, m}$ denote the arrival rate of type $\\bar{L}$ tasks that receive service from the $m$-th server. In fact, $\\lambda_{\\bar{L}, m}$ is the decomposition of $\\lambda_{\\bar{L}}$. Assuming that a server can afford at most load 1 for all local, rack-local, and remote tasks, the capacity region can be characterized as follows \\cite{BalancedPandas, yekkehkhany2017near}:\n\n\\begin{equation}\n\\begin{aligned}\n\\Lambda = & \\{ \\boldsymbol{\\lambda} = (\\lambda_{\\bar{L}} : \\bar{L} \\in \\mathcal{L}) \\ | \\ \\exists \\lambda_{\\bar{L}, m} \\geq 0, \\forall \\bar{L} \\in \\mathcal{L}, \\forall m \\in \\mathcal{M}, s.t. \\\\\n& \\lambda_{\\bar{L}} = \\sum_{m = 1}^{M} \\lambda_{\\bar{L}, m}, \\ \\forall \\bar{L} \\in \\mathcal{L}, \\\\\n& \\sum_{\\bar{L}: m \\in \\bar{L}} \\frac{\\lambda_{\\bar{L}, m}}{\\alpha} + \\sum_{\\bar{L}: m \\in \\bar{L}_k} \\frac{\\lambda_{\\bar{L}, m}}{\\beta} + \\sum_{\\bar{L}: m \\in \\bar{L}_r} \\frac{\\lambda_{\\bar{L}, m}}{\\gamma} < 1, \\forall m \\}\n\\end{aligned}\n\\end{equation}\n\nA more thorough look into the definition of the capacity region $\\Lambda$, it is easy to figure out that for finding the capacity region of the system described in Section \\ref{systemmodel}, a linear programming optimization must be solved.\n\n\\section{Affinity Scheduling}\n\\label{affinity}\nThe near-data scheduling problem is a special case of affinity scheduling \\cite{afs1}, \\cite{mandelbaumstolyar}, \\cite{intro6}, \\cite{afs4}, and \\cite{afs5}. In this section, two algorithms, Fluid Model Planning and Generalized c$\\mu$-Rule, are illustrated which are somehow the pioneer works on the scheduling problems. However, they are not practical to be used in data centers as it will be discussed in the following two subsections.\n\\subsection{{FLUID MODEL PLANNING}}\nFor routing tasks and scheduling servers, fluid model planning algorithm is proposed by Harrison and Lopez \\cite{intro5} and \\cite{intro6} which is both throughput and heavy-traffic optimal not only for three levels of data locality, but also for the affinity scheduling problem. To implement this algorithm, distinct queues are needed for different types of tasks, $\\bar{L}$. Each new incoming task is routed to the queue of its own type. In the model described, there are at most in the order of $M^3$ types of tasks (the data associated to each task type is located on 3 servers, so at most there are ${M}\\choose{3}$ $= O(M^3)$ different task types). For finding the scheduling policy, the arrival rate of each task type is required to be known in advance. By solving a linear programming problem, the algorithm distinguishes basic activities. Based on the basic activities derived from the linear programming part, tasks are assigned to idle servers. There are two main objections for this algorithm: First, there should be in the order of $M^3$ number of queues for each task type. In practice, it is not practical to have queues in the cubic order of the number of servers. It excessively complicates the system and its underlying network. Second, the algorithm assumes the arrival rate of different types of tasks to be known. However, firstly the load is not known in real applications, and secondly it changes over time. Therefore, unless the algorithm is throughput and heavy-traffic optimal, it cannot be used in real applications.\n\n\\subsection{Generalized c$\\mu$-Rule}\nStolyar \\cite{stolyar} and Mandelbaum and Stolyar \\cite{mandelbaumstolyar} proposed Generalized c$\\mu$-Rule. On contrast to fluid model planning which uses the knowledge of the arrival rate of each task type, generalized c$\\mu$-rule uses MaxWeight notion which makes the algorithm needless of knowing the arrival rates. But similar to fluid model planning, the algorithm needs one queue per task type. For routing part, each incoming task joins the queue of its own type. Assume that the cost rate incurred by the type $\\bar{L}$ tasks queued is $C_{\\bar{L}}(Q_{\\bar{L}})$, where $Q_{\\bar{L}}$ is the queue length of type $\\bar{L}$ tasks, and the cost may generally depend on the task type. The cost function should have fairly normal features among which we can mention the followings: $C_{\\bar{L}}(.)$ is convex and continuous with $C_{\\bar{L}}(0) = 0$, $C^{'}_{\\bar{L}}(.)$ to be strictly increasing and continuous with $C^{'}_{\\bar{L}}(0) = 0$. Having the cost functions for different kinds of tasks, the server $m \\in \\mathcal{M}$ is scheduled to a task type $\\bar{L}$ in the set below when it becomes idle:\n\n\\begin{equation}\n\\begin{aligned}\n\\bar{L} \\in \\underset{\\bar{L}}{ArgMax} \\ \\bigg \\{ C^{'}_{\\bar{L}} (Q_{\\bar{L}}) \\mu_{\\bar{L}, m} \\bigg \\}\n\\end{aligned}\n\\end{equation}\n\nWhere $\\mu_{\\bar{L}, m}$ is $\\alpha$, $\\beta$, or $\\gamma$ if the task type $\\bar{L}$ is local, rack-local, or remote to the idle server $m$ respectively. For instance, if the holding cost for type $\\bar{L}$ task is $\\gamma_{\\bar{L}}Q_{\\bar{L}}^{\\beta + 1}$ with $\\beta > 0$ where satisfies all the conditions for a valid cost function, the generalized c$\\mu$-rule is proved by Stolyar to asymptotically minimize the holding cost below \\cite{stolyar}:\n\n\\begin{equation}\n\\begin{aligned}\n\\sum_{\\bar{L}} \\gamma_{\\bar{L}}^{ } Q_{\\bar{L}}^{\\beta + 1}\n\\end{aligned}\n\\end{equation}\n\nAs the constant $\\beta$ should be strictly positive in order that the cost function satisfies the conditions, the algorithm is not heavy-traffic optimal in the sense defined in Section \\ref{introduction}. Besides, using generalized c$\\mu$-rule, we still need $O(M^3)$ number of servers (where M is the number of servers). However, it is not practical having a large number of queues as the system becomes more complicated. \\newline\nAll the algorithms given in the next sections do not employ the knowledge of arrival rate, and they assume the system to have the same number of queues as the number of servers, which is a more realistic structure.\n\n\\section{{TWO LEVELS OF DATA LOCALITY}}\n\\label{twolevel}\n The model described in section \\ref{systemmodel} is for the case of three levels of data locality, as a task can be served with three different service rates $\\alpha$, $\\beta$, and $\\gamma$. However, most previous theoretical work, except the very last one by Xie, Yekkehkhany, and Lu \\cite{BalancedPandas}, has been done on two levels of data locality which is actually the base of three or more levels of data locality. The model for two levels of data locality is somehow the same as the one described in section \\ref{systemmodel}; except, in two levels of data locality, there is no notion of rack and rack-local service. Assuming the two levels of data locality, tasks either get service from a server in the set $\\bar{L}$ locally with rate $\\alpha$, or get service from any other servers remotely with rate $\\gamma$. Therefore, the capacity region would be revised as $\\sum_{\\bar{L}: m \\in \\bar{L}} \\frac{\\lambda_{\\bar{L}, m}}{\\alpha} + \\sum_{\\bar{L}: m \\notin \\bar{L}} \\frac{\\lambda_{\\bar{L}, m}}{\\gamma} < 1, \\forall m \\in \\mathcal{M}$. For two levels of data locality Wang et al. \\cite{MaxWeight}, and Xie, and Lu \\cite{Pandas} respectively proposed JSQ-MaxWeight, and Pandas algorithms that are discussed in the next two subsections below.\n\n\\subsection{{Join-the-Shortest-Queue-MaxWeight (JSQ-MW)}}\nFor two levels of data locality, JSQ-MW has been proven to be throughput optimal, but heavy-traffic optimal just in a specific traffic scenario \\cite{MaxWeight}. Wang et al. \\cite{MaxWeight} assume one queue per server, where the length of queue $m$ at time $t$ is denoted by $Q_m(t)$. A central scheduler maintains the lengths of all queues to decide the routing for new incoming tasks, and scheduling for idle servers. As of routing policy, when a new task of type $\\bar{L}$ arrives to the system, the central scheduler routes the task to the shortest queue of the servers in the set $\\bar{L}$ (all ties are broken randomly all over this paper). In other words, the new task is routed to the shortest local queue. For scheduling policy, as server $m$ becomes idle, the central scheduler assigns a task queued at a queue in the set below to server $m$:\n\n\\begin{equation}\n\\underset{n \\in \\mathcal{M}}{ArgMax} \\ \\{ \\alpha Q_n(t) I_{\\{ n=m \\}}, \\beta Q_n(t) I_{\\{ n \\neq m \\}} \\}\n\\end{equation}\n\nTherefore, the idle server gives its next available service to the queue with the maximum weight as defined above. As it was stated before, JSQ-MW is not heavy-traffic optimal in all loads. The specific traffic scenario which the JSQ-MW is heavy-traffic optimal is pointed out in section \\ref{evaluation}. For more details refer to \\cite{BalancedPandas, yekkehkhany2017near}, and \\cite{MaxWeight}.\n\nNext, Pandas algorithm proposed by Xie, and Lu \\cite{Pandas} is presented, which is both throughput and heavy-traffic optimal.\n\n\\subsection{{Priority Algorithm for Near-Data-Scheduling (Pandas)}}\nPandas algorithm is both throughput optimal and heavy-traffic optimal in all loads \\cite{Pandas}. Again, assuming one queue per server, Pandas algorithm routes the new incoming task to the shortest local queue (which is the same as JSQ-MW routing). For scheduling, an idle server always give its next service to a local task queued at its own queue; unless, its queue length is zero. If the idle server's queue does not have any tasks queued, the central scheduler assigns a task from the longest queue in the system to the idle server (which the task is remote to the idle server). This assignment of the remote task from the longest queue in the system occurs when $Q_{max}(t) \\geq \\frac{\\alpha}{\\gamma}$, to make sure that the remote task experiences less service time in the remote server other than waiting and receiving its service from the local server.\n\n\nAs the conclusion of the previous work for two levels of data locality, Pandas algorithm proposed by Xie, and Lu \\cite{Pandas} is the most promising algorithm that both stabilizes the system in the whole capacity region, and minimizes mean delay for all tasks in heavy-traffic regime. However, in real applications there are usually more than two levels of data locality. The reason is that a server may have the data stored on the memory or local disk, or the server may not have the data saved locally, so it has to fetch the data from another server in the same rack, or even from another server in other racks, which results in appearance of multi levels of data locality. Therefore, it is more of interest to come up with a throughput and heavy-traffic optimal algorithm for a system with more than two levels of data locality. The model illustrated in the system model section has three levels of locality, as a task can get its service locally, rack-locally, or remotely. For the purpose of designing a throughput and heavy-traffic optimal algorithm, assuming three levels of data locality is more challenging than two levels, as a trade-off between throughput optimality and delay optimality emerges. The Pandas algorithm proposed by Xie, and Lu \\cite{Pandas} which is both throughput and heavy-traffic optimal for two levels of data locality is not even throughput optimal for three levels of data locality. Xie, Yekkehkhany, and Lu \\cite{BalancedPandas} proposed two algorithms for three levels of data locality which are discussed in the next section.\n\n\\section{{THREE LEVELS OF DATA LOCALITY}}\n\\label{threelevel}\nFor three levels of data locality of which the system model is described in section \\ref{systemmodel}, Xie, Yekkehkhany, and Lu \\cite{BalancedPandas} extended the JSQ-MaxWeight algorithm and proved it to be throughput optimal. However, the extension of JSQ-MaxWeight is still heavy-traffic optimal only in a specific traffic scenario, not in all loads (note again that JSQ-MW is not also heavy-traffic optimal in all loads for two level of data locality, except a specific load). Xie et al. \\cite{BalancedPandas} also proposed a new algorithm called the weighted-workload routing and priority scheduling algorithm which is throughput optimal for any $\\alpha > \\beta > \\gamma$, and is heavy-traffic optimal for the case that $\\beta^2 > \\alpha \\gamma$, which usually holds in real data centers. What $\\beta^2 > \\alpha \\gamma$ implies is that the rack-local service is much faster than remote service. In the next two subsections, JSQ-MaxWeight and the weighted-workload routing and priority scheduling algorithm are discussed in more details.\n\n\\subsection{{Extension of JSQ-MaxWeight}}\nAssuming one queue per server, the JSQ-MW is as follows: \\\\\n\\textbf{Routing:} An arriving task is routed to the shortest queue of the servers in the set $\\bar{L}$ (shortest local queue). \\\\\n\\textbf{Scheduling:} An idle server is scheduled to a queue in the set\n\\begin{equation}\n\\begin{aligned}\n\\underset{n \\in \\mathcal{M}}{ArgMax} \\ \\{ & \\alpha Q_n(t) I_{\\{ n = m \\}}, \\beta Q_n(t) I_{\\{ K(n) = K(m) \\}},\\\\\n&\\gamma Q_n(t) I_{\\{ K(n) \\neq K(m) \\}} \\}\n\\end{aligned}\n\\end{equation}\n\n\n\\begin{theorem}\nJSQ-MaxWeight stabilizes the system under any arrival rate vector within $\\Lambda$. Therefore, JSQ-MaxWeight is throughput optimal.\n\\end{theorem}\n\nProof outline. If $\\boldsymbol{\\lambda} \\in \\Lambda$, using $V(t) = \\sum_{n = 1}^{M} Q_n^2(t)$ as the Lyapunov function, the $T$ time slot drift of $V(t)$ is bounded in a finite subset of state space of the system, and is negative outside of this subset, which results in stability of the system by Foster-Lyapunov theorem \\cite{BalancedPandas, yekkehkhany2017near} (you can refer to \\cite{MaxWeight, Pandas, ghassami2017covert} for other uses of Foster-Lyapunov theorem for stability proof of a system).\n\n\\begin{theorem}\nThe extended JSQ-MaxWeight is heavy-traffic optimal in a special scenario of the workload, but not in all traffic scenarios.\n\\end{theorem}\n\n\\subsection{{Weighted Workload Routing and Priority Scheduling}}\nAlthough it is sufficient to have one queue per server to implement the weighted-workload routing and priority scheduling algorithm in a data center with three levels of data locality, it is easier to describe this algorithm assuming the existence of three queues per server. Therefore, assume each server to have three queues, which local, rack-local, and remote tasks to the server are queued in the three queues separately. The central scheduler keeps the vector of queue lengths $\\boldsymbol{Q}(t) = (\\boldsymbol{Q}_1(t), \\boldsymbol{Q}_2(t), \\dots, \\boldsymbol{Q}_M(t))$ where $\\boldsymbol{Q}_m(t) = (Q_m^l(t), Q_m^k(t), Q_m^r(t))$. That is, the first queue of $m$-th server keeps the tasks that are routed to server $m$ and are local to it, the second queue of the $m$-th server keeps the tasks that are routed to this server, but are rack-local to it, and finally remote tasks to the $m$-th server that are routed to this server are queued in the third queue.\n\nDefining the workload on a server, it is ready to give the routing and scheduling policies. The workload on the $m$-th server is defined as the average time needed for the server to give service to all local, rack-local, and remote tasks queued in front of it. The workload on the $m$-th server is defined below:\n\n\\begin{equation}\nW_m(t) = \\frac{Q_m^l(t)}{\\alpha} + \\frac{Q_m^k(t)}{\\beta} + \\frac{Q_m^r(t)}{\\gamma}\n\\end{equation}\n\n\n\\textbf{Weighted-Workload Routing:}\nAs a new task arrives, it joins the server with the least weighted workload. More precisely, the new task joins one of the servers in the following set, where the ties are broken randomly.\n\\begin{equation}\n\\underset{m \\in \\mathcal{M}}{ArgMin} \\ \\bigg \\{ \\frac{W_m(t)}{\\alpha} I_{\\{ m\\in \\bar{L} \\}},\\frac{W_m(t)}{\\beta} I_{\\{ m\\in \\bar{L}_k \\}}, \\frac{W_m(t)}{\\gamma} I_{\\{ m\\in \\bar{L}_r \\}} \\bigg \\}\n\\end{equation}\nIf the task is local (rack-local, or remote) to the server with the least weighted workload, it joins the first (second, or third) queue which is $Q_m^l$ ($Q_m^k$, or $Q_m^r$).\n\n\\textbf{Priority Scheduling:}\nWhen a server, say $m$, becomes idle, it gives its next service to local tasks queued in front of it at $Q_m^l$. In case that there is no local task available for idle server $m$, that is $Q_m^l = 0$, next service is assigned to rack-local tasks queued at $Q_m^k$. Finally if both local, and rack-local queues are empty, the next service goes to $Q_m^r$. In summary, the idle server gives the most priority to local, then rack-local, and finally remote tasks. If all three sub-queues of server $m$ are empty, the server remains idle; until, a new task joins any of the three sub-queues.\n\n\\begin{theorem}\nThe weighted-workload routing and priority scheduling algorithm is throughput optimal, as it stabilizes any arrival rate vector in the capacity region.\n\\end{theorem}\n\nProof outline. The $T$ time slot drift of the following Lyapunov function is bounded in a finite subset of state space, and is negative out of this subset, as long as the arrival rate vector is in the capacity region ($\\boldsymbol{\\lambda} \\in \\Lambda$), which results in the stability of the system \\cite{BalancedPandas, yekkehkhany2017near}.\n\\begin{equation}\nV(t) = ||\\boldsymbol{W}||^2 = \\sum_{m \\in \\mathcal{M}} \\bigg ( \\frac{Q_m^l(t)}{\\alpha} + \\frac{Q_m^k(t)}{\\beta} + \\frac{Q_m^r(t)}{\\gamma} \\bigg )^2\n\\end{equation}\n\n\\begin{theorem}\n\\label{htobp}\nThe weighted-workload routing and priority scheduling algorithm is heavy-traffic optimal for $\\beta^2 > \\alpha \\gamma$ \\cite{BalancedPandas}.\n\\end{theorem}\n\n\n\n\n\\section{{CONCLUSION}}\n\\label{conclusion}\nIn this paper, we first discussed the history of task routing and affinity scheduling in data centers. Two algorithms are then proposed for a system with two levels of data locality: JSQ-MaxWeight, and Pandas algorithms for Near-Data Scheduling (Pandas)- both of which are throughput optimal. However, it was shown that Pandas is the only algorithm being heavy-traffic optimal in all loads. Taking further steps to three levels of data locality, we mentioned that the Pandas algorithm known to be heavy-traffic optimal for two levels of data locality is not even throughput optimal for three levels of data locality. Then, an algorithm with weighted workload routing and priority scheduling as well as an extension of JSQ-MaxWeight are discussed for three levels of data locality. Among these two algorithms only the weighted-workload routing and priority scheduling algorithm is heavy-traffic optimal in all loads. \n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nCryptosystems based on lattices are one of the leading alternatives to RSA and ECC that are conjectured to be resistant to quantum attacks. Considered to be the genesis of the study of lattice cryptosystems, in 1996, Ajtai constructed a one-way function, and proved the average-case security related to the worst-case complexity of lattice problems \\cite{ajtai}. Later, in 2005, Regev proposed the computational problem known as ``Learning With Errors'' (LWE) and showed that LWE is as hard to solve as several worst-case lattice problems \\cite{LWE}. Whilst these problems are believed to be difficult to crack even given access to a quantum computer, their main drawback is their impracticality to implement in cryptosystems due to the large key sizes required to define them.\n\\\\ \\indent For this reason, lattices with algebraic structure are often favoured to define cryptosystems over conventional lattices. In particular, cryptosystems often employ the use of cyclotomic polynomials as their cyclic nature allows for much less cumbersome computations. Such lattices come in two main varieties: ideal lattices, whose structures are formed entirely by embedding an ideal of a number field into real or complex space, and module lattices, which are free modules defined over an algebraic ring and can be thought of as a compromise between classical and ideal lattices. Perhaps the most well-known cryptosystem based on algebraic structure is the NTRU cryptosystem. Developed in 1996 by Hoffstein, Pipher and Silverman \\cite{NTRU}, the NTRU cryptosystem uses elements of the convolution ring $\\mathbb{Z}[x]\/(x^p-1)$ and offers efficient encryption and decryption of messages, making it one of the most popular lattice-based cryptosystems even to this day. In \\cite{securentru}, noting that the ring $\\mathbb{Z}[x]\/(x^p-1)$ could be deemed insecure due to the fact that $x^p-1$ is not irreducible, Stehl\\'e and Steinfeld updated NTRU to incorporate a cyclotomic ring in place of the aforementioned ring. The computational problem involved in breaking NTRU can be thought of as a rank 2 module problem over a cyclotomic ring. More recently, an algebraic variant of LWE called Ring-LWE (RLWE) was developed by Lyubashevsky, Peikert and Regev in 2010 \\cite{RLWE}. It has been shown that the security of this scheme relies heavily on the hardness of ideal lattice reduction \\cite{RLWE-hardness}. Moreover, the work by Ajtai was also generalised to the ring case by Micciancio in 2004 \\cite{knapsack}. Using an arbitrary ring in place of a classical lattice, he managed to show that obtaining a solution to the ring-based alternative to the knapsack problem on the average was at least as hard as the worst-case instance of various approximation problems over cyclic lattices, even for rings with relatively small degree over $\\mathbb{Z}$. Whilst there are a myriad of other schemes that make use of algebras to define a cryptosystem (see for example \\cite{newhope}, \\cite{kyber}, \\cite{falcon}), concerns have been raised regarding the security of such schemes. Whilst the algebraic structure might allow for easier computations, the additional structure given by an algebra could be exploited to allow for easier reduction of lattices based on algebras.\n\\\\ \\indent As we have already mentioned, cyclotomic polynomials exhibit many properties that are desirable in cryptography. In particular, cryptographers largely favour power-of-two cyclotomic rings, that is, cyclotomic rings with conductor $N=2^n$ for some integer $n$. This is largely a consequence of a few properties exhibited by power-of-two cyclotomic rings, for example, $N\/2$ is also a power of two and arithmetic in the ring can be performed with ease using the $N$-dimensional FFT. However, restricting cryptosystems to only using power-of-two cyclotomic rings has its drawbacks. The most obvious of these drawbacks is the increase in dimension of the ideal lattice when moving from one power-of-two cyclotomic ring to the next, which doubles with each successive ring, and the cryptosystem may require a lattice of intermediate security: for example, ideal lattices of the cyclotomic ring of conductor $1024$ have dimension $512$, but the next power-of-two cyclotomic ring is of conductor $2048$, and so ideal lattices defined in this ring have dimension $1024$, which is a significant jump. For this reason amongst others, cryptographers have begun to move away from power-of-two cyclotomic rings to cyclotomic rings of more general conductor. However, this migration from power-of-two cyclotomics is relatively novel, and as such literature regarding the reduction of ideal lattices based on cyclotomic rings of general conductor is still heavily lacking.\n\\subsection{Previous Works}\nThere have been a variety of studies into the shortest vector problem (SVP) in lattices generated by ideals. In 2016, Cramer, Ducas, Peikert and Regev published a paper detailing an attack on ideals generated by principal ideals of prime-power cyclotomic rings \\cite{shortgen}. They presented a technique involving the use of the log-unit lattice, and showed that there is a polynomial time classical reduction from the problem of finding a short generator of a principal ideal (SPIP) to the problem of finding a generator of a principal ideal (PIP). Moreover, in \\cite{biasset}, it was shown how to solve PIP in polynomial time with a quantum computer by Biasset and Song. Thus, combining these results, there is a polynomial time quantum algorithm for SPIP, and we note that SPIP corresponds to finding approximate short vectors in principal ideals. Later, Cramer, Ducas, and Wesolowski gave a quantum (heuristic) reduction of the general approx-SVP algorithm to SPIP \\cite{stickelb}. Therefore a quantum polynomial algorithm for approx-SVP follows from the above three papers.\\\\\n\\indent Simultaneously, largely inspired by Bernstein's work on subfield attacks against ideal lattices \\cite{subfieldlogarithm}, Albrecht, Bai and Ducas proposed a different method by which to attack the NTRU cryptosystem with overstretched parameters - that is, the NTRU encryption scheme with much larger modulus \\cite{subfield}. Their method entailed an attack on the NTRU cryptosystem by attacking a sublattice, defined by a public key attained by ``norming down'' the public key of the original lattice to a subfield, and then ``lifting'' a solution on the sublattice to a solution for the original cryptosystem which, provided the solution is sufficiently good in the sublattice, may yield a short lattice vector in the full lattice. Indeed, there are many examples of previous works which detail lattice attacks against ideal lattices. For a detailed list regarding previous research into the reduction of ideal lattices, we refer the reader to the ``previous works'' section of \\cite{idealrandomprime}.\n\\\\ \\indent Recently, Pan, Xu, Wadleigh and Cheng pioneered a remarkable technique to approach the problem of SVP in prime and general ideal lattices, obtained from power-of-two cyclotomics \\cite{idealrandomprime}. Their method involved manipulating the decomposition group of prime ideals in order to significantly reduce the dimension of the lattice required to solve SVP for. Their primary contributions were in two parts: the first part of the paper took a number field $L$ that is Galois over $\\mathbb{Q}$ and showed that, given a prime ideal of its ring of integers $\\mathcal{O}_L$, if Hermite-SVP can be solved for a certain factor in a sublattice generated by a subideal, this yields a solution for Hermite-SVP in the original ideal lattice with a larger factor, where the factor's increase depends only on the square root of the degree of $L$ over $\\mathbb{Q}$ divided by the size of the decomposition group. The second part of their paper is dedicated to ideals over the ring of integers of cyclotomic fields of conductor $N=2^{n+1}$. Under the so-called coefficient embedding, they showed that using a subgroup of the decomposition group of a prime ideal, the shortest vector in the ideal is equivalent to the shortest vector in a subideal constructed in the paper, and so solving SVP over such ideals is easy given an oracle to solve SVP in lattices of dimension equivalent to the dimension of the ideal lattice generated by the subideal. Moreover, they showed that if such a prime ideal lies above a rational prime $p$ of the form $p \\equiv \\pm 3 \\mod 8$ then the shortest vector is of length $p$, and is very easy to determine. In the final section, they showed that their method also worked for general ideals by considering the prime decomposition of an ideal.\n\\subsection{Our Results}\nIn this paper, we generalise the results of Pan et. al., both in their work on Hermite-SVP for prime ideal lattices and solving SVP exactly for prime ideals in cyclotomic ideals. The first half of the paper is dedicated to the Hermite-SVP in ideal lattices. Whilst Pan et. al. only covered the case for lattices based on prime ideals lying above unramified primes, we extend their result to the case of general ideals whose prime ideal factors all lie over unramified primes, showing that by solving Hermite-SVP on a subideal with some factor $\\gamma$, the solution may be lifted to yield a solution for Hermite-SVP in the original lattice with factor $\\gamma^{\\prime}$, where $\\gamma^{\\prime}\/\\gamma$ depends only on the factor given by Pan et. al. multiplied by a value determined by certain properties of the ideal and its decomposition group. We take this notion even more generally, and consider a module over the ring of integers of a Galois field and provide a method where a solution to Hermite-SVP in a submodule generating a lattice of lower dimension may be lifted to a solution in the original module lattice for Hermite-SVP with an upper bound, where the new constant is given in terms of the old factor multiplied by some factor dependent only on the ideals used to describe the module in the pseudo-basis representation.\n\\\\ \\indent The second half of the paper focuses on prime ideals of cyclotomic rings. Our work extends the results of Pan et. al. to prime ideal lattices constructed from more cyclotomic rings of more general conductors, covering the cases of a general composite conductor $N=s2^{n+1}$ and $s^{\\prime} p^{n+1}$ for some odd prime $p$, odd integer $s \\geq 3$ and integer $s^{\\prime}$, $\\gcd(s^{\\prime},p)=1$, which, combined with the work of Pan et. al., covers the case for any conductor $N$. In particular, our work shows that if the prime ideal in question lies above certain primes, then the dimension in which we have to solve SVP decreases significantly.\n\\begin{theorem}\nLet $N=s2^{n+1}$, where $n$ is a positive integer and $s \\geq 3$ is an odd integer. Let $\\mathfrak{p}$ be a prime ideal in the ring $\\mathbb{Z}[\\zeta_N]$ and suppose that $\\mathfrak{p}$ contains a rational prime $\\rho$, where $\\rho^{\\phi(s)} \\equiv 3 \\mod 4$. Then, given an oracle that can solve SVP for $\\phi(s)$-dimensional lattices, a shortest nonzero vector in $\\mathfrak{p}$ can be found in time $\\text{poly}(\\phi(N),\\log_2 \\rho)$ under the canonical embedding.\n\\end{theorem}\n\\begin{theorem}\nLet $N=sp^{n+1}$, where $n$ is a positive integer, $p$ is an odd prime and $s$ is a positive integer such that $\\gcd(s,p)=1$. Let $\\mathfrak{p}$ be a prime ideal in the ring $\\mathbb{Z}[\\zeta_N]$ and suppose that $\\mathfrak{p}$ contains a rational prime $\\rho$, where $\\rho^{\\phi(s)} =lp + a$ for some integers $l,a$, $\\gcd(l,p)=\\gcd(a,p)=1$. Then, given an oracle that can solve SVP for $(p-1)\\phi(s)$-dimensional lattices, a shortest nonzero vector in $\\mathfrak{p}$ can be found in time $\\text{poly}(\\phi(N),\\log_2 \\rho)$ under the canonical embedding.\n\\end{theorem}\n\\noindent As with the case for conductor $N=2^{n+1}$, we must ask whether the ``average case'' of prime ideal SVP over such cyclotomic rings is easy. This question in itself is ill-defined, and depends on how we define the distribution from which we choose the prime ideal. As we show in section 5, if we were to pick our ideal by uniformly choosing an ideal from the set of prime ideals whose rational prime lies below a certain bound, the probability of choosing an easily solvable ideal lattice is non-negligible. However, if we are to uniformly choose from the distribution of ideals of norm less than a certain bound, the probability of choosing an easily solvable ideal lattice is negligible.\n\\\\ \\indent In the last few sections, we also cover the case of general cyclotomic ideals and modules defined over a pseudo basis of ideals and vectors. For the case of general ideals, in a similar fashion to that in Pan et. al.'s work, we analyse SVP by studying the prime decomposition of ideals, and show that the shortest nonzero vector in a general ideal can be found by finding the shortest nonzero vector in a subideal of a smaller dimension. Moreover, the algorithm used to tackle SVP in such a lattice does not use the prime decomposition of the ideal, which is the most computationally complex step after SVP. In the module case, we do not explicitly construct an algorithm to perform SVP, however we show that using the structure theorem for finitely generated modules over a principal ideal domain it is possible to construct an isomorphism such that SVP in the original module can be found by finding the shortest nonzero vector in a module which has smaller dimension as a lattice after canonically embedding the module.\n\\\\ \\indent Whilst the work on ideal-SVP may initially appear to destabilise the security of cryptosystems based on cyclotomic ideals, we must point out that this work does not break Ring-LWE. Though Ideal-SVP underpins the security of Ring-LWE, our work does not directly impact the security of these schemes, since the worst-case to average-case security reduction is one-way. However, our results on module SVP may cause concern for the security of MLWE, and hence RLWE. Unlike ideal SVP and RLWE, module SVP and MLWE are known to be (polynomially) equivalent, as are MLWE and RLWE. In section 6, we offer an algorithm that can be used to find the shortest vector in a general module lattice over a cyclotomic ring by solving SVP in a submodule defined over a cyclotomic ring of lesser degree, which decreases the dimension of the lattice required to perform SVP over under the canonical embedding significantly in some cases.\n\\subsection{Paper organisation}\nThe paper is organised as follows. Section 2 covers the mathematical preliminaries, including a definition of lattices and their various properties, some basic algebraic number theory and ideal lattices. The preliminary section ends with some useful lemmas regarding the factorisation of polynomials over finite fields. Section 3 covers a reduction Hermite-SVP in ideal lattices and module lattices based over a Galois extension of $\\mathbb{Q}$. In section 4, we present a reduction of SVP for prime ideals of cyclotomic rings of general conductor, and show that some special cases of prime ideals are much easier to perform SVP for than others. In section 5, we show that our method for prime ideal lattices may be lifted to the case of general ideal lattices of cyclotomic rings. In section 6, we show that modules over cyclotomic rings may be subject to a similar reduction of SVP by studying the decomposition group of the module, and provide an algorithm which can be used to solve SVP in module lattices over cyclotomic rings. In section 7, we discuss the average-case hardness of SVP for ideal and module lattices of cyclotomic rings. \n\\section{Mathematical Preliminaries}\n\\subsection{Lattices and the Shortest Vector Problem}\nA lattice is a discrete additive subgroup of $\\mathbb{R}^D$. A lattice $L$ has a basis $B=\\{\\mathbf{b}_1,\\dots,\\mathbf{b}_d\\}, \\mathbf{b}_i \\in \\mathbb{R}^D$ for some integer $d \\leq D$, and every lattice point may be represented by the linear sum of basis vectors over the integers, that is,\n\\begin{align*}\n L=L(B)=\\left\\{\\sum_{i=1}^dx_i \\mathbf{b}_i: x_i \\in \\mathbb{Z}\\right\\}.\n\\end{align*}\nWe say that $L$ is full-rank if $d=D$. The determinant of $L$, $\\det(L)$, is the square root of the volume of the fundamental parallelopiped generated by the lattice basis. If $L$ is full-rank, then $\\det(L)=|\\det(B)|$.\n\\\\\nIn cryptography, the security of a lattice-based cryptosystem in most cases boils down to the computational hardness of the shortest vector problem (SVP). The problem goes as follows: given a lattice $L$ with basis $B$, find the shortest nonzero vector in $L$ with respect to the Euclidean (or otherwise specified) norm. Most cryptosystems, however, loosen the requirement of finding the shortest nonzero vector, and require the assailant to find a nonzero vector within some range of the shortest vector. One such problem is known as the Hermite-SVP, and goes as follows.\n\\begin{definition}\nLet $L$ be a rank $N$ lattice. The $\\gamma$-Hermite-SVP is to find a nonzero lattice vector $\\mathbf{v} \\in L$ that satisfies\n\\begin{align*}\n \\|\\mathbf{v}\\| \\leq \\gamma \\det(L)^{1\/N},\n\\end{align*}\nfor some approximation factor $\\gamma \\geq 1$.\n\\end{definition}\n\\noindent As opposed to the shortest lattice vector, the determinant of a lattice is well-defined and can be verified easily. Moreover, as discussed in \\cite{idealrandomprime}, a solution to Hermite-SVP can be lifted to a solution for a variety of different SVP-related problems.\n\\subsection{Algebraic Number Theory}\nAn algebraic number field $L$ is a finite extension of $\\mathbb{Q}$ by some algebraic integer $\\alpha$, that is, the solution to a polynomial in $\\mathbb{Z}[x]$. The degree of $L$ over $\\mathbb{Q}$ is equivalent to the degree of the minimal polynomial of $\\alpha$ in $\\mathbb{Q}(x)$. We denote by $\\mathcal{O}_L$ the ring of integers of $L$, which is the maximal order of $L$. It is well known in algebraic number theory that any algebraic number field $L$ is a $\\mathbb{Q}$-vector space over the power basis $\\{1,\\theta,\\theta^2,\\dots, \\theta^{N-1}\\}$ for some $\\theta \\in L$, and similarly the ring of integers $\\mathcal{O}_L$ may be expressed as a $\\mathbb{Z}$-module over a power basis $\\{1,\\theta^{\\prime},\\dots, {\\theta^{\\prime}}^{N-1}\\}$ for some $\\theta^{\\prime} \\in \\mathcal{O}_K$ \\cite{algnumbertheory}.\n\\\\ \\indent For a positive integer $N$, the cyclotomic polynomial $\\Phi_N(x)$ is the polynomial given by\n\\begin{align*}\n \\Phi_{N}(x)=\\prod_{k=1: \\gcd(k,N)=1}^N\\left(x-\\exp\\left(\\frac{2\\pi i k}{N}\\right)\\right),\n\\end{align*}\nor in other words, the polynomial whose roots are all the primitive $N$th roots of unity. For ease of notation, we generally let $\\zeta_N=\\exp\\left(\\frac{2\\pi i}{N}\\right)$ denote the $N$th root of unity. The field $L=\\mathbb{Q}(\\zeta_N)$ obtained by appending $\\zeta_N$ to $\\mathbb{Q}$ is called the cyclotomic field of conductor $N$, and such a field is of degree $\\phi(N)$ over $\\mathbb{Q}$, where\n\\begin{align*}\n \\phi(N) = N\\prod_{p \\mid N: p\\hspace{0.5mm} \\text{prime}}\\left(1-\\frac{1}{p}\\right)\n\\end{align*}\nis Euler's totient function, which measures the number of integers less than or equal to $N$ which are coprime to $N$.\n\\\\ The embeddings $\\sigma$ of a number field $L$ are the injective homomorphisms from $L$ to $\\mathbb{C}$ which fix $\\mathbb{Q}$. The number of distinct embeddings is equivalent to the degree of $L$ over $\\mathbb{Q}$, and an embedding $\\sigma$ is said to be a real embedding if $\\sigma(L) \\subset \\mathbb{R}$, and is said to be a complex embedding if $\\sigma(L) \\not\\subset \\mathbb{R}$. We define the \\emph{canonical embedding} $\\Sigma_L$ from a number field $L$ of degree $N$ to $\\mathbb{C}^N$ by\n\\begin{align*}\n \\Sigma_L: L \\to \\mathbb{C}^N \\hspace{2mm} a \\mapsto (\\sigma_1(a),\\sigma_2(a),\\dots, \\sigma_N(a)).\n\\end{align*}\nMoreover, we respectively define the trace and norm of an element in $L$ by\n\\begin{align*}\n \\text{Trace}_{L\/\\mathbb{Q}}(a) \\vcentcolon= \\sum_{i=1}^N\\sigma_i(a), \\hspace{2mm} \\text{Norm}_{L\/\\mathbb{Q}}(a)=\\prod_{i=1}^N\\sigma_i(a).\n\\end{align*}\nDefining by $\\overline{a}$ the complex conjugate of an element $a \\in L$, note that $\\beta(x,y) \\vcentcolon= \\text{Trace}_{L\/\\mathbb{Q}}(x\\overline{y})$ for all $x,y \\in L$ defines a positive-definite bilinear form on the $\\mathbb{Q}$-vector space generated by $L$. \n\\\\ \\indent Another way of embedding a number field $L$ into $\\mathbb{C}^N$ is using the so-called \\emph{coefficient embedding}. By expressing $L$ by a $\\mathbb{Q}$-vector space over a power basis $\\{1,\\alpha,\\dots, \\alpha^{N-1}\\}$, we may take any element $a=\\sum_{i=0}^{N-1}a_i\\alpha^i$ where $a_i \\in \\mathbb{Q}$ and define the embedding map\n\\begin{align*}\n a \\mapsto (a_0,a_1,\\dots, a_{N-1}).\n\\end{align*}\nIt is important to note that the coefficient embedding depends on the choice of basis for $L$. Also, if $\\mathcal{O}_K=\\mathbb{Z}[\\alpha]$ then $\\mathcal{O}_K$ maps to $\\mathbb{Z}^N$ under the coefficient embedding.\n\\subsection{Ideal Lattices}\nAn ideal $\\mathcal{I}$ is a subring of the ring of integers $\\mathcal{O}_L$. We say that an ideal $\\mathfrak{p}$ is prime if, for any $ab \\in \\mathfrak{p}$ for $a,b \\in \\mathcal{O}_L$, then either $a$ or $b$ is an element of $\\mathfrak{p}$. Under the canonical embedding, an ideal forms a lattice in $\\mathbb{R}^N$, and we call lattices constructed in this way \\emph{ideal lattices}. The volume of an ideal lattice $\\mathcal{I}$ in $\\mathbb{R}^N$ is $\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{I})disc(L\/\\mathbb{Q})$, where $\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{I})$ is the norm of the ideal $\\mathcal{I}$ and is equivalent to the cardinality of $\\mathcal{O}_L\/\\mathcal{I}$ (roughly speaking, the ``density'' of the ideal in $\\mathcal{O}_L$), and $disc(L\/\\mathbb{Q})$ is the discriminant of $L$ over $\\mathbb{Q}$, which is equivalent to the volume of the lattice generated after embedding $\\mathcal{O}_L$ via the canonical embedding.\n\\\\ \\indent In lattice-based cryptography, the cyclotomic number field $L=\\mathbb{Q}(\\zeta_N)$ is frequently used to define an ideal lattice. The ring of integers of $L$ is $\\mathcal{O}_L=\\mathbb{Z}[\\zeta_N]$ for any conductor $N$ \\cite{washington}. Suppose that the cyclotomic polynomial $\\Phi_N(x)$ factors in the finite field as\n\\begin{align*}\n \\Phi_N(x)=\\prod_{i=1}^g f_i(x)^e \\mod p\n\\end{align*}\nfor some rational prime $p$, where $f_i(x)$ are irreducible mod $p$. Then the ideal $p\\mathcal{O}_L$ factors as\n\\begin{align*}\n p\\mathcal{O}_L=(\\mathfrak{p}_1\\mathfrak{p}_2\\dots \\mathfrak{p}_g)^e,\n\\end{align*}\nwhere each $\\mathfrak{p}_i=\\langle p,f(\\zeta_N) \\rangle$ are prime ideals. We say the ideal $\\mathfrak{p}_i$ \\emph{lies over} $p$. If $e>1$, then $p$ is said to be \\emph{ramified} in $\\mathcal{O}_L$, and otherwise ($e=1$) $p$ is \\emph{unramified} in $\\mathcal{O}_L$. As such, we are motivated to study the factorisation of cyclotomic polynomials over finite fields in order to better study the structure of prime ideals. However, before we delve into more technical details regarding the factorisation of polynomials over finite fields, we introduce the following definition, which will be a recurring theme throughout the paper, and will be a powerful tool to help tackle SVP in ideal lattices.\n\\begin{definition}\nLet $L\/\\mathbb{Q}$ be a finite Galois extension of degree $N$, and let $G$ be the Galois group of $L$ over $\\mathbb{Q}$. The \\emph{decomposition group} $D$ of a prime ideal $\\mathfrak{p}$ is a subgroup of $G$ satisfying\n\\begin{align*}\n D=\\{\\sigma \\in G: \\sigma(\\mathfrak{p})=\\mathfrak{p}\\},\n\\end{align*}\nthat is, the embeddings of $L$ that fix $\\mathfrak{p}$. Then the decomposition field $K$ of $\\mathfrak{p}$ is defined by\n\\begin{align*}\n K=\\{x \\in L: \\forall \\sigma \\in D, \\sigma(x)=x\\},\n\\end{align*}\nthat is, the subfield of $L$ that is fixed by the decomposition group.\n\\end{definition}\n\\subsection{Factorisation of Cyclotomic Polynomials over Finite Fields}\nThe following lemmas will be used throughout section 4 onwards. Lemmas \\ref{factornumber}-\\ref{order} are standard in the study of finite fields, and we point the reader to \\cite{finitefields} for more details, and also for any terminology regarding finite fields. Lemma \\ref{pqdivide} is stated and proved in \\cite{factorpn}.\n\\begin{lemma}\\label{factornumber}\nLet $q$ be a power of a prime and $N$ be a positive integer such that $\\gcd(q,N)=1$. Then the $N$th cyclotomic polynomial $\\Phi_{N}(x)$ can be factorised into $\\phi(N)\/m$ distinct monic irreducible polynomials of the same degree $m$ over $\\mathbb{F}_q$, where $m$ is the least positive integer such that $q^m \\equiv 1 \\mod N$.\n\\end{lemma}\n\\begin{lemma}\\label{tmonic}\nLet $f_1(x),f_2(x),\\dots,f_N(x)$ be distinct monic irreducible polynomials over $\\mathbb{F}_q$ of degree $m$ and order $e$, and let $t \\geq 2$ be an integer whose prime factors divide $e$ but not $\\frac{q^m-1}{e}$. Assume that $q^m \\equiv 1 \\mod 4$ if $t \\equiv 0 \\mod 4$. Then $f_1(x^t),f_2(x^t),\\dots, f_N(x^t)$ are all distinct monic irreducible polynomials of degree $mt$ and order $et$.\n\\end{lemma}\n\\begin{lemma}\\label{order}\nLet $f(x)$ be an irreducible polynomial over $\\mathbb{F}_q$ of degree $m$ and with $f(0) \\neq 0$. Then the order of $f(x)$ is equal to the order of any root of $f(x)$ in the multiplicative group $\\mathbb{F}_{q^m}^*$.\n\\end{lemma}\n\\begin{lemma}\\label{pqdivide}\nLet $p$ be an odd prime, and $q$ be a prime power such that $q \\equiv 1 \\mod p$. If $m,n$ are positive integers satisfying $p^n \\mid q^{p^{n-m}}-1$ and $p \\nmid \\frac{q^{p^{n-m}}-1}{p^n}$, then $p^{n+1} \\mid q^{n+1-m}-1$ and $p \\nmid \\frac{q^{p^{n+1-m}}-1}{p^{n+1}}$.\n\\end{lemma}\n\\section{Solving Hermite-SVP for general ideal lattices in a\nGalois extension}\nIn this section, we generalise the results of Pan et. al., specifically their contributions on Hermite-SVP in prime ideal lattices. We consider first general ideal lattices, and then modules with a pseudo-basis of ideals and vectors with entries in the overlying number field.\n\\begin{defn}\nLet $L\/\\mathbb{Q}$ be a finite Galois extension and $I\\subset\\mathcal{O}_L$ an ideal, expressible as $\\mathcal{I} = \\mathfrak{p}_1...\\mathfrak{p}_g$, where each $\\mathfrak{p}_i$ lies above unramified rational prime $p_i$. Let $D_\\mathcal{I} = \\{\\sigma\\in Gal(L\/\\mathbb{Q}):\\sigma(\\mathcal{I})=\\mathcal{I}\\}$, and $K_\\mathcal{I} = L^{D_\\mathcal{I}} = \\{x\\in L\\text{ : }\\sigma(x)=x, \\text{ for all }\\sigma\\in D_\\mathcal{I}\\}$. These are called the \\textit{decomposition group} and \\textit{decomposition field} of $\\mathcal{I}$, respectively.\n\\end{defn}\n\\begin{theorem}\nLet ${L} \/ \\mathbb{Q}$ be a finite Galois extension of degree $N$ and $\\mathcal{I} = \\mathfrak{p}_1...\\mathfrak{p}_g$ an ideal of $\\mathcal{O}_{{L}}$, where each $\\mathfrak{p}_i$ lies over an unramified rational prime $p_i$ such that $p_i$ has $g_i$ distinct prime ideal factors in $O_{{L}}$ and has inertial degree $f_{p_i}^L$ in $\\mathcal{O}_L$. If ${K}_\\mathcal{I}$ is the decomposition field of $\\mathcal{I}$, and each $p_i$ has inertial degree $f_{p_i}^{K_\\mathcal{I}}$ in $\\mathcal{O}_{K_{\\mathcal{I}}}$, then a solution to Hermite-$SVP$ with factor $\\gamma$ in the sublattice $\\mathfrak{c}=\\mathcal{I} \\cap \\mathcal{O}_{{K}_\\mathcal{I}}$ under the canonical embedding of ${K}_\\mathcal{I}$ will also be a solution to Hermite-SVP in $\\mathcal{I}$ with factor $\\gamma\\frac{\\sqrt{N\/r}\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{I})^{1\/r-1\/N}}{\\text{Norm}_{{K_\\mathcal{I}}\/\\mathbb{Q}}({disc}({L} \/ {K}_\\mathcal{I}))^{1\/2N}(p_1^{(f^L_{p_1}-f^{K_\\mathcal{I}}_{p_1})1\/r}...p_g^{(f^L_{p_g}-f^{K_\\mathcal{I}}_{p_g})1\/r})}$ under the canonical embedding of ${L}.$\n\\end{theorem}\n\\begin{proof}\n\\noindent Consider the following diagram:\n\\begin{center}\n\\begin{tikzcd}\n\\mathcal{O}_L \\arrow[r, hook] & L \\arrow[r, \"\\Sigma_L\"] & \\mathbb{C}^N\\\\\n\\mathcal{O}_{K_I} \\arrow[u, hook] \\arrow[r, hook] & K_I \\arrow[u, hook] \\arrow[r, \"\\Sigma_{K_I}\"] & \\mathbb{C}^{[K_I:\\mathbb{Q}]} \\arrow[u, \"\\beta^\\prime\"]\n\\end{tikzcd}\n\\end{center}\n Here $\\beta^\\prime$ is chosen to make the diagram commute. Each embedding of $K_I$ extends to $N\/[K_I:\\mathbb{Q}]$ embeddings of $L$. Then $\\beta^\\prime$ simply repeats the coordinates of $\\Sigma_{K_I}$ $N\/r$ times, for $r = [K_I:\\mathbb{Q}]$, by the definition of $K_I$. So $\\|\\beta^\\prime(x)\\| = \\sqrt{N\/r}\\|x\\|$, for any $x\\in \\Sigma_{K_I}(K_I)$. Set $\\mathfrak{c} = I\\cap\\mathcal{O}_{K_I}$. Then $det(\\mathfrak{c}) = \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})\\sqrt{|disc(K_I\/\\mathbb{Q})|}$. So Hermite-SVP solution $v\\in\\mathfrak{c}$ satisfies $\\|v\\|\\leq\\gamma(\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})\\sqrt{|disc(K_I\/\\mathbb{Q})|})^{1\/r}$. Also note ${disc}({L}\/\\mathbb{Q})={disc}({K_I}\/\\mathbb{Q})^{N \/ r} \\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))$. Then \n \\begin{align*}\n \\|\\beta^\\prime(v)\\|&\\leq\\sqrt{N\/r}\\|v\\|\\leq\\sqrt{N\/r}\\gamma\\cdot(\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})\\sqrt{|disc(K_I\/\\mathbb{Q})|})^{1\/r}\\\\\n &= \\sqrt{N\/r}\\gamma\\cdot \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})^{1\/r}\\big(\\frac{disc(L\/\\mathbb{Q})^{r\/N}}{\\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))^{r\/N}}\\big)^{1\/2r}\\\\\n &= \\sqrt{N\/r}\\gamma\\cdot \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})^{1\/r}\\frac{disc(L\/\\mathbb{Q})^{1\/2N}}{\\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))^{1\/2N}}\\\\\n &= \\gamma\\frac{\\sqrt{N\/r}}{\\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))^{1\/2N}}\\cdot \\big(\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})^{N\/r}\\sqrt{disc(L\/\\mathbb{Q})}\\big)^{1\/N}.\n \\end{align*}\n The norm is multiplicative: $\\text{Norm}_{K_I\/\\mathbb{Q}}(I) = \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathcal{P}_1)...\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathcal{P}_g)$, where $\\mathcal{P}_i=\\mathfrak{p}_i\\cap K_I$. All the $\\mathfrak{p}_i$ lie above unramified primes $p_i$, so we can write $e^{L}_{p_i}=1$. Moreover, we have $\\text{Norm}_{L\/\\mathbb{Q}}=\\text{Norm}_{K_I\/\\mathbb{Q}}\\circ \\text{Norm}_{L\/K_I}$. As a result, $\\text{Norm}_{L\/\\mathbb{Q}}(I) = p_1^{f^L_{p_1}}...p_g^{f^L_{p_g}}$. Also, $\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c}) = p_1^{f^{K_I}_{p_1}}...p_g^{f^{K_I}_{p_g}}$. Thus $\\text{Norm}_{L\/\\mathbb{Q}}(I) = p_1^{f^L_{p_1}}...p_g^{f^L_{p_g}} = (p_1^{f^{K_I}_{p_1}}...p_g^{f^{K_I}_{p_g}})\\cdot(p_1^{f^L_{p_1}-f^{K_I}_{p_1}}...p_g^{f^L_{p_g}-f^{K_I}_{p_1}}) = \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})\\cdot(p_1^{f^L_{p_1}-f^{K_I}_{p_1}}...p_g^{f^L_{p_g}-f^{K_I}_{p_g}})$, so we can rewrite\n \\begin{align*}\n \\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c}) = \\text{Norm}_{L\/\\mathbb{Q}}(I)\/(p_1^{f^L_{p_1}-f^{K_I}_{p_1}}...p_g^{f^L_{p_g}-f^{K_I}_{p_g}}).\n \\end{align*} \n Then we have, setting $\\gamma^\\prime = \\gamma\\frac{\\sqrt{N\/r}}{\\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))^{1\/2N}}$,\n\\begin{align*}\n\\|\\beta^\\prime(v)\\|&\\leq \\gamma\\frac{\\sqrt{N\/r}}{\\text{Norm}_{{K_I}\/\\mathbb{Q}}({disc}({L} \/ {K}_I))^{1\/2N}}\\cdot \\big(\\text{Norm}_{K_I\/\\mathbb{Q}}(\\mathfrak{c})^{N\/r}\\sqrt{disc(L\/\\mathbb{Q})}\\big)^{1\/N}\\\\\n&= \\gamma^\\prime\\cdot \\big(\\big(\\text{Norm}_{L\/\\mathbb{Q}}(I)\/(p_1^{f^L_{p_1}-f^{K_I}_{p_1}}...p_g^{f^L_{p_g}-f^{K_I}_{p_g}})\\big)^{N\/r}\\sqrt{disc(L\/\\mathbb{Q})}\\big)^{1\/N}\\\\\n&= \\gamma^\\prime\\cdot \\big(\\text{Norm}_{L\/\\mathbb{Q}}(I)^{N\/r}\/(p_1^{(f^L_{p_1}-f^{K_I}_{p_1})N\/r}...p_g^{(f^L_{p_g}-f^{K_I}_{p_g})N\/r})\\sqrt{disc(L\/\\mathbb{Q})}\\big)^{1\/N}\\\\\n&= \\gamma^\\prime\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(I)^{1\/r-1\/N}}{(p_1^{(f^L_{p_1}-f^{K_I}_{p_1})1\/r}...p_g^{(f^L_{p_g}-f^{K_I}_{p_g})1\/r})}\\cdot \\big(\\text{Norm}_{L\/\\mathbb{Q}}(I)\\sqrt{disc(L\/\\mathbb{Q})}\\big)^{1\/N},\n\\end{align*}\nas required.\n \\end{proof}\n \\noindent Note that when $\\mathcal{I} = \\mathfrak{p}$, $K_I$ is the regular decomposition field, and $f^{K_I}_p = 1$ and $f^{L}_p = N\/r$, so $\\text{Norm}_{L\/\\mathbb{Q}}(I)^{1\/r-1\/N}=p^{N\/r(1\/r-1\/N)} = p^{N\/r^2-1\/r}$ and $p^{(f^L_{p}-f^{K_I}_{p})1\/r}) = p^{N\/r^2-1\/r}$, and we have recovered the result of the original paper.\n \\subsection{Solving Hermite-SVP for module lattices defined over a Galois extension}\nThe above method can be extended to the case of $\\mathcal{O}_L$-modules. As before, $L$ is a field that is Galois over $\\mathbb{Q}$ with ring of integers $\\mathcal{O}_L$. Suppose $\\mathcal{I}_1,\\dots, \\mathcal{I}_d \\subseteq \\mathcal{O}_L$ are some ideals of $\\mathcal{O}_L$ and $\\mathbf{b}_1,\\dots, \\mathbf{b}_d \\in L^D$ for some integer $d \\leq D$ that are linearly independent over $L$ (that is, none of the vectors can be expressed as a linear sum of the others over $L$). We define an $\\mathcal{O}_L$-module $M$ with pseudo-basis $\\langle \\mathcal{I}_k,\\mathbf{b}_k \\rangle$ as the direct sum\n\\begin{align*}\n M=\\bigoplus_{k=1}^d \\mathcal{I}_k \\mathbf{b}_k,\n\\end{align*}\nwhich is a $\\mathbb{Z}$-module of dimension $d[L:\\mathbb{Q}]$. We define the volume of $M$ by\n\\begin{align*}\n \\text{Vol}(M)=|\\text{disc}(L\/ \\mathbb{Q})|^d\\text{Norm}_{L\/\\mathbb{Q}}(\\det(B^\\dagger B))\\prod_{k=1}^d \\text{Norm}_{L\/\\mathbb{Q}}(I_k)^2,\n\\end{align*}\nwhere $B$ is the matrix composed of the columns $\\mathbf{b}_1,\\dots, \\mathbf{b}_d$ and $\\dagger$ denotes the Hermitian transpose of a matrix. Then, for any $\\mathbf{v}=(v_1,\\dots,v_D) \\in M$, the lattice $\\mathcal{L}_M$ generated by $M$ under the canonical embedding is the lattice whose elements take the form $(\\sigma_1(v_1),\\dots,\\sigma_{1}(v_D),\\sigma_2(v_1),\\dots,\\sigma_2(v_D),\\dots,\\sigma_{[L:\\mathbb{Q}]}(v_D))$. We say a vector $\\mathbf{v}$ is a solution to the $\\gamma$-Hermite-SVP in $M$ if it satisfies $\\|\\mathbf{v}\\| \\leq \\gamma \\text{Vol}(M)^{1\/2d[L:\\mathbb{Q}]}$. By abuse of notation, we say that for any $\\mathbf{v}=(v_1,\\dots,v_D) \\in L^D$, $\\sigma(\\mathbf{v})=(\\sigma(v_1),\\dots,\\sigma(v_D))$ for all $\\sigma \\in \\text{Gal}(L\/\\mathbb{Q})$ and \\\\$\\text{Trace}_{L\/K}(\\mathbf{v})=\\sum_{\\sigma \\in \\text{Gal}(L\/K)}\\sigma(\\mathbf{v})$ for any subfield $K$ of $L$. We call the module $\\mathcal{I}_k \\mathbf{b}_k$ a \\emph{pseudo-ideal} for each $k$, and define the decomposition group $\\Delta_k=\\{\\sigma \\in \\text{Gal}(L\/\\mathbb{Q}): \\sigma(\\mathcal{I}_k\\mathbf{b}_k)=\\mathcal{I}_k\\mathbf{b}_k\\}$.\n\\begin{lemma}\\label{separate}\nLet $\\mathcal{I}_k\\mathbf{b}_k$ be a pseudo-ideal in $L^D$, and let $\\Delta_k$ denote the decomposition group of $\\mathcal{I}_k\\mathbf{b}_k$ and $K_k$ the decomposition field of $\\Delta_k$. Then there exists an $\\alpha_k \\in L$ such that $\\mathbf{b}_k=\\alpha_k\\mathbf{b}_k^{\\prime}$, where $\\mathbf{b}_k^{\\prime} \\in K_k^D$.\n\\end{lemma}\n\\begin{proof}\nLet $x_k$ be some element of $\\mathcal{I}_k$. By the definition of the decomposition group, for every $\\sigma \\in \\Delta_i$ we must have $\\sigma(x_k\\mathbf{b}_k) \\in \\mathcal{I}_k \\mathbf{b}_k$ and so there exists some $x_{\\sigma,k} \\in \\mathcal{I}_k$ such that $x_{\\sigma,k}\\mathbf{b}_k=\\sigma(x_k\\mathbf{b}_k)$. Then, if we take $x_k$ to be some element of $\\mathcal{O}_{K_k}$, we have\n\\begin{align*}\n x_k\\text{Trace}_{L\/K}(\\mathbf{b}_k)=\\text{Trace}_{L\/K}(x_k\\mathbf{b}_k)=\\left(\\sum_{\\sigma \\in \\Delta_k}x_{k,\\sigma}\\right)\\mathbf{b}_k \\in K_k^D,\n\\end{align*}\nand so setting $\\mathbf{b}_k^{\\prime}=\\left(\\sum_{\\sigma \\in \\Delta_k}x_{k,\\sigma}\\right)\\mathbf{b}_k$, $\\alpha_k=\\left(\\sum_{\\sigma \\in \\Delta_k}x_{k,\\sigma}\\right)^{-1}$, the lemma holds.\n\\end{proof}\nUsing this lemma, if we have a module equivalent to the direct sum of the pseudo-ideals $\\mathcal{I}_k\\mathbf{b}_k$ with decomposition groups $\\Delta_k$ and decomposition fields $K_k$, then we may represent the module as\n\\begin{align*}\n M=\\bigoplus_{k=1}^d \\alpha_k\\mathcal{I}_k\\mathbf{b}_k^\\prime,\n\\end{align*}\nwhere $\\mathbf{b}_k^\\prime \\in K_k^D$.\n\\begin{theorem}\nLet $M$ be a module with pseudo-basis $\\langle \\mathcal{I}_k,\\mathbf{b}_k \\rangle_{k=1}^d $, and suppose that each pseudo-ideal $\\mathcal{I}_k\\mathbf{b}_k$ has decomposition group $\\Delta_k$ and decomposition field $K_k$. Denote by $\\langle \\mathcal{I}_k \\alpha_k, \\mathbf{b}_k^{\\prime} \\rangle_{k=1}^d$ the pseudo-basis that also represents $M$ such that $\\mathbf{b}_k^{\\prime} \\in K_k^D$. Let $\\mathcal{J}_k=Q_k\\alpha_k\\mathcal{I}_k$ where $Q_k$ is a rational integer such that $\\mathcal{J}_k$ is an ideal of $\\mathcal{O}_L$ and $\\mathcal{J}_k=\\prod_{i=1}^{g_k}\\mathfrak{p}_i^{(k)}$ where $\\mathfrak{p}_i^{(k)}$ are prime ideals lying above an unramified rational prime $p_i^{(k)}$ with inertial degree $f_{p_i^{(k)}}^L$ in $\\mathcal{O}_L$ and $f_{p_i^{(k)}}^{K_k}$ in $\\mathcal{O}_{K_k}$. We let $\\mathfrak{c}_k=\\mathcal{J}_k \\cap \\mathcal{O}_{K_k}$ and let $\\mathcal{M}=\\bigoplus_{k=1}^d \\frac{1}{Q_k}\\mathfrak{c}_k\\mathbf{b}_k^{\\prime}$. If $K$ is the compositum of all $K_k$, $1 \\leq k \\leq d$, then a solution to $\\gamma$-Hermite-SVP under the canonical embedding yields a solution to $\\gamma^\\prime$-Hermite-SVP in $M$, where\n\\begin{align*}\n \\gamma^{\\prime}=\\gamma \\frac{\\sqrt{[L:K]}}{\\sqrt{|\\text{Norm}_{K\/\\mathbb{Q}}(\\text{disc}(L\/K))}^{\\frac{1}{[L:\\mathbb{Q}]}}}\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{\\frac{1}{d[K:\\mathbb{Q}]}-\\frac{1}{d[L:\\mathbb{Q}]}}}{\\prod_{i=1}^{g_k}{p_i^{(k)}}^{\\frac{1}{d[K:\\mathbb{Q}]}\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}}.\n\\end{align*}\n\\end{theorem}\n\\begin{proof}\n\\noindent Consider the following diagram:\n\\begin{center}\n\\begin{tikzcd}\nM \\arrow[r, hook] & L^D \\arrow[r, \"\\Sigma_{L^D}\"] & \\mathbb{C}^{d[L:\\mathbb{Q}]}\\\\\n\\mathcal{M} \\arrow[u, hook] \\arrow[r, hook] & K^D \\arrow[u, hook] \\arrow[r, \"\\Sigma_{K^D}\"] & \\mathbb{C}^{d[K:\\mathbb{Q}]} \\arrow[u, \"\\beta^\\prime\"]\n\\end{tikzcd}\n\\end{center}\nHere, $\\beta^\\prime$ is chosen so that the diagram commutes. Each embedding of $K$ extends to $[L:K]$ embeddings of $L$, so $\\beta^{\\prime}$ repeats the coordinates of $\\Sigma_{K^D}$ $[L:K]$ times by the definition of $K$ being the compositum of fixed fields, so $\\|\\beta^{\\prime}(\\mathbf{v})\\|=\\sqrt{[L:K]}\\|\\mathbf{v}\\|$ for any $\\mathbf{v} \\in \\Sigma_{K^D}(K)$. Since the norm is multiplicative, $\\text{Norm}_{L\/\\mathbb{Q}}=\\text{Norm}_{K\/\\mathbb{Q}} \\circ \\text{Norm}_{L\/K}$. Now, we have\n\\begin{align*}\n\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)=\\prod_{i=1}^{g_k}{p_i^{(k)}}^{f_{p_i^{(k)}}^L}\n\\end{align*} \nand \n\\begin{align*}\n\\text{Norm}_{K\/\\mathbb{Q}}(\\mathfrak{c}_k)=\\left(\\text{Norm}_{K_k\/\\mathbb{Q}}(\\mathfrak{c}_k)\\right)^{[K:K_k]}=\\prod_{i=1}^{g_k}{p_i^{(k)}}^{[K:K_k]f_{p_i^{(k)}}^{K_k}},\n\\end{align*}\nand so we have the representation\n\\begin{align*}\n\\text{Norm}_{K\/\\mathbb{Q}}(\\mathfrak{c}_k)=\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)}{\\prod_{i=1}^{g_k}{p_i^{(k)}}^{f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}}}.\n\\end{align*}\nTherefore,\n\\begin{align*}\n &\\text{Vol}(\\mathcal{M})=|\\text{disc}(K\/\\mathbb{Q})|^d\\text{Norm}_{K\/\\mathbb{Q}}(B^{\\dagger}B)\\prod_{k=1}^d\\text{Norm}_{K\/\\mathbb{Q}}(\\mathfrak{c}_k)^2\\prod_{k=1}^dQ_k^{-2[K:\\mathbb{Q}]}\n \\\\&=|\\text{disc}(K\/\\mathbb{Q})|^d\\text{Norm}_{K\/\\mathbb{Q}}(B^{\\dagger}B)\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^2}{Q_k^{2[K:\\mathbb{Q}]}\\prod_{i=1}^{g_k}{p_i^{(k)}}^{2\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}},\n\\end{align*}\nwhere $B$ is the matrix made up composed of the vectors $\\mathbf{b}_1^{\\prime},\\dots, \\mathbf{b}_d^{\\prime}$, and using the fact that $\\text{disc}(L\/\\mathbb{Q})=\\text{disc}(K\/\\mathbb{Q})^{[L:K]}\\text{Norm}_{K\/\\mathbb{Q}}(\\text{disc}(L\/K))$,\n\\begin{align*}\n &\\text{Vol}(M)=|\\text{disc}(L\/\\mathbb{Q})|^d\\text{Norm}_{L\/\\mathbb{Q}}(B^{\\dagger}B)\\prod_{k=1}^d\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^2\\prod_{k=1}^dQ_k^{-2[L:\\mathbb{Q}]}\n \\\\&=|\\text{disc}(K\/\\mathbb{Q})|^{d[L:K]}\\text{Norm}_{K\/\\mathbb{Q}}(\\text{disc}(L\/K))^d\\\\&\\cdot \\text{Norm}_{K\/\\mathbb{Q}}(B^{\\dagger}B)^{[L:K]}\\prod_{k=1}^d\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^2\\prod_{k=1}^dQ_k^{-2[L:\\mathbb{Q}]}.\n\\end{align*}\nHence, if we have a solution $\\mathbf{v} \\in \\mathcal{M}$ for the Hermite-SVP with factor $\\gamma$, then\n\\begin{align*}\n &\\|\\beta(\\mathbf{v})\\|^2=[L:K]\\|\\mathbf{v}\\|^2\\leq \\gamma^2 [L:K] \\text{Vol}(\\mathcal{M})^{1\/d[K:\\mathbb{Q}]}\n \\\\&= \\gamma^2 [L:K]|\\text{disc}(K\/\\mathbb{Q})|^{1\/[K:\\mathbb{Q}]}\\text{Norm}_{K\/\\mathbb{Q}}(B^{\\dagger}B)^{1\/d[K:\\mathbb{Q}]}\\\\ &\\cdot\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{2\/d[K:\\mathbb{Q}]}}{Q_k^{2\/d}\\prod_{i=1}^{g_k}{p_i^{(k)}}^{\\frac{2}{d[K:\\mathbb{Q}]}\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}}\n \\\\&=\\frac{\\gamma^2[L:K]}{|\\text{Norm}_{K\/\\mathbb{Q}}(\\text{disc}(L\/K))|^{1\/[L:\\mathbb{Q}]}}\\text{Vol}(M)^{1\/d[L:\\mathbb{Q}]}\\\\& \\cdot\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{\\frac{2}{d}\\left(\\frac{1}{[K:\\mathbb{Q}]}-\\frac{1}{[L:\\mathbb{Q}]}\\right)}}{\\prod_{i=1}^{g_k}{p_i^{(k)}}^{\\frac{2}{d[K:\\mathbb{Q}]}\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}},\n\\end{align*}\nas required.\n\\end{proof}\nThough the factor we have obtained for general ideal lattices and module lattices may seem somewhat convoluted and tricky to interpret, we may make two remarks. The first, if $p_i^{(k)}\\mathcal{O}_{K_k}=\\prod_{j=1}^{t_{i,k}}\\mathfrak{p}_{i,j}^{(k)}$ where $\\mathfrak{p}_{i,j}^{(k)}$ are prime ideals of $\\mathcal{O}_{K_k}$ and each $\\mathfrak{p}_{i,j}^{(k)}$ is inert in $L$ for all $i,j,k$, then $\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{\\frac{1}{d}\\left(\\frac{1}{[K:\\mathbb{Q}]}-\\frac{1}{[L:\\mathbb{Q}]}\\right)}}{\\prod_{i=1}^{g_k}{p_i^{(k)}}^{\\frac{1}{d[K:\\mathbb{Q}]}\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}}=1$ (set $d=1$ for the ideal lattice case). Secondly, we may attain an upper bound that is easier to comprehend:\n\\begin{align*}\n &\\gamma \\frac{\\sqrt{[L:K]}}{\\sqrt{|\\text{Norm}_{K\/\\mathbb{Q}}(\\text{disc}(L\/K))}^{\\frac{1}{[L:\\mathbb{Q}]}}}\\prod_{k=1}^d\\frac{\\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{\\frac{1}{d[K:\\mathbb{Q}]}-\\frac{1}{d[L:\\mathbb{Q}]}}}{\\prod_{i=1}^{g_k}{p_i^{(k)}}^{\\frac{1}{d[K:\\mathbb{Q}]}\\left(f_{p_i^{(k)}}^L-[K:K_k]f_{p_i^{(k)}}^{K_k}\\right)}}\\\\\n &\\leq \\gamma \\sqrt{[L:K]} \\prod_{k=1}^d \\text{Norm}_{L\/\\mathbb{Q}}(\\mathcal{J}_k)^{\\frac{1}{d[K:\\mathbb{Q}]}-\\frac{1}{d[L:\\mathbb{Q}]}},\n\\end{align*}\nand this value can be determined without having to know the prime decomposition of the ideals.\n\\section{Prime Ideals of Cyclotomic Fields}\n\\subsection{The Cyclotomic Field $L=\\mathbb{Q}(\\zeta_{s2^{n+1}})$}\nWe let $s$ be some positive odd integer, $s \\geq 3$. The following is Theorem 2.2 from \\cite{wangwang}. \n\\begin{theorem}\\label{s2factorisation}\nLet $q$ be an odd prime power, and let $s \\geq 3$ be any odd number such that $\\gcd(q,s)=1$, and let $q^{\\phi(s)}=m2^A+1$ for some odd $m$, $A \\geq 1$. Then, for any $A -1\\leq n$ and for any irreducible factor $f(x)$ of $\\Phi_{s2^A}(x)$ over $\\mathbb{F}_q$, then $f(x^{2^{n-A+1}})$ is also irreducible over $\\mathbb{F}_q$. Moreover, all irreducible factors of $\\Phi_{s2^{n+1}}(x)$ are obtained in this way.\n\\end{theorem}\n\\begin{theorem}\\label{s2svp}\nFor any prime ideal $\\mathfrak{p}=\\langle \\rho,f(\\zeta_{s2^{n+1}}) \\rangle$ of $\\mathcal{O}_L$ for some rational prime $\\rho$, $\\gcd(\\rho,s)=\\gcd(\\rho,2)=1$ and irreducible polynomial $f(x)$ of $\\Phi_{s2^{n+1}}$ in $\\mathbb{F}_\\rho[x]$, write $\\rho^{\\phi(s)}=m2^A+1$ where $m$ is an odd integer and $A \\geq 1$, and let $r=\\min\\{A-1,n\\}$. Then, given an oracle that can solve SVP for $\\phi(s2^{r+1})$-dimensional lattices, a shortest nonzero vector in $\\mathfrak{p}$ can be found in \\\\$\\text{poly}(\\phi(s2^{n+1}),\\log_2\\rho)$ time with the canonical embedding.\n\\end{theorem}\n\\begin{proof}\nWe assume that $n \\geq A$ otherwise the theorem is vacuously true, so $r=A-1$. Let\n\\begin{align*}\n G=\\{\\sigma_i: \\gcd(i,2)=\\gcd(i,s)=1\\}\n\\end{align*}\ndenote the Galois group of $L$ over $\\mathbb{Q}$, where\n\\begin{align*}\n &\\sigma_i: \\mathbb{Q}(\\zeta_{s2^{n+1}}) \\to \\mathbb{Q}(\\zeta_{s2^{n+1}}),\\\\\n &\\sigma_i(\\zeta_{s2^{n+1}}^k)=\\zeta_{s2^{n+1}}^{ki}.\n\\end{align*}\nBy Theorem \\ref{s2factorisation}, for any factor $f(x)$ of $\\Phi_{s2^{n+1}}(x)$ that is irreducible in $\\mathbb{F}_\\rho[x]$, there exists a polynomial $g(x)$ that is a factor of $\\Phi_{s2^{r+1}}(x)$ that is irreducible over $\\mathbb{F}_{\\rho}[x]$ such that $f(x)=g(x^{2^{n-r}})$. Then the prime ideal lattice $\\mathfrak{p}$ can be represented by\n\\begin{align*}\n \\langle \\rho,f(\\zeta_{s2^{n+1}}) \\rangle = \\langle \\rho,g(\\zeta_{s2^{r+1}}) \\rangle.\n\\end{align*}\nFor any $1 \\leq k \\leq 2^{n-r}-1$, the map $\\sigma_{ks2^{r+1}+1}$ fixes $\\zeta_{s2^{n+1}}^{l2^{n-r}}$ for any integer $0 \\leq l 1$, we must have $\\rho^{\\phi(s)} \\equiv 2l+1 \\mod 2^N$, and so for some integer $k$, we have $\\rho^{\\phi(s)} = 1+2l+2^Nk=1+2(2^{N-1}k+l)$. Since $2^{N-1}k+l$ is an odd integer and $N$ is taken totally arbitrarily, the claim holds by Theorem \\ref{s2svp}.\n\\end{proof}\n\\begin{theorem}\nLet $L=\\mathbb{Q}(\\zeta_{sp^{n+1}})$ be a cyclotomic field for some positive integer $s$ such that $\\gcd(s,p)=1$, an odd prime $p$ and some integer $n \\geq 0$. Let $\\mathfrak{p}$ denote a prime ideal lying over a positive rational odd prime $\\rho$ such that $\\rho^{\\phi(s)} =lp +a $ for some integers $l,a$, $\\gcd(l,p)=\\gcd(a,p)=1$. Then, given an oracle that can solve SVP for $(p-1)\\phi(s)$-dimensional lattices, a shortest nonzero vector in $\\mathfrak{p}$ can be found in $\\text{poly}(\\phi(sp^{n+1}),\\log_2 \\rho)$ time with the canonical embedding.\n\\end{theorem}\n\\begin{proof}\nFor some integer $N>1$, we must have $\\rho^{\\phi(s)} \\equiv lp +a \\mod p^N$, and so for some integer $k$, we have $\\rho^{\\phi(s)}=m a+pl+p^{N}k= a+p(p^{N-1}k+l)$. Since $\\gcd(p^{N-1}k+l,p)=1$ and $N$ is taken totally arbitrarily, the claim holds by Theorem \\ref{spsvp}.\n\\end{proof}\n\\section{General Ideals of Cyclotomic Rings}\n\\subsection{The Cyclotomic Field $L=\\mathbb{Q}(\\zeta_{s2^{n+1}})$}\nAs before, we set $s$ to be some odd integer greater than or equal to $3$.\n\\begin{theorem}\nLet $\\mathcal{I}$ be a nonzero ideal of $\\mathbb{Z}[\\zeta_{s2^{n+1}}]$ with prime factorisation\n\\begin{align*}\n \\mathcal{I}=\\mathfrak{p}_1\\mathfrak{p}_2\\dots \\mathfrak{p}_t,\n\\end{align*}\nwhere $\\mathfrak{p}_i=(f_i(\\zeta_{s2^{n+1}}),\\rho_i)$ for rational primes $\\rho_i$ are (not necessarily distinct) prime ideals. If $\\rho_i$ is odd, write $\\rho_i^{\\phi(s)}=m_i2^{A_i}+1$, for some integer $m_i$ such that $\\gcd(m_i,2)=1$ and let $r=\\max\\{r_i\\}$, where\n\\begin{align*}\n r_i=\n \\begin{cases}\n \\min\\{A_i-1,n\\}, \\hspace{0.5mm} &\\text{if} \\hspace{1mm} \\rho_i \\equiv 1 \\mod 2,\\\\\n n \\hspace{0.5mm} &\\text{if} \\rho_i=2.\n \\end{cases}\n\\end{align*}\nThen SVP in the lattice generated by $\\mathcal{I}$ can be solved via solving SVP in a $\\phi(s2^{r+1})$-dimensional lattice.\n\\end{theorem}\n\\begin{proof}\nIf $r=n$ the theorem vacuously holds, so we assume otherwise. We may assume WLOG that $r=r_1$. Following the notation of Theorem \\ref{s2svp}, we denote by\n\\begin{align*}\n G=\\{\\sigma_i: 1 \\leq i \\leq s2^{n+1}-1, \\gcd(i,2)=\\gcd(i,s)=1\\}\n\\end{align*}\nthe Galois group of $L$, and consider the subgroup $H=H_1 \\times H_2 \\times \\dots \\times H_{2^{n-r-1}}$, where $H_k$ is the cyclic group generated by $\\langle \\sigma_{ks2^{r+1}+1}\\rangle$, which is a subgroup of the decomposition group of every $\\mathfrak{p}_i$, since $\\sigma_{ks2^{r+1}+1}(\\rho_i)=\\rho_i$, $\\sigma_{ks2^{r+1}+1}(f_i(\\zeta_{s2^{n+1}}))=\\sigma_{ks2^{r+1}+1}(g_i(\\zeta_{s2^{r+1}}))=g_i(\\zeta_{s2^{r+1}})=f_i(\\zeta_{s2^{n+1}})$, where $g_i(x)$ is an irreducible factor of $\\Phi_{s2^{A_i}}(x)$. As shown in Theorem \\ref{s2svp}, the fixed field of $H$ is $K=\\mathbb{Q}(\\zeta_{s2^{n+1}}^{2^{n-r}})$, which has ring of integers $\\mathcal{O}_K=\\mathbb{Z}[\\zeta_{s2^{n+1}}^{2^{n-r}}]$. Let $\\mathfrak{c}= \\mathcal{I} \\cap \\mathcal{O}_K$. We claim that for any $a \\in \\mathcal{I}$, there exist $a^{(k)} \\in \\mathfrak{c}$ for $0 \\leq k \\leq 2^{n-r}-1$ such that\n\\begin{align*}\n a=\\sum_{k=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^ka^{(k)}.\n\\end{align*}\nWe prove the claim via induction. When $t=1$, the claim holds by Theorem \\ref{s2svp}, so we assume the claim holds for $t-1$. Letting $\\overline{\\mathcal{I}}=\\mathfrak{p}_1\\mathfrak{p}_2\\dots \\mathfrak{p}_{t-1}$, we have $\\mathcal{I}=\\mathfrak{p}_t\\mathcal{I}$. It suffices to show that for any $xy$, $x \\in \\overline{\\mathcal{I}}, y \\in \\mathfrak{p}_t$, there exist $b^{(k)} \\in \\mathcal{I} \\cap \\mathcal{O}_K$ for $0 \\leq k \\leq 2^{n-r}-1$ such that $xy=\\sum_{k=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^kb^{(k)}$. By the induction assumption, there exist $x^{(i)} \\in \\overline{\\mathcal{I}} \\cap \\mathcal{O}_K$, $y^{(j)} \\in \\mathfrak{p}_t \\cap \\mathcal{O}_K$, $0 \\leq i,j \\leq 2^{n-r}-1$ such that $x=\\sum_{i=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^ix^{(i)}$ and $y=\\sum_{j=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^jy^{(j)}$. Hence, we have\n\\begin{align*}\n xy&=\\sum_{i,j=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^{i+j}x^{(i)}y^{(i)}\n \\\\&=\\sum_{k=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^k\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{k=2^{n-r}}^{2\\cdot 2^{n-r}-2}\\zeta_{s2^{n+1}}^k\n \\\\&=\\sum_{k=0}^{2^{n-r}-1}\\zeta_{s2^{n+1}}^k\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{k=0}^{2^{n-r}-2}\\zeta_{s2^{n+1}}^k\\sum_{i+j=k+2^{n-r}}\\zeta_{s2^{n+1}}^{2^{n-r}}x^{(i)}y^{(j)}\n \\\\&=\\sum_{k=0}^{2^{n-r}-2}\\zeta_{s2^{n+1}}^k\\left(\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{i+j=k+2^{n-r}}\\zeta_{s2^{n+1}}^{2^{n-r}}x^{(i)}y^{(j)}\\right)\\\\&+\\zeta_{s2^{n+1}}^{2^{n-r}-1}\\sum_{i+j=2^{n-r}-1}x^{(i)}y^{(j)}.\n\\end{align*}\nBy letting \\begin{align*}\nb^{(k)}=\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{i+j=k+2^{n-r}}\\zeta_{s2^{n+1}}^{2^{n-r}}x^{(i)}y^{(j)}\n\\end{align*} \nfor $0 \\leq k \\leq 2^{n-r}-2$ and \n\\begin{align*}\nb^{(2^{n-r}-1)}=\\sum_{i+j=2^{n-r}-1}x^{(i)}y^{(j)},\n\\end{align*} \nwe have proven our claim. As in Theorem \\ref{s2svp}, we have $\\lambda_1(\\mathcal{I})=\\lambda_1(\\mathfrak{c})$, as required.\n\\end{proof}\nThe following algorithm may be used to compute the shortest vector in $\\mathcal{I}$.\n\\begin{algorithm}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \\Input{An ideal $\\mathcal{I}$.} \\Output{A shortest vector in the corresponding ideal lattice.} \\BlankLine \n\\nl \\For{$\\overline{r}=1$ \\KwTo $n$}{\n\\nl Compute a basis $(b^{(i)})_{0 \\leq i < \\phi(s2^{\\overline{r}+1})}$ of the ideal lattice $\\mathfrak{c}=\\mathcal{I} \\cap \\mathcal{O}_K$ where $K=\\mathbb{Q}(\\zeta_{s2^{n+1}}^{2^{n-\\overline{r}}})$. \\\\\n\\nl \\If{$(\\zeta_{s2^{n+1}}^jb^{(i)})_{0 \\leq i < \\phi(s2^{\\overline{r}+1}), 0 \\leq j \\leq 2^{n-\\overline{r}}}$ is exactly a basis of the ideal lattice $\\mathcal{I}$}{\n\\nl Find a shortest vector $v$ in the $\\phi(s2^{\\overline{r}+1})$-dimensional lattice $\\mathfrak{c}$.\\\\\n\\nl Output $v$.}\n}\n\\caption{SVP algorithm for general ideal lattices of $\\mathbb{Z}[\\zeta_{s2^{n+1}}]$}\n\\end{algorithm}\n\\\\\n\\subsection{The Cyclotomic Field $L=\\mathbb{Q}(\\zeta_{sp^{n+1}})$}\nAs before, $p$ is a positive, odd prime and $s$ is a positive integer such that $\\gcd(s,p)=1$.\n\\begin{theorem}\nLet $\\mathcal{I}$ be a nonzero ideal of $\\mathbb{Z}[\\zeta_{sp^{n+1}}]$ with prime factorisation\n\\begin{align*}\n \\mathcal{I}=\\mathfrak{p}_1\\mathfrak{p}_2\\dots \\mathfrak{p}_t,\n\\end{align*}\nwhere $\\mathfrak{p}_i=(f_i(\\zeta_{sp^{n+1}}),\\rho_i)$ for rational primes $\\rho_i$ are (not necessarily distinct) prime ideals. If $\\rho_i^{\\phi(s)} \\equiv a \\mod p$ for some $\\gcd(p,a)=1$, write $\\rho_i^{\\phi(s)}=m_ip^{A_i}+1$ and let $r=\\max\\{r_i\\}$, where\n\\begin{align*}\n r_i=\n \\begin{cases}\n \\min\\{A_i-1,n\\}, \\hspace{0.5mm} &\\text{if} \\hspace{1mm} \\rho_i^{\\phi(s)} \\equiv a \\mod p,\\\\\n n \\hspace{0.5mm} &\\text{if $\\rho_i=p$}.\n \\end{cases}\n\\end{align*}\nThen SVP in the lattice generated by $\\mathcal{I}$ can be solved via solving SVP in a $\\phi(sp^{r+1})$-dimensional lattice.\n\\end{theorem}\n\\begin{proof}\nIf $r=n$ the theorem vacuously holds, so we assume otherwise. We may assume WLOG that $r=r_1$. Following the notation of Theorem \\ref{spsvp}, we denote by\n\\begin{align*}\n G=\\{\\sigma_i: 1 \\leq i \\leq sp^{n+1}-1, \\gcd(i,p)=\\gcd(i,s)=1\\}\n\\end{align*}\nthe Galois group of $L$, and consider the subgroup $H=H_1 \\times H_2 \\times \\dots \\times H_{p^{n-r-1}}$, where $H_k$ is the cyclic group generated by $\\langle \\sigma_{ksp^{r+1}+1}\\rangle$, which is a subgroup of the decomposition group of every $\\mathfrak{p}_i$, since $\\sigma_{ksp^{r+1}+1}(\\rho_i)=\\rho_i$, $\\sigma_{ksp^{r+1}+1}(f_i(\\zeta_{sp^{n+1}}))=\\sigma_{ksp^{r+1}+1}(g_i(\\zeta_{sp^{r+1}}))=g_i(\\zeta_{sp^{r+1}})=f_i(\\zeta_{sp^{n+1}})$, where $g_i(x)$ is an irreducible factor of $\\Phi_{sp^{A_i}}(x)$. As shown in Theorem \\ref{spsvp}, the fixed field of $H$ is $K=\\mathbb{Q}(\\zeta_{sp^{n+1}}^{p^{n-r}})$, which has ring of integers $\\mathcal{O}_K=\\mathbb{Z}[\\zeta_{sp^{n+1}}^{p^{n-r}}]$. Let $\\mathfrak{c}= \\mathcal{I} \\cap \\mathcal{O}_K$. We claim that for any $a \\in \\mathcal{I}$, there exist $a^{(k)} \\in \\mathfrak{c}$ for $0 \\leq k \\leq p^{n-r}-1$ such that\n\\begin{align*}\n a=\\sum_{k=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^ka^{(k)}.\n\\end{align*}\nWe prove the claim via induction. When $t=1$, the claim holds by Theorem \\ref{spsvp}, so we assume the claim holds for $t-1$. Letting $\\overline{\\mathcal{I}}=\\mathfrak{p}_1\\mathfrak{p}_2\\dots \\mathfrak{p}_{t-1}$, we have $\\mathcal{I}=\\mathfrak{p}_t\\mathcal{I}$. It suffices to show that for any $xy$, $x \\in \\overline{\\mathcal{I}}, y \\in \\mathfrak{p}_t$, there exist $b^{(k)} \\in \\mathcal{I} \\cap \\mathcal{O}_K$ for $0 \\leq k \\leq p^{n-r}-1$ such that $xy=\\sum_{k=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^kb^{(k)}$. By the induction assumption, there exist $x^{(i)} \\in \\overline{\\mathcal{I}} \\cap \\mathcal{O}_K$, $y^{(j)} \\in \\mathfrak{p}_t \\cap \\mathcal{O}_K$, $0 \\leq i,j \\leq p^{n-r}-1$ such that $x=\\sum_{i=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^ix^{(i)}$ and $y=\\sum_{j=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^jy^{(j)}$. Hence, we have\n\\begin{align*}\n xy&=\\sum_{i,j=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^{i+j}x^{(i)}y^{(i)}\n \\\\&=\\sum_{k=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^k\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{k=p^{n-r}}^{2 p^{n-r}-2}\\zeta_{sp^{n+1}}^k\n \\\\&=\\sum_{k=0}^{p^{n-r}-1}\\zeta_{sp^{n+1}}^k\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{k=0}^{p^{n-r}-2}\\zeta_{sp^{n+1}}^k\\sum_{i+j=k+p^{n-r}}\\zeta_{sp^{n+1}}^{p^{n-r}}x^{(i)}y^{(j)}\n \\\\&=\\sum_{k=0}^{p^{n-r}-2}\\zeta_{sp^{n+1}}^k\\left(\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{i+j=k+p^{n-r}}\\zeta_{sp^{n+1}}^{p^{n-r}}x^{(i)}y^{(j)}\\right)\\\\&+\\zeta_{sp^{n+1}}^{p^{n-r}-1}\\sum_{i+j=p^{n-r}-1}x^{(i)}y^{(j)}.\n\\end{align*}\nBy letting \\begin{align*}\nb^{(k)}=\\sum_{i+j=k}x^{(i)}y^{(j)}+\\sum_{i+j=k+p^{n-r}}\\zeta_{sp^{n+1}}^{p^{n-r}}x^{(i)}y^{(j)}\n\\end{align*} \nfor $0 \\leq k \\leq p^{n-r}-2$ and \n\\begin{align*}\nb^{(p^{n-r}-1)}=\\sum_{i+j=p^{n-r}-1}x^{(i)}y^{(j)},\n\\end{align*} \nwe have proven our claim. As in Theorem \\ref{spsvp}, we have $\\lambda_1(\\mathcal{I})=\\lambda_1(\\mathfrak{c})$, as required.\n\\end{proof}\nThe following algorithm may be used to compute the shortest vector in $\\mathcal{I}$.\n\\begin{algorithm}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \\Input{An ideal $\\mathcal{I}$.} \\Output{A shortest vector in the corresponding ideal lattice.} \\BlankLine \n\\nl \\For{$\\overline{r}=1$ \\KwTo $n$}{\n\\nl Compute a basis $(b^{(i)})_{0 \\leq i < \\phi(sp^{\\overline{r}+1})}$ of the ideal lattice $\\mathfrak{c}=\\mathcal{I} \\cap \\mathcal{O}_K$ where $K=\\mathbb{Q}(\\zeta_{sp^{n+1}}^{2^{n-\\overline{r}}})$. \\\\\n\\nl \\If{$(\\zeta_{sp^{n+1}}^jb^{(i)})_{0 \\leq i < \\phi(sp^{\\overline{r}+1}), 0 \\leq j \\leq p^{n-\\overline{r}}}$ is exactly a basis of the ideal lattice $\\mathcal{I}$}{\n\\nl Find a shortest vector $v$ in the $\\phi(sp^{\\overline{r}+1})$-dimensional lattice $\\mathfrak{c}$.\\\\\n\\nl Output $v$.}\n}\n\\caption{SVP algorithm for general ideal lattices of $\\mathbb{Z}[\\zeta_{sp^{n+1}}]$}\n\\end{algorithm}\n\\section{Modules over Cyclotomic Rings}\nThroughout this section, we use the same notation as in section 3.1. We let $L=\\mathbb{Q}(\\zeta_N)$ be the cyclotomic field of conductor $N$, where $N$ is of the form $sq^{n+1}$ for some positive integer $s$ and some $q$ where $q$ is $2$ or an odd prime, $\\gcd(s,q)=1$. We take a module $M$ with pseudo-basis $\\langle \\mathcal{I}_k,\\mathbf{b}_k \\rangle_{k=1}^d$. As in section 3.1, we associate to each pseudo-ideal $\\mathfrak{I}_k\\mathbf{b}_k$ the decomposition group $\\Delta_{\\mathcal{I}_k}$ and the decomposition field $K_{\\mathcal{I}_k}$. As we showed in Lemma \\ref{separate}, we may alternatively represent the module $M$ using the pseudo-basis $\\langle \\alpha_k \\mathcal{I}_k,\\mathbf{b}_k^\\prime \\rangle_{k=1}^d$ where $\\mathbf{b}_k^\\prime \\in K_{\\mathcal{I}_k}^D$ and $\\alpha_k\\mathcal{I}_k$ are fractional ideals. Let $\\mathcal{J}_k=Q_k\\alpha_k\\mathcal{I}_k$ for some rational integer $Q_k$ such that $\\mathcal{J}_k$ is an ideal of $\\mathcal{O}_L$. Denote by $\\Delta_{\\mathcal{J}_k}$ the decomposition group of $\\mathcal{J}_k$ and by $r_k$ the maximum integer such that all the embeddings that fix $K_{k}=\\mathbb{Q}(\\zeta_{sq^{n+1}}^{q^{n-r_k}})$ also fix $K_{\\mathcal{I}_j}$ for $1 \\leq j \\leq d$ (that is, they must form a subgroup of $\\Delta_{\\mathcal{I}_j}$), and such that (as in section 5) we may express $\\mathcal{J}_k$ as\n\\begin{align*}\n \\mathcal{J}_k=\\bigoplus_{i=0}^{q^{n-r_k}-1}\\mathfrak{c}_k\\zeta_{sq^{n+1}}^i,\n\\end{align*}\nwhere $\\mathfrak{c}_k=\\mathcal{J}_k \\cap \\mathcal{O}_{K_k}$. Then we may express $M$ by\n\\begin{align*}\n M=\\bigoplus_{k=1}^d\\mathcal{I}_k\\mathbf{b}_k=\\bigoplus_{k=1}^d\\frac{1}{Q_k}\\mathbf{b}_k^\\prime\\bigoplus_{i=0}^{q^{n-r_k}-1}\\mathfrak{c}_k\\zeta_{sq^{n+1}}^i.\n\\end{align*}\nSince each $r_k$ are chosen so that the embeddings that fix the field $K_k$ also fix $K_{\\mathcal{I}_j}$ for $1 \\leq j \\leq d$ and under the canonical embedding of ideal lattices the coefficients next to the roots of unity in the expression above may be treated as orthogonal components (as we have already shown), under the canonical embedding of $M$, the terms multiplied by the roots of unity in the expression above may be treated as orthogonal components, and so the shortest vector of the submodule\n\\begin{align*}\n \\mathcal{M}=\\bigoplus_{k=1}^d\\frac{1}{Q_k}\\mathfrak{c}_k\\mathbf{b}_k^\\prime\n\\end{align*}\nis also the shortest vector in the module $M$. This simplifies SVP in module lattices, as the original module generated a lattice of dimension $d\\phi(sq^{n+1})$ under the canonical embedding, whilst the submodule above generates a lattice of dimension at most $d\\phi(sq^{r+1})$ where $r=\\max_k\\{r_k\\} \\leq n$. The following algorithm details how to perform SVP in module lattices over cyclotomic rings, and uses notation and conventions that we have discussed in this section.\n\\begin{algorithm}\n\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output} \\Input{A module $M$ with pseudo-basis $\\langle \\mathcal{I}_k,\\mathbf{b}_k \\rangle_{k=1}^d$.} \\Output{A shortest vector in the corresponding module lattice.} \\BlankLine \n\\nl \\For{$1 \\leq i \\leq d$}{\n\\nl Compute the decomposition group $\\Delta_{\\mathcal{I}_i}$ of the pseudo-ideal $\\mathcal{I}_i\\mathbf{b}_i$ and decomposition field $K_{\\mathcal{I}_i}$. \\\\\n\\nl Set $x_i=\\text{Norm}_{L\/K_{\\mathcal{I}_i}}(\\mathcal{I}_i)$.\\\\\n\\nl \\For{$\\sigma \\in \\Delta_{\\mathcal{I}_i}$}{\n\\nl Find $x_{\\sigma,i} \\in \\mathcal{I}_i$ that satisfies $x_{\\sigma,i}\\mathbf{b}_i=\\sigma(x_i\\mathbf{b}_i)$}\n\\nl Set $\\alpha_i=\\left(\\sum_{\\sigma \\in \\Delta_{\\mathcal{I}_i}} x_{i,\\sigma}\\right)^{-1}$, $\\mathbf{b}_i^\\prime=\\alpha_i^{-1}\\mathbf{b}_i$\\\\\n\\nl Set $Q_i$ to be smallest integer such that $Q_i\\alpha_i\\mathcal{I}_i \\subseteq \\mathcal{O}_L$, set $\\mathcal{J}_i=Q_i\\alpha_i\\mathcal{I}_i$\n}\n\\nl Compute the field $K=\\mathbb{Q}(\\zeta_{sq^{n+1}}^{q^{n-t}})$ such that $K$ is a subfield of the compositum of all $K_{\\mathcal{I}_i}$, $1 \\leq i \\leq d$.\\\\\n\\nl \\For{$1 \\leq i \\leq d$}{\n\\For{$1 \\leq \\overline{r}_i \\leq n$}{\n\\nl Compute a basis $(b_i^{(j)})_{0 \\leq j \\leq \\phi(sq^{\\overline{r}_i+1})}$ of the ideal $\\mathfrak{c}_i=\\mathcal{J}_i \\cap \\mathcal{O}_{K_i}$ where $K_i=\\mathbb{Q}(\\zeta_{sq^{n+1}}^{q^{n-\\overline{r}_i}})$ \\\\\n\\If{$(\\zeta_{sq^{n+1}}^kb_i^{(j)})_{0 \\leq j \\leq \\phi(sq^{\\overline{r}_i+1}),0 \\leq k \\leq q^{n-\\overline{r}_i}}$ is exactly a basis of $\\mathcal{J}_i$ and $K_i \\subseteq K$}{\nSet $\\mathfrak{c}_i=\\mathcal{J}_i \\cap K_i$ and break\n}\n}\n}\n\\nl Find the shortest vector $\\mathbf{v}$ in the lattice generated by the module $\\mathcal{M}=\\bigoplus_{i=1}^d \\frac{1}{Q_i}\\mathfrak{c}_i \\mathbf{b}_i^{\\prime}$. \\\\\n\\nl Output $\\mathbf{v}$.\n\\caption{SVP algorithm for a module lattice $M$ over a cyclotomic ring.}\n\\end{algorithm}\n\\section{SVP Average-Case Hardness}\nWe fix a large $M$, and select a prime ideal uniformly randomly from the set\n\\begin{align*}\n \\{\\mathfrak{p} \\hspace{2mm} \\text{a prime ideal}: N(\\mathfrak{p})0$, is there a way to design a window $g \\in L^2(\\bR^d)$ such that the ``bump with fat tail'' condition\n\t\\begin{equation}\\label{fat tail sph}\n\t|V_g f(z)| < \\frac{|\\langle f,g \\rangle|}{N}, \\quad |z|>R,\n\t\\end{equation} holds? \n\\end{quest}\n\nFrom a heuristic point of view this would amount to determine a window $g$ such that $V_gf$ shows a bump near the origin and a mild decay at infinity; that is, the energy of the signal accumulates a little near the origin and then spreads on the tail (hence a \\textit{fat tail}). This balance is unavoidable in view of the uncertainty principle, which forbids an arbitrary accumulation near the origin. \n\nA positive answer to Question \\ref{quest ft} would prove the HRT conjecture by \\cite[Theorem 3]{kreisel}. In fact we prove that the answer is negative as a consequence of the following result, which can be interpreted as a form of the uncertainty principle for the STFT \\cite{bonami,fernandez,gro up,lieb}.\n\n\\begin{theorem}\\label{maint}\n\tLet $g(t) = e^{-\\pi t^2}$ and assume that there exist $R >0$, $N>1$ and $f \\in L^2(\\bR^d)\\setminus\\{0\\}$ such that \n\t\\begin{equation}\\label{fat tail cyl}\n\t|V_g f(x,\\omega)| \\le \\frac{|\\langle f,g \\rangle|}{N}, \\quad |\\omega|=R.\n\t\\end{equation}\n\tThen\n\t\\begin{equation}\\label{R est cyl} R > \\sqrt{\\frac{\\log N}{\\pi}}. \\end{equation}\n\\end{theorem}\n\nThis result is indeed a negative answer to Question \\ref{quest ft} since $|V_gf (x,\\omega)| = |V_f g(-x,-\\omega)|$. In fact, a stronger result can be proved in the case where the cylinder in \\eqref{fat tail cyl} is replaced by a sphere.\n\n\\begin{theorem}\\label{ft ball}\n\tLet $g(t) = e^{-\\pi t^2}$ and assume that there exists $R>0$, $N>1$ and $f \\in L^2(\\bR^d)\\setminus\\{0\\}$ such that \n\t\\begin{equation}\\label{fat tail ball}\n\t|V_g f(z)| \\le \\frac{|\\langle f,g \\rangle|}{N}, \\quad |z|=R.\n\t\\end{equation}\n\tThen\n\t\\begin{equation}\\label{R est ball} R \\ge \\sqrt{\\frac{2\\log N}{\\pi}}. \\end{equation}\n\tMoreover, \\eqref{fat tail ball} holds with $R=\\sqrt{2\\log N \/ \\pi}$ if and only if $f(t)= ce^{-\\pi t^2}$ for some $c \\in \\mathbb{C}\\setminus\\{0\\}$. \n\\end{theorem} \n\n\\section{Proof of the main results and remarks}\n\\begin{proof}[Proof of Theorem \\ref{maint}] An explicit computation shows that\n\t\\[ |V_g f(x,-\\omega)| = \\left| \\int_{\\bR^d} e^{2\\pi i t \\cdot \\omega}e^{-\\pi(t-x)^2}f(t)dt \\right| = e^{-\\pi\\omega^2}|\\Phi f(z)|, \\]\n\twhere we set \n\t\\[ \\Phi f(z) = \\int_{\\bR^d} e^{-\\pi(t-z)^2}f(t)dt, \\quad z=x+i\\omega \\in \\mathbb{C}^d. \\]\n\tNotice that $\\Phi f$ is an entire function on $\\mathbb{C}^d$, since differentiation under the integral sign is allowed. Define \n\t\\[ M_{a,R} = \\sup_{z \\in Q_{a,R}} |\\Phi f(z)|, \\quad Q_{a,R}=\\{z=x+i\\omega \\in \\mathbb{C}^d : |x|\\le a, |\\omega| \\le R \\}, \\] where $a>0$ will be fixed in a moment. The maximum principle \\cite{narasi} implies that $|\\Phi f|$ takes the value $M_{a,R}$ at some point of the boundary of $Q_{a,R}$. Since $f,g\\in L^2(\\mathbb{R}^d)$, $V_gf$ vanishes at infinity (e.g.\\ \\cite[Corollary 3.10]{ct}), so that $V_g f(x,-\\omega) \\to 0$ for $|x|\\to + \\infty$, uniformly with respect to $\\omega\\in \\mathbb{R}^d$. Therefore $\\Phi f(x+i\\omega) \\to 0$ for $|x| \\to +\\infty$, uniformly with respect to $\\omega$ over compact subsets of $\\bR^d$. This shows that for sufficiently large $a>0$ we have $|\\Phi f(z_0)|=M_{a,R}$ for some point $z_0=(x_0,\\omega_0)$ with $|\\omega_0|= R$. \n\t\n\tIn view of assumption \\eqref{fat tail cyl} the following estimate holds:\n\t\\[ M_{a,R} e^{-\\pi R^2} = |V_gf(z_0)| \\le \\frac{|\\Phi f(0)|}{N}, \\] where we used the identity $\\langle f,g\\rangle = V_gf(0) = \\Phi f(0)$; therefore \n\t\\[ M_{a,R} \\le \\frac{e^{\\pi R^2}}{N}|\\Phi f(0)|. \\] \n\tAssume now that $R \\le \\sqrt{\\log N \/ \\pi}$; this would imply $M_{a,R} \\le |\\Phi f(0)|$ and thus $\\Phi f$ would be constant on $Q_{a,R}$, hence on $\\mathbb{C}^d$ by analytic continuation \\cite{narasi}. Since $\\Phi f(x+i\\omega) \\to 0$ for $|x| \\to +\\infty$ as already showed above, we could conclude that $\\Phi f \\equiv 0$, hence $V_gf \\equiv 0$ and then $f\\equiv 0$, which is a contradiction. \n\\end{proof}\n\n\\begin{remark}\n\tNotice that Theorem \\ref{maint} still holds in the case where the cylinder in \\eqref{fat tail cyl} is replaced by any other cylinder obtained from the previous one by a symplectic rotation (cf.\\ \\cite[Sec. 2.3.2]{dg}). Indeed, if $\\widehat{S}$ denotes a metaplectic operator \\cite{dg} corresponding to $S \\in \\mathrm{Sp}(d,\\mathbb{R}) \\cap \\mathrm{O}(2d,\\mathbb{R})$, condition \\eqref{fat tail cyl} with $z=(x,\\omega)$ replaced by $S^{-1}z$ is equivalent to\n\t\\[ |V_g(\\widehat{S}f)(x,\\omega)| \\le \\frac{|\\langle \\widehat{S}f,g \\rangle|}{N}. \\]\n\tThis can be easily seen by using the covariance property \\cite[Lemma 9.4.3]{gro book}\n\t\\[\n|V_g f(S^{-1}z)|=|V_{\\widehat{S}g} \\widehat{S}f(z)|,\n\t\\]\nthe fact that $\\widehat{S}$ is unitary on $L^2(\\bR^d)$ and that $\\widehat{S}g = cg$ for some $c \\in \\mathbb{C}$, $|c|=1$, if $g(t)=e^{-\\pi t^2}$ \\cite[Prop. 252]{dg}.\n\\end{remark}\n\n\\begin{remark} The estimate for $R$ in \\eqref{R est cyl} is sharp. Consider indeed a dilated Gaussian function $f_\\lambda(t) = e^{-\\pi\\lambda^2 t^2}$, $0 < \\lambda \\le 1$; a straightforward computation (see for instance \\cite[Lemma 3.1]{cn}) shows that \n\t\\[ V_g f_\\lambda (x,\\omega) = (1+\\lambda^2)^{-d\/2}e^{-2\\pi i \\frac{x\\cdot \\omega}{1+\\lambda^2}} e^{-\\pi \\frac{\\lambda^2 x^2}{1+\\lambda^2}} e^{-\\pi \\frac{\\omega^2}{1+\\lambda^2}}. \\] \nCondition \\eqref{fat tail cyl} is thus satisfied if and only if \n\\[ R \\ge \\sqrt{(1+\\lambda^2) \\frac{\\log N}{\\pi}}, \\] and letting $\\lambda \\to 0^+$ yields the bound in \\eqref{R est cyl}. \n\nIt is worth emphasizing that there is no non-zero $f \\in L^2(\\bR^d)$ such that the optimal bound in \\eqref{R est cyl} can be attained, in contrast to other uncertainty principles for the STFT. \n\\end{remark} \n\n\\begin{proof}[Proof of Theorem \\ref{ft ball}]\nRecall the connection between the STFT and the \\textit{Bargmann transform} of a function $f \\in L^2(\\bR^d)$ \\cite[Prop. 3.4.1]{gro book}:\n\\begin{equation}\\label{barg stft} V_g f (x,-\\omega) = 2^{-d\/4}e^{\\pi i x\\cdot \\omega} \\mathcal{B}f(z) e^{-\\pi |z|^2\/2}, \\quad z=x+i\\omega \\in \\mathbb{C}^d, \\end{equation} where the Bargmann transform is defined by\n\\[ \\mathcal{B}f(z) = 2^{d\/4} \\int_{\\bR^d} f(t) e^{2\\pi t\\cdot z - \\pi t^2 - \\pi z^2 \/2}dt; \\]\n(here $g(t) = e^{-\\pi t^2}$ as in the statement). This correspondence is indeed a unitary operator from $L^2(\\bR^d)$ onto the \\textit{Bargmann-Fock space} $\\mathcal{F}^2(\\mathbb{C}^d)$, i.e.\\ the Hilbert space of all entire functions $F$ on $\\mathbb{C}^d$ such that $e^{-\\pi |\\cdot|^2\/2}F \\in L^2(\\mathbb{C}^d)$, cf.\\ \\cite[Sec. 3.4]{gro book} (see also \\cite{toft1,toft2}). \n\nWe now argue as in the proof of Theorem \\ref{maint}. After setting \n\\[ M_R = \\sup_{z\\in B_R(0)} |\\mathcal{B}f(z)|, \\quad B_R(0) = \\{z \\in \\mathbb{C}^d : |z|\\le R \\}, \\] the maximum principle implies that $|\\mathcal{B}f|$ takes the value $M_R$ on some point $z$ with $|z|=R$ and moreover $M_R>0$ (otherwise by analytic continuation we would have $\\mathcal{B}f=0$ and therefore $f=0$). Condition \\eqref{fat tail ball} then implies \\[ M_R \\le \\frac{e^{\\pi R^2\/2}}{N} |\\mathcal{B}f(0)|. \n\\]\nIf $R < \\sqrt{2 \\log N\/\\pi}$ we obtain $M_R < |\\mathcal{B}f(0)|$, which is a contradiction. If $R = \\sqrt{2 \\log N\/\\pi}$ then $M_R = |\\mathcal{B}f(0)|$ and therefore\n $\\mathcal{B}f(z)= C$, $z \\in \\mathbb{C}^d$, again by the maximum principle and analytic continuation, with $C\\ne0$. On the other hand, a direct computation and the injectivity of the Bargmann transform show that $\\mathcal{B}f(z)=1$ (hence $|V_g f (z)| = 2^{-d\/4} e^{-\\pi |z|^2\/2}$) if and only if $f(t)= 2^{d\/4}e^{-\\pi t^2}$. This gives the last part of the claim. \n\\end{proof}\n\n\\section*{Acknowledgments} The authors wish to thank Professor Elena Cordero for fruitful discussions. \\\\ The present research was partially supported by MIUR grant \"Dipartimenti di Eccellenza\" 2018\u20132022, CUP: E11G18000350001, DISMA, Politecnico di Torino.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{s:intro}\nPlanetesimal belts appear to be a common feature of planetary systems.\nThere are two main belts in the solar system: the asteroid belt and\nthe Kuiper belt.\nThese belts inhabit the regions of the solar system where planetesimal\norbits can remain stable over the 4.5 Gyr age of our system (Lecar et al. 2001).\nThe larger planetesimals in the belts are continually grinding\ndown feeding the smaller bodies in a process known as a collisional cascade\nwhich is slowly eroding the belts (Bottke et al. 2005).\nThe smallest dust in the asteroid belt is acted on by radiation forces;\nP-R drag makes the dust spiral in toward the Sun making a disk known as\nthe zodiacal cloud that the Earth sits in the middle of (Leinert \\& Gr\\\"{u}n 1990).\nA dust cloud is also predicted to arise from collisions amongst Kuiper belt\nobjects (Liou \\& Zook 1999), although our information\non this population is sparse (Landgraf et al. 2002)\nbecause its emission is masked by the zodiacal emission\n(Backman, Dasgupta \\& Stencel 1995) and few dust grains make it into the\ninner solar system (Moro-Mart\\'{\\i}n \\& Malhotra 2003).\n\nMany extrasolar systems also have such planetesimal belts, known as\ndebris disks.\nThese have been detected from their dust content (Aumann et al. 1984) from which\nit has been inferred that larger planetesimals must exist to replenish\nthe dust disks because of the short lifetime of this dust (Backman \\& Paresce 1993).\nThe collisional cascade scenario is supported by modeling of the\nemission spectrum of the dust which shows a size distribution similar\nto that expected for dust coming from a collisional cascade (Wyatt \\& Dent 2002, hereafter\nWD02).\nHowever, the issue of how these disks evolve has recently come under\nclose scrutiny.\n\nFrom a theoretical point view, Dominik \\& Decin (2003; hereafter DD03)\nshowed that if P-R drag is not important then a planetesimal belt\nevolving in quasi-steady state would lose mass due to collisional grinding\ndown giving a disk mass (and dust luminosity) that falls off $\\propto t^{-1}$.\nThis is in broad agreement with the observed properties of debris disks:\nthe mean dust luminosity at a given age falls off $\\propto t^{-1.8}$\n(Spangler et al. 2001);\nthe mass inferred from detection statistics falls off $\\propto t^{-0.5}$\n(Greaves \\& Wyatt 2003), while the mass of the detected disks falls off\n$\\propto t^{-1}$ (Najita \\& Williams 2005);\nthe upper limit in luminosity of the detected disks also falls off $\\propto t^{-1}$\n(Rieke et al. 2005).\nWhile these trends can be viewed as a success of the steady-state model,\nit has yet to be proved that a steady state evolution model fits the data\nin more than just general terms (Meyer et al. 2006).\nSeveral puzzling observations also remain to be explained.\n\nDecin et al. (2003) noted that the maximum fractional luminosity of debris disks remains\nconstant at $f=L_{\\rm{ir}}\/L_\\star \\approx 10^{-3}$ up to the oldest stars,\nwhere $L_{\\rm{ir}}$ and $L_\\star$ are the disk and stellar luminosities respectively\n(see also Table \\ref{tab:symb} for definitions of the parameters used in the text), and this\nwas explained by DD03 as a consequence of delayed stirring.\nA delay in the ignition of a collisional cascade is expected if \nit is the formation of Pluto-sized objects which trigger the cascade, since\nsuch massive bodies take longer, up to several Gyr, to form\nfurther from the star (Kenyon \\& Bromley 2002).\nHowever, that interpretation predicts that the radius of the belts should increase with stellar \nage, and this is not observed (Najita \\& Williams 2005).\nThere is also recent evidence that the dust content of some systems is transient.\nThe discovery of a population of dust grains around Vega in the process of removal by \nradiation pressure indicates that this system cannot have remained in steady state for the full \n350 Myr age of the star (Su et al. 2005).\nRieke et al. (2005) used their statistics on A stars, which showed a wide variety of properties \namong the debris disks, to suggest that much of the dust we see is produced episodically in \ncollisions between large planetesimals.\nThere is also an emerging population of debris disks detected around sun-like\nstars with dust at a few AU (Gaidos 1999; Beichman et al. 2005; Song et al. 2005;\nSmith, Wyatt \\& Dent in prep.).\nThere is debate over whether these are atypically massive asteroid belts or the\nconsequence of a rare transient event (e.g., Beichman et al. 2005).\n\nA stochastic element to the evolution of debris disks would fit with \nour understanding of the evolution of the dust content of the inner \nsolar system.\nThis is believed to have been significantly enhanced for timescales of a few\nMyr following collisions between objects $\\sim 100$ km in size in the asteroid \nbelt (Nesvorn\\'{y} et al. 2003; Farley et al. 2006).\nHowever, it is not known whether the aftermath of individual collisions would be\ndetectable in a debris disk, or indeed whether such events would happen frequently\nenough to explain the statistics (WD02; Telesco et al. 2005).\nSuch events have a dramatic effect on the amount of dust in the solar system\nbecause there is relatively little around during the quiescent periods.\nPlanetesimal belts of equivalent mass to those in the solar system would\nnot have been detected in the current debris disk surveys.\nHowever, there is evidence to suggest that both belts were $\\sim 200$ times more\nmassive in the past (e.g., Stern 1996; Bottke et al. 2005).\nPeriods analogous to the heavy bombardment experienced in the solar system up to\n$\\sim 700$ Myr after its formation have also been invoked to explain the fact that debris\ndisks are most often detected around stars $<400$ Myr old (Habing et al. 1999).\n\nIn the light of this controversy we revisit a simple analytical model\nfor the steady state collisional evolution of planetesimal belts which was\noriginally explored in DD03.\nThe model we derive for that evolution is given in \\S \\ref{s:model},\nand differs in a subtle but important way from that of DD03, since it affects\nthe dust production as a function of collision velocity.\nThis model shows that there is a maximum possible\ndisk mass (and dust luminosity) at any given age.\nIn \\S \\ref{s:hot} confrontation with the few hot planetesimal belts discovered recently\nshows that the majority of these cannot be explained as massive asteroid belts,\nrather these must be systems undergoing a transient event.\nThe possibility that these are caused by a recent collision within a planetesimal\nbelt is also discussed, as is the possibility that the dust originates in a planetesimal\nbelt in the terrestrial planet region.\nThe implications of these results are discussed in \\S \\ref{s:conc}.\nApplication of the model to the statistics of detected debris disks will be\nconsidered in a later paper (Wyatt et al., in prep.).\n\n\\section{Analytical collisional evolution model}\n\\label{s:model}\nIn this section a simple analytical model is developed for the\nevolution of a planetesimal belt due to collisions amongst\nits members.\nThe parameters used in this model are summarized in the table \\ref{tab:symb}\nwhich also gives the units assumed for these parameters throughout the paper.\n\n\\subsection{The planetesimal belt size distribution}\n\\label{ss:pb}\nThe planetesimal belt is assumed to be in collisional\nequilibrium with a size distribution defined by:\n\\begin{equation}\n n(D) = K D^{2-3q}, \\label{eq:nd}\n\\end{equation}\nwhere $q=11\/6$ in an infinite collisional cascade (Dohnanyi 1969)\nand the scaling parameter $K$ is called $f_a$ by DD03.\nThat distribution is assumed to hold from the largest planetesimal\nin the disk, of diameter $D_{\\rm{c}}$, down to the size below which particles\nare blown out by radiation pressure as soon as they are created,\n$D_{\\rm{bl}}$.\nIf we assume that $q$ is in the range 5\/3 to 2 then most of the\nmass is in the largest planetesimals while the cross-sectional area is\nin the smallest particles such that:\n\\begin{eqnarray}\n \\sigma_{\\rm{tot}} & = & 3.5 \\times 10^{-17} K(3q-5)^{-1} (10^{-9}D_{\\rm{bl}})^{5-3q} \\label{eq:stot} \\\\\n M_{\\rm{tot}} & = & 8.8 \\times 10^{-17} K \\rho (6-3q)^{-1} D_{\\rm{c}}^{6-3q}, \\label{eq:mtot1} \\\\\n & = & 2.5 \\times 10^{-9} \\left( \\frac{3q-5}{6-3q} \\right)\n \\rho \\sigma_{\\rm{tot}}D_{\\rm{bl}}\n \\left( \\frac{10^9 D_{\\rm{c}}}{D_{\\rm{bl}}} \\right)^{6-3q}, \n \\label{eq:mtot2}\n\\end{eqnarray}\nwhere spherical particles of density $\\rho$ have been assumed and $M_{\\rm{tot}}$ is\nin $M_\\oplus$ if the units of table \\ref{tab:symb} are used for the other parameters.\n\nThe planetesimal belt is assumed to be at a radius $r$, and to have a width $dr$ (in AU).\nOne of the observable properties of a planetesimal belt is its fractional luminosity,\n$f=L_{\\rm{ir}}\/L_\\star$, i.e., the infrared luminosity from the disk divided by the stellar\nluminosity.\nAssuming that the grains act like black bodies and so absorb all the radiation\nthey intercept we can write:\n\\begin{equation}\n f = \\sigma_{\\rm{tot}}\/(4\\pi r^2). \\label{eq:f}\n\\end{equation}\nIn other words, in this model $\\sigma_{\\rm{tot}}$, $M_{\\rm{tot}}$ and $f$ \nare all proportional to each other and just one is needed to define the scaling\nfactor $K$ in equation (\\ref{eq:nd}).\nAssuming the particles act like black bodies also allows us to derive\nthe following relation:\n\\begin{equation}\n D_{\\rm{bl}} = 0.8(L_\\star\/M_\\star)(2700\/\\rho), \\label{eq:dbl}\n\\end{equation}\nwhere $D_{\\rm{bl}}$ is in $\\mu$m, $L_\\star$ and $M_\\star$ are in solar units, and\n$\\rho$ is in kg m$^{-3}$.\n\nRelaxing the black body assumption is easily achieved (e.g., WD02).\nHowever, this would result in relatively small changes in the way $f$ \nscales with $M_{\\rm{tot}}$, and so for its heuristic simplicity we keep this assumption\nthroughout this paper.\nProbably the most important simplification within this model is that of the\ncontinuous size distribution.\nFor example, we know that the cut-off in the size distribution at $D_{\\rm{bl}}$ would\ncause a wave in the size distribution at sizes just larger than this (Th\\'{e}bault,\nAugereau \\& Beust 2003), that large quantities of blow-out grains can also affect\nthe distribution of small size particles (Krivov, Mann, \\& Krivova 2000), and that\nthe dependence of planetesimal strength on size can result in $q \\ne 11\/6$ as well\nas a wave in the distribution at large sizes (Durda et al. 1998;\nO'Brien \\& Greenberg 2003).\nAlso, since the largest planetesimals would not be in collisional equilibrium\nat the start of the evolution, their initial distribution may not be the same\nas that of a collisional cascade, although distributions with $q \\approx 11\/6$\nhave been reported from planet formation models (e.g., Stern \\& Colwell 1997;\nDavis \\& Farinella 1997; Kenyon \\& Luu 1999) meaning this is a reasonable starting\nassumption.\nDespite these simplifications, we believe this model is adequate to explore to \nfirst order the evolution of planetesimal belts which can later be studied in\nmore depth.\n\n\\subsection{Collisional evolution}\n\\label{ss:ce}\nIn a collisional cascade material in a bin with a given size range $D$ to $D+dD$ is replaced\nby fragments from the destruction of larger objects at the same rate that it is destroyed\nin collisions with other members of the cascade.\nThe long-timescale evolution is thus determined by the removal of mass from the top end of\nthe cascade.\nIn this model the scaling factor $K$ (and so the total mass and fractional luminosity\netc) decreases as the number of planetesimals of size $D_{\\rm{c}}$ decreases.\nThe loss rate of such planetesimals is determined by their collisional lifetime, which in\nthe terminology of WD02 is given by:\n\\begin{equation}\n t_{\\rm{c}} = \\sqrt{r^3\/M_\\star} (r dr\/\\sigma_{\\rm{tot}}) [2I\/f(e,I)] \/ f_{\\rm{cc}}, \n\\label{eq:tc1}\n\\end{equation}\nwhere maintaining the units used previously gives $t_{\\rm{c}}$ in years, $I$ is the mean \ninclination\nof the particles' orbits (which determines the torus height), $f(e,I)$ is the ratio of\nthe relative velocity of collisions to the Keplerian velocity ($=v_{\\rm{rel}}\/v_{\\rm{k}}$, \nalso called $\\nu$ by DD03),\nand $f_{\\rm{cc}}$ is the fraction of the total cross-sectional area in the belt which\nis seen by planetesimals of size $D_{\\rm{c}}$ as potentially causing a catastrophic \ncollision.\n\nFrom hereon we will use the assumption that $f(e,I) = \\sqrt{1.25e^2+I^2}$, where $e$\nis the mean eccentricity of the particles, which is valid for Rayleigh distributions\nof $e$ and $I$ (Lissauer \\& Stewart 1993; Wetherill \\& Stewart 1993).\nAn expression for $f_{\\rm{cc}}$ was given in WD02, however, here we will\nignore the gravitational focussing effect, which is important in the\naccumulation phase but not during the destruction\nphase of a planetesimal belt (see \\S \\ref{ss:pc}),\nand so derive an expression that is the\nsame as that given in Wyatt et al. (1999):\n\\begin{equation}\n f_{\\rm{cc}} = (10^{-9} D_{\\rm{bl}}\/D_{\\rm{c}})^{3q-5}G(q,X_{\\rm{c}}), \\label{eq:fcc}\n\\end{equation}\nwhere $X_{\\rm{c}}=D_{\\rm{cc}}\/D_{\\rm{c}}$, $D_{\\rm{cc}}$ is the smallest planetesimal \nthat has enough energy to catastrophically destroy a planetesimal of size $D_{\\rm{c}}$ (which \nis\ncalled $\\epsilon$ in DD03), and:\n\\begin{eqnarray}\n G(q,X_{\\rm{c}}) & = & [(X_{\\rm{c}}^{5-3q}-1)+ (6q-10)(3q-4)^{-1}(X_{\\rm{c}}^{4-3q}-1) \n\\nonumber \\\\\n & & + (3q-5)(3q-3)^{-1}(X_{\\rm{c}}^{3-3q}-1)]. \\label{eq:qgxc}\n\\end{eqnarray}\n\nThe factor $X_{\\rm{c}}$ can be worked out from the dispersal threshold, $Q_{\\rm{D}}^\\star$, \ndefined\nas the specific incident energy required to catastrophically destroy a particle such \nthat (WD02):\n\\begin{eqnarray}\n X_{\\rm{c}} & = & (2Q_{\\rm{D}}^\\star\/v_{\\rm{rel}}^2)^{1\/3}, \\label{eq:xc1} \\\\\n & = & 1.3 \\times 10^{-3} [Q_{\\rm{D}}^\\star r M_\\star^{-1} f(e,I)^{-2} ]^{1\/3}, \n\\label{eq:xc2}\n\\end{eqnarray}\nwhere $Q_{\\rm{D}}^\\star$ is in J kg$^{-1}$ (called $S$ in DD03\\footnote{Equation 25 in DD03 differs \nfrom our\nequation (\\ref{eq:xc1}) because we define $Q_{\\rm{D}}^\\star$ to be the specific incident \nkinetic\nenergy so that $0.5M_2v_{\\rm{rel}}^2=M_1 Q_{\\rm{D}}^\\star$ whereas DD03 define $S$ to \nbe the specific binding energy of the two objects (giving their equation 24).\nIn the limit of $S \\ll v_{\\rm{rel}}^2\/8$ the two equations are the same, since $X_{\\rm{c}} \n\\ll \n1$.})\n\nCombining the above equations gives for the collisional lifetime of the planetesimals\nof size $D_{\\rm{c}}$:\n\\begin{eqnarray}\n t_{\\rm{c}} & = & \\left( \\frac{r^{2.5} dr}{M_\\star^{0.5} \\sigma_{\\rm{tot}}} \\right) \n \\left( \\frac{2[1+1.25(e\/I)^2]^{-0.5}}{G(q,X_{\\rm{c}})} \\right)\n \\left( \\frac{10^{-9}D_{\\rm{bl}}}{D_{\\rm{c}}} \\right)^{5-3q}, \n\\label{eq:tcstot} \\\\\n & = & \\left( \\frac{3.8\\rho r^{2.5} dr \n D_{\\rm{c}}}{M_\\star^{0.5} M_{\\rm{tot}}} \\right) \n \\left( \\frac{(12q-20)[1+1.25(e\/I)^2]^{-0.5}}{(18-9q)G(q,X_{\\rm{c}})} \\right).\n \\label{eq:tcmtot}\n\\end{eqnarray}\nAssuming that collisions are the only cause of mass loss in the belt, the evolution of\nthe disk mass $M_{\\rm{tot}}(t)$ (or equivalently of $K$, $\\sigma_{\\rm{tot}}$, or $f$)\ncan be worked out by solving $dM_{\\rm{tot}}\/dt = -M_{\\rm{tot}}\/t_{\\rm{c}}$ to give:\n\\begin{equation}\n M_{\\rm{tot}}(t) = M_{\\rm{tot}}(0)\/[1+t\/t_{\\rm{c}}(0)], \\label{eq:mtott}\n\\end{equation}\nwhere $M_{\\rm{tot}}(0)$ is the initial disk mass and $t_{\\rm{c}}(0)$ is the collisional \nlifetime at that initial epoch;\nthis solution is valid as long as mass is the only parameter of the planetesimal belt\nthat changes with time.\nThis results in a disk mass which is constant at $M_{\\rm{tot}}(0)$ for $t \\ll t_{\\rm{c}}(0)$, \nbut which falls off $\\propto 1\/t$ for $t \\gg t_{\\rm{c}}(0)$ (as noted, e.g., in DD03).\n\n\\begin{figure*}\n \\centering\n \\begin{tabular}{cc}\n \\hspace{-0.15in} \n \\includegraphics[width=3.2in]{f1a.ps} &\n \\hspace{-0.15in} \n \\includegraphics[width=3.2in]{f1b.ps}\n \\end{tabular}\n \\caption{The dependence of \\textbf{(left)} $G(11\/6,X_{\\rm{c}})$ and \\textbf{(right)}\n $X_{\\rm{c}}$ on planetesimal eccentricity ($e$) for planetesimals of different\n strengths ($Q_{\\rm{D}}^\\star$) and at different distances from the star ($r$).}\n\\label{fig:gvse}\n\\end{figure*}\n\nHowever, another interesting property of this evolution is that, since the\nexpression for $t_{\\rm{c}}(0)$ includes a dependence on $M_{\\rm{tot}}(0)$, the disk\nmass at late times is independent of initial disk mass.\nThis is because more massive disks process their mass faster.\nThis means that for any given age, $t_{\\rm{age}}$, there is a maximum disk mass \n$M_{\\rm{max}}$\n(and also infrared luminosity, $f_{\\rm{max}}$) that can remain due to collisional processing:\n\\begin{eqnarray}\n M_{\\rm{max}} & = & \\left( \\frac{3.8 \\times 10^{-6} \\rho r^{3.5} (dr\/r) \n D_{\\rm{c}}}{M_\\star^{0.5}t_{\\rm{age}}} \\right) \\times \\nonumber \\\\\n & & \\left( \\frac{(12q-20)[1+1.25(e\/I)^2]^{-0.5}}{(18-9q)G(q,X_{\\rm{c}})} \\right), \n \\label{eq:mmax1} \\\\\n f_{\\rm{max}} & = & \\left( \\frac{10^{-6} r^{1.5}(dr\/r)}{4\\pi M_\\star^{0.5} t_{\\rm{age}}} \\right)\n \\left( \\frac{10^{-9} D_{\\rm{bl}}}{D_{\\rm{c}}} \\right)^{5-3q}\n \\times \\nonumber \\\\ \n & & \\left( \\frac{2[1+1.25(e\/I)^2]^{-0.5}}{G(q,X_{\\rm{c}})} \\right).\n \\label{eq:fmax1} \n\\end{eqnarray}\nIn this model, the present day disk mass (or luminosity) is expected to\nbe equal to this \"maximum\" disk mass (or luminosity) for disks in which the\nlargest planetesimals are in collisional equilibrium.\nThis corresponds to disks around stars that are older than the collisional\nlifetime of those planetesimals given in equation (\\ref{eq:tcmtot}).\n\nFor example, with the further assumptions that $q=11\/6$, $e \\approx I$, and\n$\\rho = 2700$ kg m$^{-3}$, we find:\n\\begin{eqnarray}\n M_{\\rm{max}} & = & 0.009 r^{3.5} (dr\/r)\n D_{\\rm{c}} M_\\star^{-0.5} t_{\\rm{age}}^{-1}\/G(11\/6,X_{\\rm{c}}), \\label{eq:mmax2} \\\\\n f_{\\rm{max}} & = & 0.004 r^{1.5} (dr\/r)\n D_{\\rm{c}}^{0.5} L_\\star^{-0.5} t_{\\rm{age}}^{-1}\/G(11\/6,X_{\\rm{c}}), \\label{eq:fmax2}\n\\end{eqnarray}\nwhere $M_{\\rm{max}}$ is in $M_\\oplus$, $r$ in AU, $D_{\\rm{c}}$ in km, $t_{\\rm{age}}$ in Myr, \nand\n$G(11\/6,X_{\\rm{c}})=X_{\\rm{c}}^{-0.5}+0.67X_{\\rm{c}}^{-1.5}+0.2X_{\\rm{c}}^{-2.5}-1.87$,\nwith $X_{\\rm{c}}=10^{-3}(rQ_{\\rm{D}}^\\star\/e^2)^{1\/3}$ ($Q_{\\rm{D}}^\\star$ is in J kg$^{-1}$).\n\nPlots of $G(11\/6,X_{\\rm{c}})$ and $X_{\\rm{c}}$ for typical planetesimal belts are\nshown in Fig.~\\ref{fig:gvse}.\nHowever, for many disks the approximation that $X_{\\rm{c}} \\ll 1$ is valid, and\nso $G(11\/6,X_{\\rm{c}}) \\approx 0.2X_{\\rm{c}}^{-2.5} =\n6.3 \\times 10^6 r^{-5\/6}{Q_{\\rm{D}}^\\star}^{-5\/6}e^{5\/3}M_\\star^{5\/6}$, giving:\n\\begin{eqnarray}\n M_{\\rm{max}} & = & 1.4 \\times 10^{-9} r^{13\/3} (dr\/r)\n D_{\\rm{c}} {Q_{\\rm{D}}^\\star}^{5\/6} \\times \\nonumber \\\\\n & & e^{-5\/3} M_\\star^{-4\/3} t_{\\rm{age}}^{-1}, \\label{eq:mmax3} \\\\\n f_{\\rm{max}} & = & 0.58 \\times 10^{-9} r^{7\/3} (dr\/r)\n D_{\\rm{c}}^{0.5} {Q_{\\rm{D}}^\\star}^{5\/6}\\times \\nonumber \\\\\n & & e^{-5\/3} M_\\star^{-5\/6} L_\\star^{-0.5} t_{\\rm{age}}^{-1}.\n \\label{eq:fmax3}\n\\end{eqnarray}\n\n\n\\subsection{Comparison with DD03}\n\\label{ss:dd03}\nSince DD03 produced a very similar analytical model, our results were\ncompared with those of DD03.\nThe results of disk evolution for a planetesimal belt close to their nominal\nmodel were computed using the parameters:\n$r=43$ AU, $dr=15$ AU, $D_{\\rm{c}}=2$ km, $\\rho=2700$ kg m$^{-3}$, $f(e,I)=0.1$, $e\/I=1$, \n$Q_{\\rm{D}}^\\star=200$ J kg$^{-1}$, $M_{\\rm{tot}}(0)=10M_\\oplus$, A0 star (for which $L_\\star=54L_\\odot$,\n$M_\\star=2.9M_\\odot$, $D_{\\rm{bl}}=15$ $\\mu$m).\nEach of the parameters $M_{\\rm{tot}}(0)$, $r$, $f(e,I)$, $D_{\\rm{c}}$ and spectral type \nwere also varied to make the plots shown in Fig.~\\ref{fig:dd03} which are\nequivalent to Figs 1b-1f of DD03.\n\n\\begin{figure*}\n \\centering\n \\begin{tabular}{cc}\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f2a.ps} &\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f2b.ps} \\\\[-0.0in]\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f2c.ps} &\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f2d.ps} \\\\[-0.0in]\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f2e.ps} &\n \\end{tabular}\n \\caption{The collisional evolution of a planetesimal belt with parameters similar\n to the nominal model of DD03 \n [$r=43$ AU, $dr=15$ AU, $D_{\\rm{c}}=2$ km, $\\rho=2700$ kg m$^{-3}$, $f(e,I)=0.1$,\n $e\/I=1$, $Q_{\\rm{D}}^\\star=200$ J kg$^{-1}$, $M_{\\rm{tot}}(0)=10M_\\oplus$, A0 star]\n showing the effect of changing:\n \\textbf{(top left)} starting disk mass $M_{\\rm{tot}}(0)$,\n \\textbf{(top right)} disk radius $r$,\n \\textbf{(middle left)} collision velocity $v_{\\rm{rel}}\/v_{\\rm{k}}$,\n \\textbf{(middle right)} maximum planetesimal size $D_{\\rm{c}}$, and\n \\textbf{(bottom left)} stellar spectral type.\n These plots can be directly compared to figs. 1b-f of DD03.}\n \\label{fig:dd03}\n\\end{figure*}\n\nThe results are very similar in most regards:\nmore massive disks start out with higher $f$, but the turnover from constant to $1\/t$\nevolution is later for lower mass disks meaning that at late times all disks converge\nto the same maximum value (Fig.~\\ref{fig:dd03}a);\nputting the same mass at larger distances reduces the initial dust luminosity $f$, but the\nresulting lower surface density and longer orbital timescales there combine to make the\nturnover happen later which means that at late times more distant belts are more \nmassive (Fig.~\\ref{fig:dd03}b);\nputting the same mass into larger planetesimals reduces the cross-sectional area\nof dust (equation \\ref{eq:mtot2}) and so the initial dust luminosity $f$, but increases\nthe collisional lifetime of those planetesimals (equation \\ref{eq:tcmtot}) which means\nthat at late times belts with larger planetesimals retain their mass for longer\n(Fig.~\\ref{fig:dd03}d);\nlater spectral types have higher starting dust luminosities because the cascade extends\ndown to smaller sizes (equation \\ref{eq:dbl}), and the longer orbital times mean that\nthey keep their mass for longer (Fig.~\\ref{fig:dd03}e).\n\nWhere the models differ is in the exact way $M_{\\rm{tot}}$ is used to get $f$ and \n$t_{\\rm{c}}$, and\nin the way the evolution is affected by changing $v_{\\rm{rel}}\/v_{\\rm{k}}$ \n(Fig.~\\ref{fig:dd03}c).\nThis is because the models make different assumptions.\nHere we assume that the size distribution is continuous between $D_{\\rm{c}}$ and \n$D_{\\rm{bl}}$,\nwhereas in DD03 the large planetesimals feeding the cascade are seen as separate from\nthe cascade.\nThis means that for us $M_{\\rm{tot}}$ gives a direct estimate of $K$ (equation \n\\ref{eq:mtot1})\nand so the amount of dust $f$, while for DD03 they equate the mass flow through the cascade\nwith the mass input from the break-up of planetesimals meaning that while their scaling\nparameter is proportional to $M_{\\rm{tot}}$ (as is ours), it also includes a dependence on \nthe\nparameter we call $X_{\\rm{c}}$ which affects the mass flow rate in the cascade.\nThis explains all of the differences:\nthe details of the scaling explain the slightly different initial $f$ values in all\nthe figures,\nand the fact that for us planetesimals of size $D_{\\rm{c}}$ are destroyed by planetesimals\ndown to size $X_{\\rm{c}}D_{\\rm{c}}$ means that our collisional lifetimes are always shorter \nthan \nthose in\nDD03, since they assume that planetesimals only collide with same size planetesimals.\nFor us changing $v_{\\rm{rel}}\/v_{\\rm{k}}$ does not affect the initial $f$ parameter as \ndescribed\nabove, but it does affect the collisional lifetime of the largest planetesimals\nwhich can survive longer if $v_{\\rm{rel}}\/v_{\\rm{k}}$ is reduced (since this means that fewer\nplanetesimals in the cascade cause destruction on impact).\nThe opposite is the case for the DD03 model: changing $v_{\\rm{rel}}\/v_{\\rm{k}}$ does not\naffect the collisional lifetime of the largest planetesimals, since they only collide\nwith each other, but a lower collision velocity does increase the initial dust\nluminosity because the cascade must have more mass in it to result in a mass flow\nrate sufficient to remove mass introduced by the large planetesimals.\nWhile the difference is subtle, it is important, since $v_{\\rm{rel}}\/v_{\\rm{k}}$ may be \nimportant\nin determining the presence of dust at late times (DD03; section \\ref{s:hot}).\n\nOn the face of it, it seems that our model provides a more accurate description of the disk.\nThe reason is that in a collisional cascade the mass flow does not need to be taken into\naccount, since it results in the $q=11\/6$ size distribution (Tanaka et al. 1996).\nIn other words the dependence of the scaling of the cascade with $X_{\\rm{c}}$ found by\nDD03 should have been removed if the largest planetesimals had been allowed to\ncollide with smaller planetesimals (since increasing $X_{\\rm{c}}$ would have both\nrestricted mass flow within the cascade and slowed down the mass input from\nthe destruction of large planetesimals).\nHowever, it is also true that the $q=11\/6$ distribution only applies in an\ninfinite cascade, and since both models have truncated the size distribution\nat $D_{\\rm{c}}$, this would affect the evolution.\nAlso, the effect of the variation of $Q_{\\rm{D}}^\\star$ with $D$ on the size distribution\nand its evolution are not yet clear, and neither is the evolution of the size \ndistribution while the collisional cascade is being set up.\nThese issues will be discussed only briefly in this paper, in which the simple\nevolution model described above is applied to some of the latest observational\nresults on debris disks.\n\n\\section{Application to rare systems with hot dust}\n\\label{s:hot}\n\n\\begin{deluxetable*}{cccccccc}\n \\tabletypesize{\\scriptsize}\n \\tablecaption{Main sequence sun-like (F, G and K) stars in the literature\n with evidence for hot dust at $<10$ AU.\n \\label{tab:hot} }\n \\tablewidth{0pt}\n \\tablehead{\n \\colhead{Star name} & \\colhead{Sp. Type} & \\colhead{Age, Myr} & \\colhead{Radius, AU} &\n \\colhead{$f_{\\rm{obs}} = L_{\\rm{ir}} \/ L_\\star $} & \\colhead{$f_{\\rm{max}}$} &\n \\colhead{Transient?} & \\colhead{Reference} }\n \\startdata\n HD98800\\tablenotemark{c} & K4\/5V & $\\sim$ 10 & 2.2 & $220\\times10^{-3}$ & \n $270 \\times 10^{-6}$ & Not req & Low et al. (2005) \\\\\n HD113766\\tablenotemark{ac} & F3V & 16 & 3 & $2.1\\times10^{-3}$ & \n $45 \\times 10^{-6}$ & Not req & Chen et al. (2005) \\\\\n HD12039 & G3\/5V & 30 & 4-6 & $0.1\\times10^{-3}$ &\n $200 \\times 10^{-6}$ & Not req & Hines et al. (2006) \\\\ \n BD+20307\\tablenotemark{a} & G0V & 300 & 1 & $40\\times10^{-3}$ &\n $0.36 \\times 10^{-6}$ & Yes & Song et al. (2005) \\\\ \n HD72905\\tablenotemark{a} & G1.5V & 400 & 0.23\\tablenotemark{b} & $0.1\\times10^{-3}$ & \n $0.011 \\times 10^{-6}$ & Yes & Beichman et al. (2006a) \\\\ \n $\\eta$ Corvi\\tablenotemark{a} & F2V & 1000 & 1-2\\tablenotemark{b} & $0.5\\times10^{-3}$ & \n $0.15 \\times 10^{-6}$ & Yes & Wyatt et al. (2005) \\\\ \n HD69830\\tablenotemark{a} & K0V & 2000 & 1 & $0.2\\times10^{-3}$ &\n $0.13 \\times 10^{-6}$ & Yes & Beichman et al. (2005)\n \\enddata\n \\tablenotetext{a}{infrared silicate feature}\n \\tablenotetext{b}{also has cool dust component at $>10$ AU}\n \\tablenotetext{c}{binary star}\n\\end{deluxetable*}\n\nVery few main sequence stars exhibit hot dust within $\\sim 10$ AU,\ni.e., in the region where we expect planets may have formed.\nFour surveys have searched for hot dust around sun-like stars\n(main sequence F, G or K stars) by looking for a 25 $\\mu$m flux in\nexcess of photospheric levels using IRAS (Gaidos 1999), ISO (Laureijs\net al. 2002) and Spitzer (Hines et al. 2006; Bryden et al. 2006).\nAll concluded that only $2 \\pm 2$\\% of these stars have hot dust with\ninfrared luminosities $f=L_{\\rm{ir}}\/L_\\star > 10^{-4}$, finding a total\nof 3 candidates.\nOther hot dust candidates exist in the literature, however some IRAS\nexcess fluxes have turned out to arise from chance alignments with background \nobjects (e.g., Lisse et al. 2002), including the candidate HD128400 from\nthe hot dust survey of Gaidos (1999) (Zuckerman, priv. comm.).\nThus confirmation of the presence of dust centred on the star using\nground- and space-based mid-IR imaging is vitally important (Smith, Wyatt\n\\& Dent, in prep.).\nThe tally of confirmed hot dust sources now stands at seven, and these\nare summarized in Table \\ref{tab:hot} which also gives the estimated\nradial location of the dust based on fitting of the spectral energy\ndistribution of the excess emission;\nfor all stars the dust is predicted to lie at $<10$ AU.\n\nWhile the frequency of the presence of such emission is low, there is\nas yet no adequate explanation for its origin and why it occurs in so few \nsystems.\nAnalogy with the solar system suggests that these are systems in which\nwe are witnessing the collisional grinding down of atypically massive\nasteroid belts.\nHowever, other scenarios have also been proposed in which the dust\nis transient, having been produced in some stochastic process.\nSuch a process could be a recent collision between two massive\nprotoplanets in an asteroid belt (Song et al. 2005), the sublimation\nof one supercomet (Beichman et al. 2005), or the sublimation of a swarm\nof comets, possibly scattered in from several tens of AU in an\nepisode analogous to the period of Late Heavy Bombardment in the\nsolar system (Gomes et al. 2005).\n\n\\subsection{Are these massive asteroid belts?}\n\\label{ss:qe}\n\nHere we consider the possibility that these are atypically massive\nasteroid belts, and show that for the majority of the known systems this\nis unlikely to be the case.\nThe reason is that given in \\S \\ref{ss:ce}, which is that more massive\nasteroid belts are not necessarily more dusty at late times, and there\nis a maximum dust luminosity we can expect for a belt of a given age,\ngiven its radial location (equations \\ref{eq:mmax1}-\\ref{eq:fmax3}).\nTo arrive at a rough estimate of the maximum possible $f_{\\rm{max}}$\nwe assume the following parameters:\nthe largest possible planetesimal is $D_{\\rm{c}} = 2000$ km, since this is above the\nlargest members of the asteroid and Kuiper belts, and fits with the\nexpectation that planetesimal growth is halted once the largest planetesimals\nreach this size due to the resulting gravitational perturbations (Kenyon \\& Bromley \n2002);\nbelt width is $dr = 0.5r$;\nplanetesimal strength is $Q_{\\rm{D}}^\\star = 200$ J kg$^{-1}$, the canonical\nvalue used in DD03, although gravity strengthening can give rise to higher\nvalues for planetesimals larger than $\\sim 1$ km (see \\S \\ref{ss:pc});\nplanetesimal eccentricity is $e = 0.05$, typical for planetesimal belts\nlike the asteroid belt that are undergoing a collisional cascade, and close\nto that expected from stirring by 2000 km planetesimals within such belt.\n\\footnote{Equating the velocity dispersion in the belt with the escape\nvelocity of a planetesimal of size $D_{\\rm{c}}$ gives\n$e \\approx 2.6 \\times 10^{-7} \\rho^{0.5} r^{0.5} M_\\star^{-0.5} D_{\\rm{c}}$. }\nSubstituting in these nominal values into equation (\\ref{eq:fmax3}) and\napproximating $M_\\star=L_\\star=1$ gives:\n\\begin{equation}\n f_{\\rm{max}} = 0.16 \\times 10^{-3} r^{7\/3} t_{\\rm{age}}^{-1}. \\label{eq:fmax4}\n\\end{equation}\nPlots analogous to those in Fig.~\\ref{fig:dd03} are presented in Fig.~\\ref{fig:hotevol}\nwhich shows the evolution for a planetesimal belt with the nominal parameters\ndescribed above (and with a nominal starting mass of $M_{\\rm{tot}}(0)=1M_\\oplus$) along\nwith the consequence for the evolution of changing any of those parameters.\nNote that it is most appropriate to refer to Fig.~\\ref{fig:hotevol}, rather than\nFig.~\\ref{fig:dd03}, when considering the evolution of planetesimal belts close to\nsun-like stars.\n\n\\begin{figure*}\n \\centering\n \\begin{tabular}{cc}\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f3a.ps} &\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f3b.ps} \\\\[-0.0in]\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f3c.ps} &\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f3d.ps} \\\\[-0.0in]\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f3e.ps} &\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f3f.ps}\n \\end{tabular}\n \\caption{The collisional evolution of a planetesimal belt at $r=1$ AU around a sun-like\n star ($L_\\star=M_\\star=1$) of initial mass $M_{\\rm{tot}}(0)=1M_\\oplus$ assuming that belt\n can be described by the parameters used in \\S \\ref{ss:qe} (i.e., $dr\/r=0.5$,\n $D_{\\rm{c}}=2000$ km, $\\rho=2700$ kg m$^{-3}$, $e=0.05$, $e\/I=1$,\n $Q_{\\rm{D}}^\\star=200$ J kg$^{-1}$).\n All panels show dust luminosity $f=L_{\\rm{ir}}\/L_\\star$ as a function of time, and\n the evolution with the above nominal parameters is shown with a solid line.\n The different panels show the effect of changing the following parameters:\n \\textbf{(top left)} starting disk mass $M_{\\rm{tot}}(0)$,\n \\textbf{(top right)} disk radius $r$,\n \\textbf{(middle left)} planetesimal eccentricity $e$,\n \\textbf{(middle right)} maximum planetesimal size $D_{\\rm{c}}$,\n \\textbf{(bottom left)} planetesimal strength $Q_{\\rm{D}}^\\star$, and\n \\textbf{(bottom right)} stellar spectral type.}\n \\label{fig:hotevol}\n\\end{figure*}\n\nThe value of $f_{\\rm{max}}$ is quoted in Table \\ref{tab:hot} under the assumption\nthat the planetesimal belt has the same age as the star.\nThe quoted value for each star is that from equation (\\ref{eq:fmax2}) for its\nspectral type, but is within a factor of three of that given in equation\n(\\ref{eq:fmax4}), indicating that this equation may be readily applied to\nobserved belts in the future.\nThe four oldest systems (BD+20307, HD72905, $\\eta$ Corvi and HD69830)\nhave $f_{\\rm{obs}} \\gg 10^{3}f_{\\rm{max}}$.\nWe show in \\S \\ref{ss:pc} that even with a change in parameters it is not\npossible to devise asteroid belts in these systems that could survive to the age of\nthe stars giving rise to the observed dust luminosities.\nThus we conclude that this period of high dust luminosity started\nrelatively recently.\nThe timescale over which a belt can last above a given\nluminosity, $f_{\\rm{obs}}$, is $t_{\\rm{age}}f_{\\rm{max}}\/f_{\\rm{obs}}$,\nsince collisions would grind a belt down to this level on such a timescale.\nThis implies that belts this luminous only last between a few thousand years\n(BD+20307 and HD72905) and a few Myr ($\\eta$ Corvi and HD69830).\nHowever, the true duration of this level of dust luminosity depends\non the details of the process causing it, and moreover there is still\nup to two orders of magnitude uncertainty in $f_{\\rm{max}}$ (see \\S \\ref{ss:pc}).\nThus this calculation should not yet be used to infer from the $\\sim 2$\\% of\nsystems with hot dust that, e.g., every sun-like star must undergo 10-1000 such\nevents in its lifetime (or fewer systems must undergo even more events).\nFor now the conclusion is that these systems cannot be planetesimal\nbelts that have been evolving in a collisional cascade for the full age of the\nstar.\n\nThis leaves open the possibility that the collisional cascade in these systems\nwas initiated much more recently, perhaps because a long timescale was required\nto form the 2000-3000 km sized planetesimals necessary to stir the planetesimal\nbelt and cause the switch from accretion to collisional cascade (Kenyon \\& Bromley\n2004).\nHowever, we consider this to be unlikely, because the timescale for the formation\nof objects of this size at 1 AU from a solar mass star was given in\nKenyon \\& Bromley (2004) to be $\\sim 0.6 dr\/M_{\\rm{tot}}$ Myr, where\n$M_{\\rm{tot}}$ is the mass of material in an annulus of width\n$dr$, just as in the rest of the paper.\nThis means that the cascade can only be delayed for 100-1000 Myr at 1 AU for planetesimal\nbelts of very low mass, which would also be expected to have low dust luminosities when\nthe cascade was eventually ignited.\nFor example, a delay of $>500$ Myr would require $<0.6 \\times 10^{-3}M_\\oplus$ in the\nannulus at 1 AU of 0.5 AU width, a mass which corresponds to a fractional luminosity\nof $<5 \\times 10^{-6}$ (equations \\ref{eq:mtot2} and \\ref{eq:f} with $\\rho = 2700$ kg \nm$^{-3}$ and $q=11\/6$), much lower than that observed in all systems.\nOne can also consider the same argument in the following way:\nthe observed luminosity $f_{\\rm{obs}}$ implies a planetesimal belt mass which current\nplanet formation theories indicate would result in the growth of 2000 km planetesimals\nwhich would ignite a collisional cascade on a timescale of\n$3 \\times 10^{-3}(dr\/r)\/f_{\\rm{obs}}$ Myr if this was placed at 1 AU from a solar mass star.\nThe conclusion at the end of the last paragraph also considers the collisional cascade to evolve \nin quasi-steady state, and it is possible that collisions between large members of the cascade \nmay have recently introduced large quantitites of small dust;\nthat possibility is discussed in \\S \\ref{ss:singlecoll}.\n\nFor the three youngest systems the conclusions are less clear.\nThe dust luminosities of HD12039 and HD113766 are, respectively, close to\nand fifty times higher than the maximum allowed value for collisionally evolved \nplanetesimal belts.\nHowever, given the uncertainties in the parameters in the model (described in \\S \n\\ref{ss:pc}), we conclude that it is not possible to say that these could not be massive \nasteroid belts.\nThe main reason that firm conclusions cannot be drawn is the large radial location\nof the dust at $>2$ AU.\nThe strong dependence of $f_{\\rm{max}}$ on $r$ means that it is easiest to\nconstrain the nature of belts within a few AU which evolve very rapidly.\nFor the youngest system (HD98800), while its dust luminosity lies a factor of \n800 above the maximum for the age of the star, we do not infer\nthat this must be transient, since the high dust luminosity and low age imply\nthat this system is in a transitional phase and the collisional cascade in\nthis debris disk is likely to have only recently been ignited.\nRather we note that this model implies that due to collisional processing\nthis debris disk cannot maintain this level of dust emission beyond the\nnext $\\sim 10,000$ years (albeit with an additional two orders of magnitude\nuncertainty, \\S \\ref{ss:pc}).\n\n\\subsection{Possible caveats}\n\\label{ss:pc}\nGiven the large number of assumptions that went into the estimate for\n$f_{\\rm{max}}$, it is worth pointing out that this model is in excellent\nagreement with the properties of the asteroid belt in the solar system,\nsince for a 4500 Myr belt at 3 AU the model predicts\n$M_{\\rm{max}}=0.4 \\times 10^{-3}M_\\oplus$, which is close to the inferred\nmass of the asteroid belt of $0.6 \\times 10^{-3}M_\\oplus$ (Krasinsky et al. 2002).\nThe model also predicts $f_{\\rm{max}}=5 \\times 10^{-7}$, which is consistent\nwith the estimate for the zodiacal cloud of\n$L_{\\rm{ir}}\/L_\\star = 0.8 \\times 10^{-7}$ (Backman \\& Paresce 1993).\n\\footnote{In planetesimal belts as tenuous as the asteroid belt, the\neffect of P-R drag is important (Wyatt 2005) meaning that the cross-sectional\narea of dust in the zodiacal cloud is dominated by $\\sim 100$ $\\mu$m sized grains\nrather than grains of size $D_{\\rm{bl}}$ as assumed in the simple model of\n\\S \\ref{ss:pb}.\nTaking this into account would reduce the fractional luminosity predicted by the model\nby an order of magnitude.}\nIt is also necessary to explore if there is any way in which the parameters\nof the model could be relaxed to increase $f_{\\rm{max}}$ and so change the\nconclusions about the transience of the hot dust systems.\nEquation (\\ref{eq:fmax3}) indicates one way in which\n$f_{\\rm{max}}$ could be increased, which is by either reducing the eccentricities of the \nplanetesimals, $e$, or increasing their strength, $Q_{\\rm{D}}^\\star$, both of which could \nincrease $X_{\\rm{c}}$ and so decrease the rate at which mass is lost from the cascade\n(e.g., fig.~\\ref{fig:hotevol}).\nThe other way is to change the size distribution so that a given disk mass results\nin a significantly larger dust luminosity, e.g., by increasing $q$.\n\nIn fact Benz \\& Asphaug (1999) found a value of $Q_{\\rm{D}}^\\star$ that is higher \nthan $2 \\times 10^{5}$ J kg$^{-1}$ for planetesimals as large as 2000 km for both ice\nand basalt compositions.\nThis would result in an increase in $f_{\\rm{max}}$ by a factor of $\\sim 170$\n(e.g., Fig.~\\ref{fig:hotevol}).\nHowever, such a high value of $Q_{\\rm{D}}^\\star$ is possible only due to gravity \nstrengthening\nof large planetesimals, and the dependence in this regime of $Q_{\\rm{D}}^\\star \\propto \nD^{1.3}$ \n(Benz \\& Asphaug 1999) would result in an equilibrium size distribution with\n$q_{\\rm{g}} \\approx 1.68$, since when $Q_{\\rm{D}}^\\star \\propto D^s$ then $q = (11+s)\/(6+s)$\n(O'Brien \\& Greenberg 2003).\nIf such a distribution was to hold down to the smallest dust grains the net result\nwould be a decrease in $f_{\\rm{max}}$ by $\\sim 200$.\nThis is not the case, however, since objects in the size\nrange $D1$ (e.g., Fig.~\\ref{fig:gvse}b).\nIn such a regime mutual collisions do not result in the destruction of \nplanetesimals, rather in their merger and growth.\nAt this point $G(11\/6,X_{\\rm{c}})<0$, i.e., $f_{\\rm{max}}$ is infinite since, in this\nsimple model, whatever the starting conditions there is no evolution (although\nin practice the size distribution would evolve due to planetesimal growth).\nAt $\\sim 1$ AU, this means $e$ must be larger than\n0.0005 (for $Q_{\\rm{D}}^\\star = 200$ J kg$^{-1}$, appropriate for $D_{\\rm{c}}=0.15$ km)\nor 0.014 (for $Q_{\\rm{D}}^\\star = 2 \\times 10^5$ J kg$^{-1}$, appropriate for $D_{\\rm{c}}=2000$ km)\nto initiate a collisional cascade, values which are consistent with those quoted by\nmore detailed planet formation models (e.g., Kenyon \\& Bromley 2002).\nSuch eccentricities would be expected through stirring either by $>1000$ km\nplanetesimals which formed within the belt, or by more massive perturbers which formed \noutside the belt, both of which can be expected to occur within 10-100 Myr\n(Kenyon \\& Bromley 2006).\nThis was considered in \\S \\ref{ss:qe} where it was shown that the cascade would\nbe initiated following the growth of $\\sim 2000$ km planetesimals on timescales\nthat are much shorter than the age of the system for the disk masses required to\nproduce a dust luminosity at the observed level.\n\nThe only route which could plausibly maintain the hot dust systems in Table \\ref{tab:hot}\nin collisional equilibrium over the age of the stars might be to invoke some mechanism\nwhich maintains the eccentricity at a level which the cascade is only just being eroded.\nHowever, Fig.~\\ref{fig:gvse}a shows that $G(11\/6,X_{\\rm{c}})$ is a strong function\nof $e$ when $G(11\/6,X_{\\rm{c}}) < 1$, since the range $G(11\/6,X_{\\rm{c}})=0-1$ is\ncovered by a factor of less than two in eccentricity.\nThus we consider it reasonable to assume that the best possible combination of \n$Q_{\\rm{D}}^\\star$ and $e$ in this regard would result in \n$G(11\/6,X_{\\rm{c}}) \\approx 1$ (corresponding to $X_{\\rm{c}}=0.69$);\nlower values of $G(11\/6,X_{\\rm{c}})$ are possible, but only within a very narrow\nrange of eccentricity.\nSince in the above example with a realistic $Q_{\\rm{D}}^\\star$ prescription extending\nup to 2000 km we assumed $e=0.05$ which already resulted in $G(q_{\\rm{g}},X_{\\rm{c}}) < 1$, \nwe consider that it is not reasonable to fine tune the eccentricity further \nto increase $f_{\\rm{max}}$;\ne.g., decreasing to $e=0.03$ results in some disks not evolving and the rest\nwith $f_{\\rm{max}}$ higher than that quoted in Table \\ref{tab:hot} by a factor $\\sim\n150$.\nThus we conclude that the estimate given in Table \\ref{tab:hot} (and e.g., equation\n\\ref{eq:fmax1}) underestimates $f_{\\rm{max}}$ by at most a factor of $\\sim 100$, unless\nthe eccentricity happens to lie within $\\pm 10\\%$ of a critical value.\n\nIt is also worth noting that low levels of eccentricity would result in large \ngravitational focussing factors for large planetesimals which would enhance\n$f_{\\rm{cc}}$ and so decrease the time for these planetesimals to be catastrophically\ndestroyed, something which is compounded by the higher collision velocity\nin gravitationally focussed collisions which reduces $X_{\\rm{c}}$ because collisions\nwith smaller planetesimals can cause catastrophic disruption (e.g., equation\n\\ref{eq:xc2}).\nHowever, we do not need to account for this here, since gravitational focussing\nbecomes important when $v_{\\rm{rel}} < v_{\\rm{esc}} \\approx \\sqrt{(2\/3)\\pi \\rho G}(10^{-3}D)$\nand so when $e<4\\times 10^{-7}\\sqrt{\\rho r M_\\star^{-1}[1.25+(I\/e)^2]^{-1}}D$\n(where $D$ is in km);\ni.e., when $e<2 \\times 10^{-6}$ for $D=0.15$ km and $e<0.027$ for $D=2000$ km\nat 1 AU from a $1M_\\odot$ star, both of which occur close to or below the level\nat which collisions result in accumulation rather than destruction.\n\n\n\\subsection{Are these the products of single collisions?}\n\\label{ss:singlecoll}\nOne possible origin for the hot dust which is quoted in the literature is that\nit is the product of a single collision (Song et al. 2005).\nOur model can be used to make further predictions for the likelihood of massive collisions\noccurring within an asteroid belt.\nThe maximum number of parent bodies (i.e., planetesimals) larger than $D_{\\rm{pb}}$ \nremaining at late times occurs when $M_{\\rm{tot}}=M_{\\rm{max}}$ and so is given by:\n\\begin{eqnarray}\n n(D>D_{\\rm{pb}}) & = & \\left( \\frac{5.6 \\times 10^{10}r^{3.5} (dr\/r)}\n {M_\\star^{0.5} D_{\\rm{c}}^2 t_{\\rm{age}}} \\right)\n \\left[ \\left( \\frac{D_{\\rm{c}}}{D_{\\rm{pb}}} \\right)^{3q-3}-1 \\right]\n \\nonumber \\\\\n & & \\times \\left( \\frac{(3q-5)[1+1.25(e\/I)^2]^{-0.5}}{(3-3q)G(q,X_{\\rm{c}})} \\right).\n \\label{eq:ndgtdpb}\n\\end{eqnarray}\nThe collision timescale for planetesimals of size $D_{\\rm{pb}}$ is\n\\begin{eqnarray}\n t_{\\rm{c}}(D_{\\rm{pb}}) & = & t_{\\rm{c}}(D_{\\rm{c}}) \n f_{\\rm{cc}}(D_{\\rm{c}})\/f_{\\rm{cc}}(D_{\\rm{pb}}) \\nonumber \\\\ \n & = & 10^6 t_{\\rm{age}}(D_{\\rm{pb}}\/D_{\\rm{c}})^{3q-5}, \n\\label{eq:tcd}\n\\end{eqnarray}\nnoting that the collisional lifetime of the largest planetesimals, $t_{\\rm{c}}(D_{\\rm{c}})$, \nis the age of the star for a planetesimal belt at maximum luminosity for this\nage.\nThese can be combined to give the destructive collision rate for planetesimals larger\nthan $D_{\\rm{pb}}$:\n\\begin{eqnarray}\n dN_{\\rm{c}}(D>D_{\\rm{pb}})\/dt & = & 1000r^{13\/3}(dr\/r)t_{\\rm{age}}^{-2}D_{\\rm{c}}\n D_{\\rm{pb}}^{-3}\\times \\nonumber \\\\ \n & & M_\\star^{-4\/3}{Q_{\\rm{D}}^\\star}^{5\/6}e^{-5\/3},\n \\label{eq:dncdt}\n\\end{eqnarray}\nin Myr$^{-1}$, where the assumptions that $q=11\/6$, $e=I$ and $X_{\\rm{c}} \\ll 1$ have been\nused in deriving this equation.\n\nWe now assume that we are considering collisions capable of reproducing the\nobserved dust level, $f_{\\rm{obs}}$, so that the lifetime of the resulting collision\nproducts can be estimated from the collisional lifetime of that dust, assumed to be\nof size $D_{\\rm{bl}}$ (WD02):\n\\begin{equation}\n t_{\\rm{c}}(D_{\\rm{bl}}) = 0.04 r^{1.5}M_\\star^{-0.5}(dr\/r)f_{\\rm{obs}}^{-1}, \\label{eq:tcdust}\n\\end{equation}\nin years, noting that collisions would remove the dust on a faster timescale than P-R drag\n(Wyatt 2005; Beichman et al. 2005).\nCombining equations (\\ref{eq:dncdt}) and (\\ref{eq:tcdust}) gives the fraction\nof time that collisions are expected to result in dust above a given level of\n$f_{\\rm{obs}}$:\n\\begin{eqnarray}\n P(f>f_{\\rm{obs}}) & = & 4 \\times 10^{-5}r^{35\/6}(dr\/r)^2t_{\\rm{age}}^{-2}\n D_{\\rm{c}}D_{\\rm{pb}}^{-3} \\times \\nonumber \\\\\n & & M_\\star^{-11\/6}f_{\\rm{obs}}^{-1}\n {Q_{\\rm{D}}^\\star}^{5\/6}e^{-5\/3}.\n \\label{eq:pffobs}\n\\end{eqnarray}\n\nTo estimate the minimum size of the parent body, $D_{\\rm{pb}}$, responsible for this\ndust, we consider how large a planetesimal must be to reproduce $f_{\\rm{obs}}$ if a\ndestructive collision resulted in one fragment with half the mass of the original\nplanetesimal (i.e., the definition of a destructive collision), with the remaining\nmass in particles of size $D_{\\rm{bl}}$:\n\\begin{equation}\n D_{\\rm{pb}} = 890[D_{\\rm{bl}}r^2f_{\\rm{obs}}]^{1\/3}. \\label{eq:dpb}\n\\end{equation}\n\n\\begin{deluxetable*}{ccccccc}\n \\tabletypesize{\\scriptsize}\n \\tablecaption{Parameters in the model for the hot dust systems of Table \\ref{tab:hot}\n used to determine whether the observed dust can be the outcome of a single\n collision in a massive asteroid belt which is itself not normally bright\n enough to be detected.\n \\label{tab:hot2} }\n \\tablewidth{0pt}\n \\tablehead{ \\colhead{Star name} & \\colhead{$D_{\\rm{pb}}$, km} & \\colhead{$N(D>D_{\\rm{pb}})$} & \n \\colhead{$dN_{\\rm{c}}(D>D_{\\rm{pb}})\/dt$, Myr$^{-1}$} &\n \\colhead{$t_{\\rm{c}}(D_{\\rm{bl}})$, yr} & \n \\colhead{$P(f>f_{\\rm{obs}})$} & \\colhead{Single collision?} }\n \\startdata\n HD98800 & 530 & 200 & 41 & 0.36 & $15 \\times 10^{-6}$ & No \\\\\n HD113766 & 280 & 890 & 150 & 41 & $6100 \\times 10^{-6}$ & Not imposs \\\\\n HD12039 & 110 & 77,000 & 12,000 & 2300 & 27\\tablenotemark{*} & Not imposs \\\\\n BD+20307 & 320 & 0.47\\tablenotemark{*} & 0.0039 & 0.49 & $0.0019 \\times 10^{-6}$ & No \\\\\n HD72905 & 15 & 1.4 & 0.036 & 22 & $0.79 \\times 10^{-6}$ & No \\\\\n $\\eta$ Corvi & 110& 7.8 & 0.033 & 59 & $2.0 \\times 10^{-6}$ & No \\\\\n HD69830 & 39 & 19 & 0.068 & 110 & $7.7 \\times 10^{-6}$ & No\n \\enddata\n \\tablenotetext{*}{For disks with $P(f>f_{\\rm{obs}})>1$, this value indicates the number of\n collisions at that level we can expect to see in the disk at any one time.\n Likewise, for disks with $N(D>D_{\\rm{pb}})<1$, this value indicates the probability\n that there is an object of this size remaining in the disk.}\n\\end{deluxetable*}\n\nTable \\ref{tab:hot2} lists the parameters for the hot dust systems assuming the\ncanonical parameters of $Q_{\\rm{D}}^\\star=200$ J kg$^{-1}$, $D_{\\rm{c}}=2000$ km and \n$e=0.05$.\nTo determine whether a system could have been reproduced by a single collision, the\nfinal value of $P(f>f_{\\rm{obs}})$ was compared with the statistic that 2\\% of systems\nexhibit hot dust (which therefore considers the optimistic case where all stars\nhave planetesimal belts at a few AU).\nFor the systems which were inferred in Table \\ref{tab:hot} to be transient,\nall are extremely unlikely ($<0.001$\\%) to have been caused by a single collision amongst \nplanetesimals in a planetesimal belt which has undergone a collisional cascade since the star\nwas born.\n\nWhile this statistic is subject to the uncertainties in the model parameters described\nin \\S \\ref{ss:pc}, and so could be in error by around two orders of magnitude,\nit must also be remembered that the most optimistic assumptions were used to arrive at\nthis figure.\nFor example, it is unlikely that the destruction of planetesimals of size $D_{\\rm{pb}}$ would\nrelease half of the mass of the planetesimal into dust $D_{\\rm{bl}}$ in size.\n\\footnote{Such an optimistic assumption should not be dismissed out of hand, however, since \nthe large amount of collisional processing that must have taken place means that\nplanetesimals more than a few km would be rubble piles.\nThese would have undergone \nshattering and reaccumulation numerous times meaning that they could have deep dusty\nregolith layers which could be preferentially ejected in a collision.}\nOn the other hand, one might consider that the lifetime of the observed dust, \n$t_{\\rm{c}}(D_{\\rm{bl}})$, is an\nunderestimate of the duration of dust at the level of $f>f_{\\rm{obs}}$, since the dust\ncould be replenished from the destruction of larger particles.\nIndeed Farley et al. (2006) model the destruction of a 150 km planetesimal in the\nasteroid belt and inferred a dust peak that lasted $\\sim 1$ Myr, precisely because\nlarge fragments produced in the collision replenished the dust population.\nHowever, it should be cautioned that the dust peak inferred by Farley et al. (2006)\nwould not have been detectable as an infrared excess since it only caused a factor\n$\\sim 10$ enhancement in the luminosity of the zodiacal cloud (i.e., to\n$f \\approx 0.8 \\times 10^{-6}$), and that in the context of our model, invoking a\npopulation of larger grains that result from the collision would lead to a \nlarger parent body (i.e., a larger $D_{\\rm{pb}}$) required to reproduce the observed luminosity \n$f_{\\rm{obs}}$ and so less frequent collisions;\ni.e., it may be possible (even desirable) to increase $t_{\\rm{c}}(D_{\\rm{bl}})$, but only at the\nexpense of decreasing $dN_{\\rm{c}}(D>D_{\\rm{pb}})\/dt$ leading to little change in \n$P(f>f_{\\rm{obs}})$.\nWe note that $t_{\\rm{c}}(D_{\\rm{bl}})$ given in Table (\\ref{tab:hot2}) is sufficiently short\nthat a measurement of the variability of the infrared excess on realistic (few year)\ntimescales could lead to constraints on the size of the grains feeding the\nobserved phenomenon, since if a population of larger grains existed then the\nluminosity would fade on much longer timescales.\n\nA further argument against the transient disks being caused by single collisions\nis the fact that the probability of seeing the outcome of a collision, $P(f>f_{\\rm{obs}})$,\nfalls off $\\propto t_{\\rm{age}}^{-2}$, which means that we would expect to see more\ntransient disks around younger stars than around older stars\n(because young stars have more massive disks with more large\nplanetesimals and so more frequent collisions).\nThere is some evidence from Table \\ref{tab:hot} that transience is \nmore common around young systems, since none of the transient systems is older than\n2 Gyr, whereas sun-like stars in the solar neighborhood would be expected to\nhave a mean age of $\\sim 5$ Gyr.\nHowever, while the statistics are poor, a $t_{\\rm{age}}^{-2}$ dependence does seem\nto be ruled out;\ne.g., we would have expected to have detected 10 times more transient disks\ncaused by single collisions in the age range 50-500 Myr\\footnote{It is not reasonable\nto extend the age range to younger systems, since, as noted in \\S \\ref{ss:qe} it is\nhard to discern whether or not dust detected in such systems is transient.}\nthan in the age range 0.5-5 Gyr, whereas 2 transient disks are known in the younger age bin, \nand 2 in the older age bin which is more consistent with a $t_{\\rm{age}}^{-1}$ dependence.\n\n\\begin{deluxetable*}{ccccc}\n \\tabletypesize{\\scriptsize}\n \\tablecaption{Parameters in the model for the transient hot dust systems of Table \\ref{tab:hot}\n used to determine whether the observed dust could originate in the destruction of a\n planetesimal belt coincident with the dust:\n $dM_{\\rm{loss}}\/dt$ is the observed mass loss rate,\n $M_{\\rm{max}}$ is the maximum mass of a planetesimal belt that is coincident with the dust given\n the age of the star,\n $t(f>f_{\\rm{obs}})$ is the length of time such a planetesimal belt could sustain the observed\n dust luminosity,\n and $r_{\\rm{out}}(\\rm{100Myr})$ is the radius of a planetesimal belt which \n would still have enough mass\n to sustain the observed dust luminosity for 100 Myr. \\label{tab:hot3} }\n \\tablewidth{0pt}\n \\tablehead{ \\colhead{Star name} &\n \\colhead{$dM_{\\rm{loss}}\/dt$, $10^{-6}M_\\oplus$Myr$^{-1}$} &\n \\colhead{$M_{\\rm{max}}$, $10^{-6}M_\\oplus$} &\n \\colhead{$t(f>f_{\\rm{obs}})$, Myr} & \n \\colhead{$r_{\\rm{out}}(\\rm{100Myr})$, AU} }\n \\startdata\n BD+20307 & $8.0 \\times 10^6$ & 53 & $6.7 \\times 10^{-6}$ & 45 \\\\\n HD72905 & 19 & 0.072 & $3.7 \\times 10^{-3}$ & 2.4 \\\\\n $\\eta$ Corvi & 2500 & 57 & 0.023 & 9.6 \\\\\n HD69830 & 64 & 12 & 0.18 & 4.5\n \\enddata\n\\end{deluxetable*}\n\n\n\nIn fact, within the context of this model, all of the disks which we infer to be\ntransient would also be inferred to not be the product of single\ncollisions.\nThis is evident by substituting $D_{\\rm{pb}}$ from equation (\\ref{eq:dpb}) and \n$f_{\\rm{max}}$ from equation (\\ref{eq:fmax2}) into equation (\\ref{eq:pffobs}) to get:\n\\begin{equation}\n P(f>f_{\\rm{obs}}) = 0.2 \\times 10^6 (f_{\\rm{max}}\/f_{\\rm{obs}})^2\n (M_\\star e^2 r^{-1} {Q_{\\rm{D}}^\\star}^{-1})^{5\/6},\n \\label{eq:pff2}\n\\end{equation}\nwhich reduces to $P(f>f_{\\rm{obs}})=16(f_{\\rm{max}}\/f_{\\rm{obs}})^2(M_\\star r^{-1})^{5\/6}$\nfor the canonical parameters used before.\nSince transient disks are defined by $f_{\\rm{obs}}\/f_{\\rm{max}} \\gg 1000$, this means\nthey cannot also have a high probability of having their origin in single collisions.\nIt would only be inferred that disks with $f_{\\rm{obs}}\/f_{\\rm{max}} \\ll 1000$ could\nhave their origin in single collisions, but since it is also possible that these disks\nare the result of steady state collisional evolution there is no need to invoke\na single collision to explain their presence, which is why Table \\ref{tab:hot2}\nsimply concluded that it is \"not impossible\" that the disks of\nHD113766 and HD12039 are the product of single collisions.\nWhat equation (\\ref{eq:pff2}) does indicate, however, is that it is possible for single \ncollisions to cause disks to spend some fraction of their time at a\nluminosity enhanced above the nominal maximum value $f_{\\rm{max}}$, and\nthat this occurs more readily for disks at smaller radii and around\nhigher mass stars.\nHowever, whether single collisions really do achieve an observable increase in\nluminosity depends on the size distribution of the collisional fragments, for\nwhich it must be remembered that equation (\\ref{eq:pff2}) used an unrealistically\noptimistic estimate. \n\n\n\\subsection{Are parent planetesimals coincident with dust?}\n\\label{ss:coincident}\n\nFor similar reasons to those in \\S \\ref{ss:singlecoll} it is also possible to\nshow that the parent planetesimals of the dust are extremely unlikely to\noriginate in a planetesimal belt that is coincident with the dust.\nThe reason is that the mass remaining in such a belt would be insufficient to\nreplenish the dust for a length of time commensurate with the statistic that\n2\\% of stars show this phenomenon.\nThe observed dust luminosity, assuming this is comprised of\ndust of size $D_{\\rm{bl}}$ which has a lifetime of $t_{\\rm{c}}(D_{\\rm{bl}})$\n(equation \\ref{eq:tcdust}), implies a mass loss rate due to mutual collisions\nbetween the dust grains of:\n\\begin{equation}\n dM_{\\rm{loss}}\/dt = 1700 f_{\\rm{obs}}^2 r^{0.5} L_\\star M_\\star^{-0.5} (r\/dr),\n \\label{eq:mloss}\n\\end{equation}\nin $M_\\oplus$\/Myr, and this is independent of the collisional evolution\nmodel of \\S \\ref{s:model}.\nHowever, due to the collisional evolution of a planetesimal belt's largest\nmembers, there is a maximum mass that can remain in a belt at the same radius\nas the dust at this age, and this is given in equation (\\ref{eq:mmax1}).\nThis means that if the observed dust originates in an event which, for\nwhatever reason, is causing planetesimals in a belt at the same radius as\nthe dust to be converted into dust, then this can last a maximum time of\n$t(f>f_{\\rm{obs}})=M_{\\rm{max}}\/dM_{\\rm{loss}}\/dt$ before the planetesimal\nbelt is completely exhausted.\nThese figures are given in Table \\ref{tab:hot3} which shows that the longest\nthe type of transient event observed could be sustained in these systems is\nunder 1 Myr, under the assumptions about the planetesimal belts employed in the\nrest of the paper.\n\nA maximum duration of 1 Myr is not sufficient to explain the statistic that\n2\\% of sun-like stars exhibit this phenomenon, since the median age of such\nstars is 5 Gyr, indicating a typical duration (even if this occurs\nin multiple, shorter, events) of around 100 Myr.\nClearly a reservoir of mass is required in excess of that which it\nis possible to retain so close to the star.\n\n\\subsection{Constraints on parent planetesimal belt}\n\\label{ss:constraints}\nIf we assume that the observed mass of hot dust originates in planetesimals that\nwere initially in a belt at a radius $r_{\\rm{out}}$ which has properties like those\nassumed in the rest of the paper,\nand a fractional luminosity of $f_{\\rm{out}}$,\nthen there are two main constraints on that belt.\nFirst, assuming that this belt has been collisionally evolving for the age of\nthe star, then this belt cannot have more mass (or luminosity) than the maximum\nthat could possibly remain due to collisional processing, i.e., $f_{\\rm{out}} < f_{\\rm{max}}$\n(equation \\ref{eq:fmax1}).\nSecond, it must have sufficient mass remaining to feed the observed mass loss rate\nfor long enough to reproduce the statistic that 2\\% of stars exhibit this phenomenon\nwhich implies a total duration of $>100$ Myr.\nFor a belt to have enough mass to feed the observed hot dust\nluminosity of $f_{\\rm{obs}}$ at a radius $r$ for a total time of $t_{\\rm{hot}}$ in Myr\nrequires the belt to have a luminosity of:\n\\begin{equation}\n f_{\\rm{out}} > 710 t_{\\rm{hot}} f_{\\rm{obs}}^2 r_{\\rm{out}}^{-2} r^{0.5}\n D_{\\rm{c}}^{-0.5} L_\\star^{0.5} (dr\/r)^{-1},\n \\label{eq:fout}\n\\end{equation}\nor rather, this is the luminosity it must have had before it was depleted.\n\n\\begin{figure*}\n \\centering\n \\begin{tabular}{cc}\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f4a.ps} &\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f4b.ps} \\\\[-0.0in]\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f4c.ps} &\n \\hspace{-0.15in} \\includegraphics[width=3.2in]{f4d.ps} \\\\[-0.0in]\n \\end{tabular}\n \\caption{Constraints on the fractional luminosity and radius of the\n planetesimal belt feeding the observed transient hot dust (shaded region)\n for the systems: \\textbf{(top left)} $\\eta$ Corvi, \\textbf{(top right)} HD72905,\n \\textbf{(bottom left)} HD69830 and \\textbf{(bottom right)} BD+20307.\n The solid lines are the constraints imposed by the far-IR detection limits (assuming\n black body emission), the maximum luminosity possible in a belt at this radius due to\n erosion by collisional processing, and the luminosity from a belt of sufficient mass\n to feed the observed mass loss rate for 100 Myr.\n The properties of the hot dust in these systems is shown with a diamond, and those\n of the cold dust, where known (the top two figures), shown with a triangle.}\n \\label{fig:transsum}\n\\end{figure*}\n\nComparing this with the maximum mass possible at this age indicates that the parent\nbelt must have a minimum radius of:\n\\begin{eqnarray}\n r_{\\rm{out}}(t_{\\rm{hot}}) & > & 615t_{\\rm{hot}}^{3\/13}t_{\\rm{age}}^{3\/13}f_{\\rm{obs}}^{6\/13}\n r^{3\/26} (dr\/r)^{-6\/13} \\times \\nonumber \\\\\n & & D_{\\rm{c}}^{-3\/13}{Q_{\\rm{D}}^\\star}^{-5\/26}e^{5\/13}L_\\star^{3\/13}M_\\star^{5\/26}.\n \\label{eq:rout}\n\\end{eqnarray}\nTable \\ref{tab:hot3} gives an estimate of the minimum radial location of such a\nplanetesimal belt, under the assumption that the event (or multiple events) of\nhigh hot dust luminosity last $t_{\\rm{hot}} = 100$ Myr.\nThese values indicate that the planetesimal belts must be at least a few AU from the star.\nIt must be cautioned that this conclusion is relatively weak in the case of HD69830,\nsince the uncertainty in the properties of the planetesimals still leaves two orders\nof magnitude uncertainty in the maximum luminosity, $f_{\\rm{max}}$,\nand so also in the maximum mass $M_{\\rm{max}}$ (see \\S \\ref{ss:pc}).\nThis means that, with suitable planetesimal belt properties, a belt in this\nsystem that is coincident with the dust at 1 AU may be able to replenish the\nobserved phenomenon for 20 Myr.\nHowever, we still consider this to be an unlikely scenario, since it would require\nthat the mass of the planetesimal belt is depleted at a constant rate for the\nfull 100 Myr, whereas most conceivable scenarios would result in a mass loss rate\nwhich decreases with time as the planetesimal population is depleted thus requiring\nan even larger starting mass.\n\nThese two constraints are summarized for the 4 systems with transient hot dust in\nFig.~\\ref{fig:transsum}, which shows the shaded region of parameter space in\n$f_{\\rm{out}}$ and $r_{\\rm{out}}$ where the parent planetesimal belt can lie.\nThis figure also shows the location of the hot dust at $f_{\\rm{obs}}$ and $r$,\nillustrating the conclusion of \\S \\ref{ss:qe} that this lies significantly\nabove $f_{\\rm{max}}$, the maximum fractional luminosity expected for a planetesimal\nbelt at the age of the parent star.\nNote that the value $r_{\\rm{out}}(100\\rm{Myr})$ given in Table \\ref{tab:hot3}\ndenotes the intersection of the limits from $f_{\\rm{max}}$ and from equation\n(\\ref{eq:fout}).\n\nA third constraint for the parent planetesimal belt comes from far-IR observations\nof these systems.\nFor 2\/4 of the transient dust systems a colder dust component has already been detected:\n$\\eta$ Corvi has a planetesimal belt with a resolved radius of $\\sim 100$ AU (Wyatt et al. 2005),\nand HD72905 has one inferred to be at $\\sim 14$ AU (Beichman et al. 2006a).\nIn both cases these outer planetesimal belts have been inferred to be at a\ndifferent spatial location from the hot dust either because of imaging constraints\n(Wyatt et al. 2005) or from analysis of the SED (Beichman et al. 2006a).\nThe properties inferred for these planetesimal belts are indicated on Fig.~\\ref{fig:transsum}\nand lie within the shaded region, implying that these planetesimal belts\ndo not have to be transiently regenerated, and also provide a plausible source\npopulation for the hot dust found closer in.\nHowever, no such excess emission has been seen toward HD69830 at either 70 $\\mu$m\n(Beichman et al. 2005) or 850 $\\mu$m (Sheret, Dent \\& Wyatt 2004) indicating a \nplanetesimal belt with a mass at most 5-50 times greater than our own Kuiper belt.\nLikewise, BD+20307 does not have a detectable excess in IRAS 60 $\\mu$m\nobservations (Song et al. 2005).\n\nA low mass reservoir of planetesimals does not necessarily rule out the presence\nof an outer planetesimal belt which is feeding the hot dust for two reasons.\nFirst, the shaded region of Fig.~\\ref{fig:transsum} actually constrains the properties of\nthe planetesimal belt at the time at which depletion started;\ni.e., this population may have already been severely depleted by the same event\nwhich is producing the dust and we are now nearing the end of the hot dust episode.\nSecond, the constraints imposed by a non-detection in the far-IR do not eliminate\nthe whole of the parameter space in which an outer planetesimal belt can lie.\nFig.~\\ref{fig:transsum} includes the constraints on the outer planetesimal belt\nimposed by the non-detection of excess in the far-IR, assuming that the\ndust emits like a black body.\nThe resulting detection limit is then given by:\n\\begin{equation}\n f_{\\rm{det}} = 3.4 \\times 10^9 F_{\\rm{det}}(\\lambda)\n d^2 r_{\\rm{out}}^{-2}\/B_\\nu(\\lambda,T_{\\rm{bb}}), \n \\label{eq:fdet}\n\\end{equation}\nwhere $F_{\\rm{det}}$ is the detection limit in Jy, $d$ is the distance to the star in pc,\nand $B_\\nu(\\lambda,T_{\\rm{bb}})$ is in Jy\/sr.\nFor BD+20307 the non-detection is limited by the sensitivity of\nIRAS, and so lower limits should be achievable with Spitzer.\nFor the two systems with non-detections, the shaded region already takes\nthe far-IR constraint into account.\n\nThe simplification that the emission comes from black body-type grains\nmeans that equation (\\ref{eq:fdet}) underestimates the upper limit\nfrom the far-IR fluxes.\nThis is because the majority of the luminosity comes from \nsmall grains which emit inefficiently at long wavelengths.\nIndeed, the black body assumption would require the hot dust\nof HD69830 and BD+20307 to have been detected in the far-IR,\nwhereas this is not the case.\nWe modeled the emission from non-porous silicate-organic refractory\ngrains in a collisional cascade size distribution at 1 AU from these\nstars to find that the black body assumption used in equation (\\ref{eq:fdet})\nunderestimates the limit by a factor of $3-5$ meaning that non-detection\nof the hot dust in these systems in the far-IR is to be expected.\nThis also means that slightly more luminous outer planetesimal belts\nthan those indicated by the shaded region on Fig.~\\ref{fig:transsum}\nmay still have escaped detection in the far-IR.\n\nUntil now we have not proposed a mechanism which converts the planetesimals\ninto dust.\nWhereas Beichman et al. (2005) invoke sublimation of comets as the origin of\nthe hot dust, and use this to estimate the mass of the parent planetesimal\nbelt, we consider a scenario in which a significant fraction of material\nof all sizes in the parent planetesimal belt is placed on orbits either\nentirely coincident with the hot dust, or with pericenters at that location.\nIn this scenario the dust is reproduced in collisions and the material\nmaintains a collisional cascade size distribution.\nSimply moving material from $r_{\\rm{out}}$ to $r$ would result in an\nincrease in fractional luminosity from $f_{\\rm{out}}$ to\n$f_{\\rm{out}}(r_{\\rm{out}}\/r)^2$.\nThis indicates that the parent planetesimal belt responsible for\nthe hot dust could have originally been on the line on Fig.~\\ref{fig:transsum}\ntraced by $f_{\\rm{out}} = f_{\\rm{obs}} (r\/r_{\\rm{out}})^2$.\nSince this is parallel to the mass loss limit line (equation \\ref{eq:fout}),\nand for all but HD69830 the observed hot dust component lies below this line,\nthis indicates that parent planetesimal belts in the shaded region\ncould be responsible for the hot dust observed, as long as a large fraction\nof their mass is scattered in to the inner regions.\nHowever, it is to be expected that only a fraction of the outer planetesimal\nbelt ends up in the hot dust region, and so it is more likely that the\nparent planetesimal belt started on a line which falls off less steeply\nthan $\\propto r_{\\rm{out}}^{-2}$, and this is consistent with the ratio of the\nhot and cold components of $\\eta$ Corvi and HD72905 which indicate a\ndependence of $f_{\\rm{out}} = f_{\\rm{obs}} (r\/r_{\\rm{out}})^{0.5 \\pm 0.2}$;\nit is also interesting to note that both have $r_{\\rm{out}}\/r =60-70$.\nWe defer further consideration of the expected properties of the\nparent planetesimal belt to a more detailed model of the dynamics\nof the types of events which could cause such a perturbation,\nbut simply note here that the existence of an outer planetesimal belt\nis not ruled out by the current observational constraints in any\nof the systems.\n\n\n\\section{Discussion}\n\\label{s:conc}\nA simple model for the steady state evolution of dust luminosity for planetesimal\nbelts evolving due to collisions was described in \\S \\ref{s:model}.\nThis showed how at late times the remaining planetesimal belt mass and so\ndust luminosity is independent of the initial mass of the belt.\nThis has important implications for the interpretation of the properties of detected disks.\nThis paper discussed the implications for the population of sun-like stars with hot dust\nat $<10$ AU;\nthe implications for the statistics will be discussed in a forthcoming paper (Wyatt\net al., in prep.).\n\nIt was shown in \\S \\ref{ss:qe} that for 4\/7 of the systems with hot dust their radius and age \nare incompatible with a planetesimal belt which has been evolving in quasi-steady state over the \nfull age of the star, and in \\S \\ref{ss:pc} it was shown that this is the case\neven when uncertainties in the model are taken into account.\nThis implies either that the cascade was started recently (within the last Myr or so), or that\nthe dust arises from some other transient event.\nRecent ignition of the collisional cascade seems unlikely, since the mass required\nto feed the observed luminosity would result in the growth of 2000 km planetesimals\nwhich would stir the belt and ignite the cascade on timescales much shorter than\nthe age of the stars.\nPossible origins for the transient event that have been proposed in the literature are:\nrecent collision between massive planetesimals in a planetesimal belt which introduces\ndust with a size distribution $q \\gg 11\/6$ and so can be detected above a collisional\ncascade which is too faint to detect;\none supercomet $\\sim 2000$ km in diameter that was captured into a circular orbit\nin the inner system replenishing the dust through sublimation (Beichman et al. 2005);\na swarm of comets scattered in from the outer reaches of the system (Beichman et al. 2005).\nIn \\S \\ref{ss:singlecoll} the collisional model was used to show that the transient\ndisks are very unlikely ($<0.001$\\% for the most optimistic estimate for any of the\nstars compared with a detection probability of 2\\% for transient hot dust) to\nhave their origin in a recent collision;\nsuch collisions occur too infrequently.\nIn \\S \\ref{ss:coincident} it was also shown that the parent planetesimals of the\nobserved dust must originate in a planetesimal belt much further from the star\nthan the observed dust, typically at $\\gg 2$ AU.\nThis is because collisional processing means that the mass that can remain so\nclose to the star at late times is insufficient to feed the observed phenomenon.\n\nThe most likely scenario is thus a recent event which provoked one or more planetesimals to\nbe scattered in from further out in the disk (Beichman et al. 2005).\nThe observed dust could have been produced from such a scattered planetesimal population\nthrough their grinding down in mutual collisions (\\S \\ref{ss:constraints}), although \nsublimation close to the pericenters of the planetesimals' orbits is a further possible source \nof dust.\nMore detailed study of the scattering and consequent dust production processes is\nrequired to assess these possibilities.\nHowever, this scenario is supported by the presence of far-IR emission originating from\na colder outer planetesimal belt component in 2\/4 of the transient dust systems.\nThe constraints on the outer planetesimal belt which is feeding the\nphenomenon are discussed in \\S \\ref{ss:constraints}, showing that\nthe outer planetesimal belts already found in $\\eta$ Corvi and HD72905 provide a\nplausible source population for the hot dust found closer in, and that the current\nnon-detection of cold dust around the remaining two systems does not rule out the\npresence of an outer planetesimal belt capable of feeding the observed hot dust luminosity.\n\nOne clue to the origin of the parent planetesimals of the dust may be the composition\nof that dust.\nSilicate features have been detected in the mid-IR spectrum of all of the transient\nhot dust stars (Song et al. 2005; Beichman et al. 2005; Beichman et al. 2006a;\nChen et al. 2006).\nDetailed modeling of the spectrum of HD69830 indicates that the mineralogical\ncomposition of its dust is substantially different from that of comets, rather\nthere is a close match to the composition of P- or D-type asteroids found mainly in\nthe 3-5 AU region of the solar system (Lisse et al. 2006).\nWhile the radial location at which planetesimals of this composition form in the\nHD69830 system will depend on the properties of its protostellar nebula, which may be \nsignificantly different to that of the protosolar nebula, as well as on the structure\nand evolution of its planetary system, evidence for water ice in the dust spectrum\nindicates that the parent body formed beyond the ice-line in this system\n(Lisse et al. 2006), i.e., beyond $2-5.5$ AU (Lecar et al. 2006; Alibert et al. 2006).\nThus the compositional data supports the conclusion that the dust is not produced by a \nplanetesimal that formed in situ.\nHowever, it is worth noting that the same compositional data also\nfinds evidence for differentiation in the parent body (inferred from\nabundance differences between the dust and the star) and for heating of\nits rocky material to $>900$ K (inferred from the absence of amorphous\npyroxene), which would also have to be explained in the context of an outer\nplanetesimal belt origin for the dust.\n\nAn analogous transient event is thought to have happened in the solar system resulting in\nthe period known as the Late Heavy Bombardment (LHB) when the terrestrial planets\nwere subjected to an abnormally high impact rate from asteroids and comets.\nThis is believed to have been triggered by a dynamical instability in the\nplanetary system resulting from Jupiter and Saturn crossing the 1:2 resonance\nduring their slow migration (inwards for Jupiter, outwards for Saturn)\ndue to angular momentum exchange with the primordial Kuiper belt\n(Gomes et al. 2005).\nIn this scenario both the asteroid and Kuiper belts were depleted with\na large fraction of these objects being scattered into the terrestrial planet\nregion during an event which lasted $10-150$ Myr (Gomes et al. 2005), i.e.,\nexactly the type of event required to explain the observed hot dust in the scenario\nproposed here (\\S \\ref{ss:constraints}).\nDynamical instabilities in extrasolar planetary systems can also arise from\nmutual gravitational perturbations between giant planets which formed close\ntogether (Lin \\& Ida 1997; Thommes, Duncan \\& Levison 1999).\nIn both scenarios slow diffusion of the orbits of the planets\nmeans that the dynamical instability can occur up to several Gyr after\nthe formation of the planetary system.\nThe delay to the onset of the instability is determined by the separation\nof the outer planet from the outer planetesimal belt (Gomes et al. 2005), or\nfrom the separation between the planets (Lin \\& Ida 1997), with larger separations\nresulting in longer timescales.\n\nLittle is known about the planetary systems of four of the hot dust systems.\nHowever, three Neptune mass (or Jupiter mass if the system is seen face-on)\nplanets have recently been discovered orbiting the star HD69830 at $<1$ AU\non nearly circular orbits (Lovis et al. 2006).\nDynamical simulations showed that the detected planetary system is stable on\ntimescales of 1 Gyr.\nThis does not, however, rule out the possibility of a dynamical instability having occurred.\nWhile no mean motion or secular resonances are immediately identifiable within\nthe detected planetary system which could have have been crossed recently invoking\nsuch a catastrophic event, it is possible that the instability arose with another\nplanet further out which has yet to be detected with longer timescale observations.\nIt is also possible that a fourth planet which existed in the region 0.19-0.63 AU\nbetween the planets HD69830c and HD69830d has recently been scattered out due to a\ndynamical instability (e.g., Thommes et al. 1999).\nThe region 0.3-0.5 AU was identified in Lovis et al. (2006) as being marginally stable,\nand to encompass several mean motion resonances with the outer planet, including\nthe 1:2 resonance at 0.4 AU;\ni.e., a putative fourth planet could have remained in this region for the past 2 Gyr\nuntil the slow migration\/diffusion of the outer planet (HD69830d) caused the 1:2\nresonance to coincide with the orbit of the putative planet which was then scattered\noutward thus promoting the depletion of an outer planetesimal belt much of which\nwas scattered into the inner regions of the system.\nAlibert et al. (2006) considered that the most plausible formation scenario for the\nplanetary system of HD69830 included the inward migration of the outer planets\nfrom beyond the ice-line at a few AU.\nThis would put a substantial distance between the outer planet (HD69830d) and any\nouter planetesimal belt which favors a delay of 2 Gyr before the onset of the\ninstability.\nSearches for further planetary companions in this system, and for the \nrelic of its outer planetesimal belt, are clearly necessary to constrain the\nevolutionary history of this system.\n\nIn conclusion, $\\sim 2$\\% of sun-like stars exhibit transient hot\ndust in the terrestrial planet region;\nthis dust must originate in a planetesimal belt located further from\nthe star than the dust, typically at $\\gg 2$ AU.\nJust four members of this class are currently known, although it seems\nreasonable to assume that our own solar system would have been placed\nin this class during the LHB.\nThe frequency of this phenomenon indicates that either all stars are\nsubjected to an epoch of similar duration (lasting $\\sim 100$ Myr assuming\na typical age of 5 Gyr) or that a smaller fraction of stars undergo much\nlonger (or multiple) events.\nThe distribution of the ages of the stars in this class indicate that the likelihood of\nthese events occurring falls off roughly inversely proportional to the\nage of the stars.\nAn origin for these events in a dynamical instability as proposed for the LHB\nin the solar system is supported by the recent discovery of a multiple\nplanet system coincident with the dust in one of the systems currently in\nthis class.\nHowever, since the LHB in the solar system is thought to have lasted just\n$\\sim 100$ Myr, it remains to be seen whether we are to infer that dynamically\nunstable planetary systems form around all stars, or that the LHB event in\nother systems lasted much longer than in our own, or perhaps that there is in\nfact more than one mechanism causing this hot dust signature.\nObservations that further constrain the planet, planetesimal and dust\ncomplements of the transient hot dust systems are needed to ascertain the \nsimilarities and dissimilarities within this population.\n\n\n\\acknowledgements\nWe are grateful for support provided by the Royal Society (MCW) and PPARC (RS).\nWe are also grateful to Ben Zuckerman, Joseph Rhee and Inseok Song for pointing\nout that there is a strong (unrelated) infrared source in the vicinity of\nHD128400 which is causing the excess identified by Gaidos (1999).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSolid-state scintillators are widely used as particle detectors at room temperature, in fields as diverse as national security, medicine, and particle physics. \nThe light they emit when a particle interacts in them is a proxy for the energy deposited by the particle. \nCompared to other types of particle detectors, scintillators offer a wide range of target nuclei, and for some materials, very fast response. \nIn addition, in certain cases, the quantity of emitted light (light yield, or LY) or its timing, can also provide information on the nature of the interacting particle. \nCsI, pure or doped, is a frequently used scintillator, for instance in neutrino physics~\\cite{Akimov:2017}, or in searches for hypothetical dark matter particles~\\cite{Kim:2012rza} that could account for most of the matter in our Universe~\\cite{schnee_introduction_2011}.\nThere has been recent interest in the use of CsI as a cryogenic scintillating calorimeter for the detection of direct dark matter interactions with regular matter~\\cite{Derenzo:2016fse,mikhailik_2014,cerdeno_scintillating_2014,Nadeau:2014kta,Zhang:2016}. \nCsI is an appealing target for such a detector because of its high LY at low temperature and the possibility of probing new WIMP interaction parameter space~\\cite{cerdeno_scintillating_2014}. \nAs an alkali halide material, it could also help to shed some light on the long-standing DAMA\/LIBRA detection claim~\\cite{Bernabei:2008ec,Bernabei:2013ax}, by performing a dark matter experiment with a similar material.\n\nIn comparison to the simple scintillation detectors used by DAMA, which are\nincapable of event-by-event background discrimination, scintillating calorimeters can provide more insight into the nature of the interacting particle. \nA scintillating calorimeter is a particle detector consisting of a scintillating crystal held at cryogenic temperatures ($\\lesssim$~50~mK) as the target medium, read-out by a light detector and thermal sensors, such that, for a given particle interaction, both scintillation and phonon signals can be observed~\\cite{schnee_introduction_2011}. \nThe phonon signal is a good proxy for the energy deposited by the particle.\nFor a given energy deposit, ionizing radiation in the form of alphas ($\\alpha$), gammas ($\\gamma$), and betas ($\\beta$) will produce different amounts of scintillation light compared to the nuclear recoils expected to be caused by dark matter particles and neutrons, and possibly compared to one another.\nThis process, known as quenching, allows for very powerful discrimination against background events. The CRESST experiment, for example, uses an array of scintillating calorimeters to detect nuclear recoils from dark matter particles~\\cite{Angloher:2012tx}.\nThe possible sensitivity of such an alkali halide detector has been explored in simulations~\\cite{nadeau_cryogenic_2015} and by the COSINUS experiment~\\cite{Angloher:2016CsI}, which is planning on using cryogenic undoped NaI to evaluate the DAMA\/LIBRA claim~\\cite{Angloher:2016}. \nThis approach is complementary to larger, room-temperature searches, \nincluding \nANAIS~\\cite{amare_preliminary_2014},\nCOSINE~\\cite{Adhikari:2017,Cherwinka:2014xta},\nPICO-LON~\\cite{Fushimi:2016},\nand SABRE~\\cite{Xu:2014}.\n\nThe scintillation of CsI (with and without doping) has been studied over a range of temperatures~\\cite{lamatsch1971kinetics,Kubota:1988,Schotanus:1990tk,Nishimura:1995,Amsler:2002kq,Moszynski:2005hu,mikhailik_2014}. \nThe LY tends to increase as the temperature of the scintillator decreases. \nThe ratio between $\\alpha$ and $\\gamma$ light emission is an interesting value to probe as it can allow for particle discrimination between $\\alpha$ and $\\gamma$ interactions. \nThe difference in the ionization density between these two particles can affect the portion of deposited energy that is converted to light in the scintillator.\n\nIn the following, we report on our time-resolved measurements of the light yield of CsI between 300--3.4~K, under both $\\alpha$ and $\\gamma$ excitations for the first time, and over an unexplored time window~\\cite{nadeau_cryogenic_2015}.\nWe use a novel zero-suppression data-acquisition technique enabling a large acquisition window of 1 millisecond to fully resolve the decays and reduce bias from losing light from the end of the window.\nOur simultaneous measurement of $\\alpha$ and $\\gamma$ light from undoped CsI provides a determination of the ratio of light emission between $\\alpha$ and $\\gamma$ excitations, hereafter referred to as the $\\alpha\/\\gamma$ ratio ($R_{\\alpha\/\\gamma}$), over this temperature range.\nOur measurements are motivated by the possible use of CsI in rare-event searches at cryogenic temperatures.\n\n\n\\section{Experimental setup and sample preparation}\nAn optical cryostat produced by ColdEdge Technologies\\footnote{Cold Edge Technologies, Allentown PA, www.coldedgetech.com} is used to cool the samples to any temperature down to 3.4~K, as has been described in previous publications~\\cite{verdier_2.8_2009,Verdier:2011pb,Verdier:2012ie,Veloce:2015slj}. \nThe crystal sample is mounted inside the cryostat and excited by $\\alpha$ particles and $\\gamma$ 60~keV quanta from an internal collimated $^{241}$Am source mounted to the sample holder. An external $^{57}$Co source can also be used to test $\\gamma$ energies of 122~keV.\nThe $^{241}$Am $\\alpha$ source used in this work was adapted from a common smoke detector where a protective film covers the radioactive material. Using a silicon detector, we have measured the degraded mean energy of the $\\alpha$ particles coming from the source to be 4.7~MeV~\\cite{vonSivers:2015}.\n\nIn order to facilitate our study of hygroscopic crystals we have built a glove box around the cryostat that was constantly flushed with compressed ``extra dry'' air ($<10$~ppm H$_2$O) and contained several bags of desiccant.\nA photograph of the cryostat and glove box setup is shown in Figure~\\ref{fig:setuppic}.\nThe nominally pure CsI crystal samples studied in this work were purchased from Hilger Crystals\\footnote{www.hilgercrystals.co.uk} and have square cuboid geometries with dimensions $5\\times5\\times2$~mm$^3$ and all sides polished. \nThe samples were adhered to a custom-made sample holder using silver paint instead of being mechanically held in place since the alkali halides are slightly brittle. \nWe determined the optimal amount of adhesive to avoid cracking the crystal through trial and error on other samples, as there is differential contraction between the sample and holder during thermal cycling. \nThe sample and holder were then moved into the glove box, installed onto the cold finger of the cryostat, and put under vacuum as quickly as possible. \nNo changes in the optical quality of the sample were observed by eye during this procedure.\n\nAs illustrated in Fig.~\\ref{fig:setup}, the crystal sample was at an angle of {30\\textdegree} with respect to the $^{241}$Am source so that the emitted $\\alpha$ particles are incident with one of the smooth, large faces of the sample. \nAt the same time, this face is well exposed to a 28~mm diameter PMT (Hamamatsu R7056) through a fused quartz window, with a second exposed to the rear face, partially obstructed by the sample holder. \nThese PMTs have sensitivity to wavelengths down to 185~nm (the emission spectrum of CsI \\cite{Kubota:1988} is known to have UV components) and a maximum quantum efficiency of 25\\% at 420~nm. \nThe PMT that is in view of the unobstructed face of the sample is used as the primary PMT for data acquisition, while the other is used to assist triggering. \nBased on the solid angle of the crystal exposed to the primary PMT and transmittance of the windows, we expect a light collection efficiency of roughly $10\\%$~\\cite{nadeau_cryogenic_2015}.\nNote that no optical filters were used, so all light within the sensitivity of the PMT contributed to the light yield, as is standard in particle detection.\n\n\\begin{figure}\n\t\\centering \n\t\t\\includegraphics[width=.5\\columnwidth]{fig1}\n\t\t\\caption{\\label{fig:setuppic}The optical cryostat and glove box.}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering \n\t\t\\includegraphics[width=\\columnwidth]{fig2}\n\t\t\\caption{\\label{fig:setup}A schematic of the setup for scintillation studies. A $5\\times5\\times2$~mm$^3$ sample is secured at the centre of the cryostat optical chamber on a holder, and is observed by two photomultiplier tubes (PMT). The volume of the optical cryostat is held at vacuum throughout the experiment whereas the glove box is filled with dessicated air.\nThe measurement of $\\alpha$ particles (dotted arrow) and $\\gamma$ quanta (incoming wavy arrows) is achieved using an internally mounted $^{241}$Am source. Optionally, other $\\gamma$ sources can be mounted to the outside of the cryostat. The coincidence of the two PMTs provides a trigger for a digitizer that records photon signals from the PMT that is in view of the completely uncovered side of the crystal (the other PMT is partially masked by the holder).}\n\\end{figure}\n\n\\section{Data acquisition}\n\\label{sec:data_acq}\nData acquisition is based on the multiple photon counting coincidence technique (MPCC), where the trigger for acquisition is produced when both PMTs detect photons within a given coincidence time window, allowing the measurement of both the light yield and the decay time constants of a scintillator \\cite{Kraus:2005vj}. \nWe have previously adapted this technique to the study of bismuth germanate (BGO) \\cite{Verdier:2011pb} and ZnWO$_4$ \\cite{Verdier:2012ie,di_stefano_counting_2013}. \nSignals from the PMTs pass through a series of Phillips Scientific Nuclear Instrumentation Modules (NIM) to produce a trigger pulse for the digitizer: first, a fan-out module to copy the signal; then, a discriminator to discard any low amplitude events below a given threshold, producing a flat logic pulse with length equal to the desired coincidence time window; and finally, a logic module that performs an AND operation on the incoming logic pulses on both channels and accepts any events where the pulses overlap, thus creating the coincidence condition. \nThe logic module provides the trigger pulse to an 8-bit National Instruments digitizer (PXI-5154), which then records the incoming pulses from the PMTs at a 1~GHz sampling rate.\nThe full setup is described in Fig.~\\ref{fig:setup}.\n\nThe coincidence window was set to 30~ns and an acquisition window of 1~ms (50~$\\mu$s pre-trigger and 950~$\\mu$s post-trigger) was used for all events at all temperatures. \nA long acquisition window was chosen so that we are sure that the full scintillation pulse could be observed at all temperatures.\nTo deal with the large amounts of data, a custom-made LabVIEW interface records only samples of the incoming signals that exceed a certain threshold to save hard-disk space and analysis time.\nThis introduces a light-yield and time-constant dependent bias on the estimation of the light yield which has been corrected. This correction is described in detail in Section~\\ref{sec:data_anal}.\nThis is particularly important to ensure an unbiased comparison between $\\alpha$ and $\\gamma$ interactions.\n\nThe scintillation pulses from CsI have been characterized to have at least one very prominent fast decay time ($\\sim$10-100~ns) \\cite{Kubota:1988,Schotanus:1990tk} and other slower decay times ($>$1~$\\mu$s)~\\cite{Schotanus:1990tk}. \nAt the start of a high light-yield scintillation event, many photons arrive at the PMTs faster than can be resolved individually; as a result, the PMTs output a high amplitude spike several $\\mu$s wide (``pile-up'' or analog regime). \nBy contrast, the slow component is slow enough such that single photons can be resolved easily by the PMTs (counting regime). \nDue to the vertical resolution of our digitizer, we therefore cannot use the technique we have developed based on a streaming time-to-digital converter to measure these pulses, since it depends on being able to resolve individual photons~\\cite{di_stefano_counting_2013}. \nIn order to measure the full pulse (fast $+$ slow components) with our digitizer, two copies of the signal from the primary PMT (facing the unobstructed side of the sample) are obtained from the Fan-Out module, as shown in Figure~\\ref{fig:setup}. \nThe signals are then sent to separate channels on the digitizer, where one channel is set to the maximum vertical range (5~V) so that the full amplitude of the initial fast spike can be measured without saturation, and the other channel is set to a low vertical range (0.5~V) in order to resolve the slow, individual photoelectrons that follow.\n\nData from both digitizer channels are recorded and then combined off-line to reconstruct the full pulse by replacing all of the saturated samples on the low range channel with their corresponding unsaturated samples on the high range channel after applying the scaling factor and offset. \nThis results in a single channel of data with complete, unsaturated, reconstructed pulses with a larger dynamic range than possible with a single channel. \nAn example of a single pulse reconstruction is shown in Figure~\\ref{fig:pulse}. \nThe reconstructed data are saved in an ASCII file as arrays of time and amplitude for each digitizer sample above the LabVIEW threshold in each triggered scintillation event.\n\n\\begin{figure}\n \\centering\n\t\\includegraphics[width=\\columnwidth]{fig3}\n\t\\caption{\\label{fig:pulse} Demonstration of pulse reconstruction. Two channels with different ranges are used to record the same pulse, and are later combined through software (red line). Saturation in the low-range channel (blue dot) is fixed using properly scaled values from the high-range channel (black cross), and pulses below precision in the high-range are still caught in the low-range channel. Pulse begins after a 50~$\\mu$s pre-trigger and continues until 1~ms, though only the first 5~$\\mu$s are shown here.}\n\\end{figure}\n\n\\section{Data analysis methods}\n\\label{sec:data_anal}\nThe standard MPCC technique determines the light yield and the decay time components based on the number of photons in a given event and the known arrival time of each photon~\\cite{Kraus:2005vj}. \nA set of cuts are applied to the data in order to reject spurious events such as events where multiple scintillations occur in the acquisition window, and events where photons from a previous event straggle into the pre-trigger. \nEvents with more than a single scintillation will have more photons in the later part of the time window, and thus can be identified by comparing the mean arrival time of the photons in a given scintillation event to the distribution of the mean arrival time of all the events. \nEvents with photons in the pre-trigger region can be rejected using the distribution of the first photon in each event.\n\nIn the standard MPCC technique, the spectrum from a given source is obtained from the histogram of the number of photons in each event; after cutting the spectrum on a particle type and energy, the histogram of all photons with respect to their arrival time gives an average pulse shape~\\cite{Kraus:2005vj}. \nWe have adapted this technique to account for the early burst of unresolvable photons in the alkali halide pulses by building a histogram of the total charge in each event, instead of the arrival time of each photon, to produce the spectra shown in Figure~\\ref{fig:spectrum}. \nA fit to the peak in the spectrum from a given particle at a given energy provides a measurement of the light yield (LY) of the scintillator as described in Section~\\ref{sec:data_anal_ly}.\nA secondary cut is applied to the data set on the total charge to separate the events that are a result of $\\alpha$ or $\\gamma$ interactions.\nWe use $\\pm$ 1-$\\sigma$ from the mean of each spectrum to determine where these cuts will be placed.\n\nWe can also reconstruct the average pulse for an $\\alpha$ or $\\gamma$ interaction by combining the charge vs. time of all pulses that are within the corresponding spectrum. \nThis allows us to determine the scintillation decay time constants separately for excitations from the different particles. \nThe combined pulse is then fit with a series of exponentials to extract the expected exponential decay time constants.\nThis process is described in Section~\\ref{sec:data_anal_ts}.\n\n\\begin{figure}\n \\centering\n\t\\includegraphics[width=\\columnwidth]{fig4}\n\t\\caption{\\label{fig:spectrum}Example of a spectrum of detected photons, here obtained at 77~K from the $\\alpha$s, $\\gamma$s and X-rays of $^{241}$Am and $^{57}$Co.}\n\\end{figure}\n\n\n\\subsection{Light Yield}\n\\label{sec:data_anal_ly}\nThanks to our cryostat and our new data treatment, we are simultaneously sensitive to both $\\alpha$ and $\\gamma$ interactions at all temperatures, even with their large energy difference as illustrated in Figure~\\ref{fig:spectrum}.\nIt allows us to measure charge over a large spread of particles (60~keV--4700~keV) and different temperatures.\nThe charge detected for a given particle varies by two orders of magnitude over the temperature range.\n\nWe note that at temperatures for which the LY enables sufficient resolution, an asymmetry in the $\\alpha$ line becomes evident as a secondary peak that appears 15\\% lower than the main peak.\nWe have verified with a Si detector that it does not come from the source, and we attribute the shape to small scratches on the polished surface, visible under microscope, changing the energy deposition of the $\\alpha$ particles, which interact primarily with the surface of the crystal (interaction depth 24~$\\mu$m~\\cite{ASTAR}). \nThe $\\gamma$ LY is unaffected by the surface condition, as 60~keV $\\gamma$s have a mean free path of 280~$\\mu$m~\\cite{XCOM}\nWe have therefore modeled the $\\alpha$ line by two gaussians with fixed ratios of amplitudes, means and widths based on a free fit at the temperature with the greatest light yield (60~K).\nFor the $\\alpha$-event selection cuts, we consider the higher peak to be representative of the $\\alpha$ LY.\n\nAs mentioned in Section~\\ref{sec:data_acq}, we use an on-line threshold to reduce the amount of data that we are required to read and save to disk by excluding all voltage data below the threshold. \nWe found, however, that doing so affected our measurement efficiency differently for different LYs.\nFigure~\\ref{fig:thresh}a-i shows the single photon response of the PMTs used, determined from a dedicated experiment without the use of the threshold.\nWhen the threshold is applied, we lose a small amount of charge on each side of the distribution.\nIf photons arrive sufficiently spaced out in time as in Figure~\\ref{fig:thresh}a-ii, the same portion of the response will be lost for each photon.\nHowever, if several photons arrive in quick succession the responses can pile-up on each other, allowing the charge to exceed the threshold and be measured (as shown in Figure~\\ref{fig:thresh}a-iii), which would increase the efficiency of that measurement.\nThe likelihood that photons will pile-up on each other depends on both the LY of the interaction as well as the decay constant of the scintillation, both of which change as a function of temperature. \nWe correct for this effect for each particle interaction and each temperature point to be able to globally relate the LY of these interactions.\n \nWe define $Q_{total}$ as the real total amount of charge that the scintillation event has induced in the PMT. \nThis is the value that we want to have access to in order to compare the LY under $\\alpha$s and $\\gamma$s at all temperatures.\n$Q_{meas}$ is defined as our actual measured charge.\nThe efficiency of a measurement $\\epsilon = Q_{meas}\/Q_{total}$ will change at each temperature and for each type of particle interaction. \nSince $Q_{total}$ is inaccessible, we define $\\rho=Q_{above}\/Q_{meas}$, the ratio of the integral above the threshold $Q_{above}$ to the total measured integral $Q_{meas}$, as a proxy for the efficiency $\\epsilon$.\nIn Figure~\\ref{fig:thresh}b, we plot the variable $\\rho$ against $Q_{meas}$ which is representative of the emitted light, for an example temperature. \n$\\alpha$ interactions can be seen in the top right of the plot, with high emitted light and high $\\rho$. \n$\\gamma$ interactions are seen in the middle of the plot, with a lower $\\rho$ and lower emitted light. \nFrom this plot, we can see that the efficiencies for $\\alpha$ and $\\gamma$ interactions are systematically different because of the very different LY causing differing amounts of pile-up between photons.\n\nWe carried out simulations to relate $\\rho$ to the efficiency $\\epsilon$. \nIn the simulations, we used different numbers of photons distributed in time by an exponential distribution with varying decay time constants to produce different values of $\\rho$. \nDetermining the relationship between these two variables in simulated data allows us to make an estimate on the efficiency in our recorded data. \nThe relationship between $\\rho$ and $\\epsilon$ is shown in Figure~\\ref{fig:thresh}c, allowing us to make corrections to our LY and ratio $R_{\\alpha\/\\gamma}$. \n\n\\begin{figure}\n \\centering\n\t\\includegraphics[width=\\columnwidth]{fig5}\n\t\\caption{\\label{fig:thresh}Effect of threshold on measured LY. Fig.~\\ref{fig:thresh}a-i shows that a portion of a single photoelectron charge is lost because the amplitudes are below the detection threshold. \n $Q_{meas}$, shaded in blue, is the amount of charge that we record for a given event.\n $Q_{above}$, vertical cross-hatched, is the area under the curve but above the threshold. \n This allows us to create the variable $\\rho=Q_{above}\/Q_{meas}$, the ratio of charge above threshold, to estimate the fraction of lost charge, as described in Section~\\ref{sec:data_anal_ly}.\n If multiple photons are resolved in time as in Fig.~\\ref{fig:thresh}a-ii, the fraction of lost charge is the same as for the single photoelectron. \n If there is pileup of the photoelectrons as in Fig.~\\ref{fig:thresh}a-iii, the fraction of lost charge is smaller. \n At a given temperature, the data presented in Fig.~\\ref{fig:thresh}b of $\\rho$ vs. $Q_{meas}$ show that the effect is detrimental to the pulses from $\\gamma$s compared to $\\alpha$s, since the former emit fewer photons. \n Fig.~\\ref{fig:thresh}c shows simulations (blue spots) allowing to reconstruct the actual LY from the measured one using the median (black line); error bars are taken from spread of points.}\n\\end{figure}\n\n\\subsection{Time Structure}\n\\label{sec:data_anal_ts}\n\nOur data acquisition system is able to sample the charge induced in the PMTs at a sampling rate of 1~ns.\nUsing this time-resolved data, we are able to create an ``average pulse'' by combining the charge vs. time data for a set of individual traces that correspond to the $\\alpha$ or $\\gamma$ population.\nWe chose to use the coincidence trigger time as the beginning of each pulse; because we are only recording the signal from one PMT, and the trigger is set by a 30~ns coincidence window, the actual first pulse recorded may be anywhere from 0-30~ns before the trigger time.\nThus, we do not expect to be sensitive to time structure below this order of magnitude.\n\nFor each population, the average pulse histogram is fit to five exponential decays plus a constant background to model the scintillation decay and background noise. \nWe use the same functional fit for all temperatures to fit all pulses in the same unbiased way.\nWe expect at least three exponential decay components~\\cite{mikhailik_2014} at low temperature, and additional exponentials allow for a good fit of extra-slow components, though we don't attempt to interpret them.\n\nAs mentioned in Section~\\ref{sec:data_acq}, the application of a threshold changes the efficiency of detection for individual photons versus pile-up photons. \nThis has an effect on the average pulse shape separate from the measurement of LY.\nThe effect can be seen in Figure~\\ref{fig:avepulse}a as a kink in the pulse, at a time where the likelihood of a photon arriving at the PMT decreases to a point where pile-up is no longer likely (see Figure~\\ref{fig:thresh}). \nWe have confirmed through simulation, shown in Figure~\\ref{fig:avepulse}b, that the change in efficiency due to photons piling-up only impacts the pulse in this transition region. \nBefore and after, the shape of the pulse follows the underlying distribution.\nTo deal with this in the fit of exponentials, the kink time section has been excluded, and the fit keeps the same time constant before and after the kink time. \nLastly, since the pulse shape spans multiple orders of magnitude, non-uniform binning has been utilized to reduce bin-to-bin statistical fluctuations in the late portion of the pulse.\n\n\\begin{figure}\n \\centering\n\t\\includegraphics[width=\\columnwidth]{fig6}\n\t\\caption{\\label{fig:avepulse}a) Example of average $\\alpha$ pulses for different temperatures (note irregular binning). At least two exponential decays are evident for each temperature, spanning nearly 6 orders of magnitude in intensity. Kink in pulse, caused by variable charge-reconstruction efficiency, is visible at 60~K between 52--56~$\\mu$s.\n %\n b) Simulations of the kink (Sec.~\\ref{sec:data_anal_ts}). Simulated data for a single time constant with no threshold applied (blue circles); the threshold applied to the total pulse, with photons piling up on each other, as done in the real data (green triangles); and the threshold applied to every arriving photon as if they arrived separately (red squares). Kink occurs in transition between regions where photoelectrons pile up and where they don't, but does not affect time constant on either side of transition. }\n\\end{figure}\n\n\n\n\n\\section{Results and discussions}\n\\label{sec:disc}\n\n\\subsection{Light yield and $\\alpha\/\\gamma$ ratio}\n\\label{sec:disc:LYQ}\nThe temperature response curves for the LY of CsI under $\\alpha$ and $\\gamma$ excitation were measured for several stabilized temperatures while cooling from 300~K to 3.4~K.\nOur method allows us to measure LY without the use of a shaping time, making the measurement independent of any changes in decay time constant vs. temperature.\nAs a proxy for the number of emitted photons, we take the number of photons detected in the front PMT.\nThis number of photons is determined from the charge detected by the PMT thanks to a separate measurement with a low-light source providing the average charge induced in the PMT by a single photon.\n\nIdeally, these results would take into account possible temperature-dependent shifts of the emission spectra relative to the fixed quantum efficiency of the PMTs.\nComparing emission spectra at low temperatures with the quantum efficiency of our PMT, we estimate that, relative to room temperature, our detection efficiency changes by at most 10\\% over the entire temperature range~\\cite{Nishimura:1995,mikhailik_2014}.\n\nThe number of detected photons for the various particles and energies are shown as a function of temperature in Figure~\\ref{fig:LYQF}.\nExcept for $\\gamma$s at temperatures above 200~K, at least 10~photons are detected per event. \nThe temperature dependence of the number of detected photons over energy, a proxy for LY, is also shown in Figure~\\ref{fig:LYQF}b. \nThis LY has been corrected for efficiency as described in Section~\\ref{sec:data_anal_ly}\nWe see the $\\alpha$ LY increase by a factor of 100 from 300~K to 30~K, and decrease to a factor of 30 above 300~K at 3.4~K, our lowest temperature. \nThe $\\gamma$ LY for both 60~keV and 122~keV does not see as much of an increase as for the $\\alpha$s, rising by a factor 20-30 above that at 300~K below 100~K.\nThe increase in LY is consistent between the two different $\\gamma$ energies, within uncertainty.\nOverall, the LY is relatively constant at temperatures below 7~K. We assume it remains so at even lower temperatures.\n\nThough comparison is difficult because of uncertainties in our light collection efficiency and quantum efficiency of the PMTs, previous results of the absolute LY of CsI by Amsler et al.~\\cite{Amsler:2002kq} give $50\\pm5$~ph\/keV at 80~K under $\\gamma$ excitation.\nAnother measurement under $\\gamma$ excitation by Moszynski et al.~\\cite{Moszynski:2003} gives $107\\pm10$~ph\/keV for undoped CsI at 77~K.\nAt 77~K, we detect $83\\pm5$ photons from a 60~keV $\\gamma$-excitation. \nAssuming an efficiency of $\\sim 10 \\%$ due to numerical aperture and reflection of windows~\\cite{verdier_2.8_2009}, and a PMT quantum efficiency of $\\sim 15\\%$ at the mean emission of CsI, this gives a rough estimate of $90$~ph\/keV for the LY , which is compatible with the result from Amsler et al.\nMore recently, Sch\\\"affner et al.~\\cite{Schaffner:2012ei} quote an absolute detected energy of 7.1\\% from undoped CsI at 10~mK.\nUsing the mean energy of an emitted photon from CsI (3.9~eV~\\cite{nadeau_cryogenic_2015}), and taking the light collection efficiency of the CRESST type light detectors to be 31\\%~\\cite{Kiefer:2012va}, this would imply an absolute LY of $60$~ph\/keV.\nAt our lowest temperature, using the same method as above, we estimate the LY of CsI to be roughly $65$~ph\/keV at 3.4~K, which appears to be compatible with the result from Sch\\\"affner et al.\nThe agreement of estimated LY values with previous measurements indicates that the corrections described in Section~\\ref{sec:data_anal_ly} are performing correctly.\n\nTo compare with recent results from Mikhailik et al.~\\cite{mikhailik_2014} and Gridin et al.~\\cite{Gridin:2015}, we have scaled their reported LY values to ours at 77~K. \nAlthough Gridin et al. do not report time-resolved LY, we believe that our results are comparable because of our long acquisition window capturing full scintillation events.\nWe see the same general trend of LY increase as previous measurements, but different relative LY at high temperatures; this could be due to a difference in incidental Tl impurities in the crystals.\nThe light emission from Tl can be seen in the room temperature spectra of Mikhailik et al., but disappears at lower temperature; this could increase the LY preferentially at high temperatures in a crystal with a larger amount of Tl.\n\nAs our setup allows the measurement of the lines from $\\alpha$ particles and $\\gamma$ quanta emitted by $^{241}$Am, we can determine the $\\alpha$\/$\\gamma$ ratio, $R_{\\alpha\/\\gamma}$, defined by the ratio of LY for $\\alpha$s over the LY for $\\gamma$s at a given energy $E$:\n\\begin{equation}\\label{eq:qf}R_{\\alpha\/\\gamma} (E) = \\frac{{\\rm LY}_{\\alpha}(E)}{{\\rm LY}_{\\gamma}(E)} = \\frac{\\mu_\\alpha (E) \/ E}{\\mu_\\gamma (E) \/E} = \\frac{\\mu_\\alpha (E) }{\\mu_\\gamma (E)} ,\n\\end{equation}\nwhere $\\mu_i (E)$ is the measured response to particles of type $i$ depositing energy $E$.\n$R_{\\alpha\/\\gamma}$ quantifies the relative difference in light produced by both types of interactions for an equivalent energy deposit, and thus is independent of our overall light collection efficiency barring effects related to the interaction depth of the particle.\n\nThe temperature dependence of $R_{\\alpha\/\\gamma}$ is shown in Figure~\\ref{fig:LYQF}c, as determined using the 60~keV $\\gamma$ and 4.7~MeV degraded $\\alpha$ from $^{241}$Am.\nRemarkably in our data, $R_{\\alpha\/\\gamma}$ becomes greater than 1 for temperatures between 10-100~K.\nA value of $R_{\\alpha\/\\gamma}$ greater than 1 is unexpected in general because of the higher ionization density of $\\alpha$ particles leading to a higher probability of non-radiative recombination of electrons and holes before formation of self-trapped excitons~\\cite{Sysoeva:1998}, though this behavior has previously been observed in alkali halide crystals grown by the Kyropoulos method~\\cite{Birks:1964-11}, and in pure or doped ZnSe~\\cite{Arnaboldi:2011ce,Sysoeva:1998,Klamra:2002bg,Nagorny:2017}.\nWatts et al.~\\cite{Watts:1962} observed an anomalous value of the $\\alpha\/\\beta$ ratio (which should be equivalent to $R_{\\alpha\/\\gamma}$) of $3.5\\pm0.3$ at 77~K and 4~K for a single CsI crystal, that was not reproduced in other crystals, though they presented no explanation.\n\nBirks suggests that a large temperature dependence in $R_{\\alpha\/\\gamma}$ is associated with a high defect density, as the lower ionization density of $\\gamma$-excitation allows electrons and holes to drift and become trapped in defects instead of forming self-trapped excitons~\\cite{Birks:1964-11}.\nThere is evidence for this in ZnSe, where thermal treatment has been seen to affect $R_{\\alpha\/\\gamma}$~\\cite{Nagorny:2017}.\nAnother hypothesis is that there are significant excitation-dependent changes in the emission spectra, causing our detection efficiency to change between particles at various temperatures.\nLastly, there are three effects that could be considered, though they are unlikely because they would have to be strongly temperature dependent to explain the data: the first being that the light collection efficiency differs significantly between the surface and the bulk of the sample, the second being that the scintillation properties of the surface differ from those of the bulk, and the third being the non-proportionality of the LY.\n\nSince in practice our measurement involves $\\alpha$ and $\\gamma$ interactions of different energies (4.7~MeV and 60~keV respectively), the non-proportionality ($NP$) of the scintillation response could also have an affect on our measured light ratio:\n\\begin{equation}\n\\label{eq:qf_e}\nR_{\\alpha\/\\gamma} (4.7 \\mbox{ MeV}) =\\frac{{\\rm LY}_{\\alpha}(4.7 \\mbox{ MeV})}{{\\rm LY}_{\\gamma}(4.7 \\mbox{ MeV})}= \\underbrace{ \\frac{{\\rm LY}_{\\gamma}(60 \\mbox{ keV})}{{\\rm LY}_{\\gamma}(4.7 \\mbox{ MeV})} }_{NP} \\underbrace{ \\frac{{\\rm LY}_{\\alpha}(4.7 \\mbox{ MeV})}{{\\rm LY}_{\\gamma}(60 \\mbox{ keV})}}_{\\text{measurement}}\n\\end{equation}\nThere is some evidence that the non-proportionality of the LY in pure CsI depends on temperature.\nLu et al~\\cite{Lu:2015} report that at 295~K, the LY of a 60~keV $\\gamma$-interaction is greater than that of a 1~MeV $\\gamma$ by approximately 15\\%, whereas at 100~K there is almost no difference. \nIt is unlikely the observed changes in $R_{\\alpha\/\\gamma}$ can be explained by this effect alone, however, as $R_{\\alpha\/\\gamma}$ varies by a factor of 4 over the measured temperature range. \nAdditionally, to be consistent with a value of $R_{\\alpha\/\\gamma} < 1$ at all temperatures, the non-proportionality would have to decrease an additional 20\\% between 100~K and 30~K. \nWe are not aware of any experimental data at other temperatures for pure CsI, or for $\\gamma$-interactions higher than $\\sim 2$~MeV.\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=0.6\\columnwidth]{fig7}\n \\end{center}\n \\caption{\\label{fig:LYQF}a) Detected photons as a function of temperature, for 4.7~MeV $\\alpha$s (circles) and $\\gamma$s (60~keV: squares; 122~keV: triangles). A correction has been applied (original: light ; corrected: solid) to compensate for the loss in charge detected from our threshold (see Sec.~\\ref{sec:data_anal} for details).\n b) Detected photons (corrected) per unit deposited energy as a function of temperature for $\\alpha$s and $\\gamma$s, showing results for 60~keV and 122~keV $\\gamma$s are consistent within errorbars. Results of $\\alpha$-excitation from Mikhailik et al.~\\cite{mikhailik_2014} are shown by the dashed red and of $\\gamma$-excitation from Gridin et al.~\\cite{Gridin:2015} are shown by the solid green line, scaled to be equal to our result at 77~K.\n c) $\\alpha$\/$\\gamma$ ratio (corrected) as a function of temperature, showing a factor 4 variation and reaching values greater than 1. In a) and b), conversion of detected photons to emitted ones requires a multiplication by roughly 70, though this does not affect ratio in c).} \n\\end{figure}\n\n\\subsection{Time Structure}\n\\label{sec:disc_TS}\nThe results of the time constant fits mentioned in Section~\\ref{sec:data_anal_ts} are summarized in Figure~\\ref{fig:tau}.\nTime constants from the fits that contribute at least 10\\% of the total LY at that temperature are plotted in Figure~\\ref{fig:tau}a for $\\alpha$ excitation, and Figure~\\ref{fig:tau}b for $\\gamma$ excitation.\nThe area of the marker represents the contribution of that time constant to the total integral of the pulse, the largest points showing the most prominent time constant at a given temperature.\n\nFirst, for $\\alpha$ excitation, above 100~K there appears to be two prominent time constants, both shorter than 1~$\\mu$s.\nAs the temperature decreases, the main time constant increases from 0.4~$\\mu$s at 300~K to 1~$\\mu$s at 100~K, where it becomes the only important time constant.\nWe also see a fast component that begins at 0.01~$\\mu$s at 300~K, increasing in length to 0.1~$\\mu$s at 150~K.\nThis behaviour is consistent with the measurements of Amsler et al.~\\cite{Amsler:2002kq} from 100-300~K.\n\nFrom 100~K to 10~K, there remains only one time constant at around 1~$\\mu$s.\nThis component is consistent with previous measurements of undoped CsI scintillation at low temperature~\\cite{mikhailik_2014,Watts:1962}\nBelow 10~K, the structure becomes more complicated, with several components contributing equal amounts of light to the pulse.\nWe see a short component of the order 100~ns from both $\\alpha$s and $\\gamma$s, consistent with the 290~nm emission,\nand a long component that is consistent with the 338~nm emission, which are described by previous studies as a system of three excitonic absorption bands~\\cite{lamatsch1971kinetics,Gridin:2015}.\nThere are several very long time constants present at high temperatures that seem to hold a constant value of 10~$\\mu$s and 100~$\\mu$s, which could be attributed to coincidence within our acquisition window of $\\alpha$ and $\\gamma$ events.\nAlternatively, these could be attributed to the presence of shallow traps or defects.\n\nFor the $\\gamma$ excitation, the main time constants show a very similar pattern to the $\\alpha$ excitation data.\nThis indicates that the same light production mechanism is being excited by the two different radiation sources, which has also been seen previously with $\\alpha$s and X-rays~\\cite{mikhailik_2014}.\nLong time constants visible by $\\alpha$ excitation are absent in the $\\gamma$ excitation data, which could indicate a larger concentration of defects near the surface of the crystal, as expected~\\cite{Nishimura:1995}.\n\nWe have compared our results to the results of fits to a model of various STE decays by Nishimura et al.~\\cite{Nishimura:1995} in Figure~\\ref{fig:tau}. For both $\\alpha$ and $\\gamma$ excitation, the data corresponds well with the model at low temperatures. At high temperatures, two exponential components are seen in $\\alpha$ excitation, corresponding to the singlet and triplet STE states~\\cite{Nishimura:1995} seen as the solid red lines in Figure~\\ref{fig:tau}. At low temperature, the evolution of the two decay rates of the off-center STE, seen as the dotted black lines, follow the same trend in our data. The on-center STE decay rate can also be seen as the dashed black line in Figure~\\ref{fig:tau}, which we also see because of our lack of spectral discrimination between the two states.\n\n\\begin{figure}\n \\centering\t\\includegraphics[width=0.9\\columnwidth]{fig8}\n\t\\caption{\\label{fig:tau} \n\tEvolution of time constants with temperature. At a given temperature, only time constants composing at least 10\\% of LY are shown. Marker size at a given temperature is proportional to the contribution of that time constant to the total LY at that temperature. Models of STE decay from Nishimura et al.~\\cite{Nishimura:1995} are shown for the off-center STE state (dotted black line), on-centre STE state (dashed black line), and singlet and triplet STE state at high temperatures (solid red line).\n a) Time constants from $\\alpha$ excitation \n b) Time constants from $\\gamma$ excitation\n }\n\\end{figure}\n\n\\section{Conclusion}\nWe present the first measurement of the scintillation properties of undoped CsI over a wide range of temperature under both $\\alpha$ and $\\gamma$ excitation using a time-resolved zero-suppression measurement technique.\nFor the first time to our knowledge in CsI, a long measurement time window was used to capture the full scintillation event at each trigger, removing any bias from long or short scintillation times.\n\nThe LY under $\\alpha$ excitation increases by nearly two orders of magnitude from 300--30~K, to a maximum value of roughly 120~ph\/keV.\nAt the lowest measured temperature of 3.4~K, the LY remains a factor 30 above the level at room temperature.\nUnder $\\gamma$ excitation, a factor 20 increase in LY is observed below 100~K when compared with the room temperature yield, with a maximum at roughly 90~ph\/keV.\nThe ratio between $\\alpha$ and $\\gamma$ excitations fluctuates significantly over the temperature range, and was found to be greater than one for temperatures between 10--100~K, which could be an indicator of a large amount of defects in the crystal.\nIt also may be an artifact of particle specific emission changes as the temperature decreases.\nMeasurements of the time constants for both $\\alpha$s and $\\gamma$s agree with previous measurements, and follow previous models of STE decay~\\cite{Nishimura:1995}.\n\nThese measurements further establish undoped CsI as an effective cryogenic scintillator with a high light yield at low temperatures. \nWith such a light yield, above that of most other cryogenic scintillators~\\cite{Mikhailik:2010}, low thresholds can be reached in experimental applications that require it, such as searches for dark matter. In such searches, the high $\\alpha$\/$\\gamma$ ratio at low temperatures indicates that the $\\alpha$ background should remain separated from the nuclear recoil signal region due to the low nuclear recoil quenching factor of 0.1 for Cs and I~\\cite{Angloher:2016CsI}. More generally, light yield and $\\alpha\/\\gamma$ ratio are high at liquid nitrogen temperatures, conditions that lend themselves well to practical applications.\n\n\\section{Acknowledgements}\nThis work has been funded in Canada by NSERC (Grant SAPIN 386432), CFI-LOF and ORF-SIF (project 24536).\n\n\\section{References}\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe computation of a measure of similarity (or dissimilarity) between pairs of objects is a crucial subproblem \nin several applications in Computer Vision \\cite{Rubner98,Rubner2000,Pele2009}, Computational Statistic \\cite{Bickel2001}, Probability \\cite{Bassetti2006,Bassetti2006b},\nand Machine Learning \\cite{Solomon2014,Cuturi2014,Frogner2015,Arjovsky2017}.\nIn mathematical terms, in order to compute the similarity between a pair of objects, we want to compute a {\\it distance}.\nIf the distance is equal to zero the two objects are considered to be equal; the more the two objects are different,\nthe greater is their distance value. For instance, the Euclidean norm is the most used distance function to compare a pair of points in $\\mathbb{R}^d$.\nNote that the Euclidean distance requires only $O(d)$ operations to be computed. \nWhen computing the distance between complex discrete objects, \nsuch as for instance a pair of discrete measures, a pair of images,\na pair of $d$-dimensional histograms, or a pair of clouds of points, the Kantorovich-Wasserstein distance \\cite{Villani2008,Vershik2013} has proved to be \na relevant distance function \\cite{Rubner98}, which has both nice mathematical properties and useful practical implications.\nUnfortunately, computing the Kantorovich-Wasserstein distance requires the solution of an optimization problem.\nEven if the optimization problem is polynomially solvable, the size of practical instances to be solved is very large,\nand hence the computation of Kantorovich-Wasserstein distances implies an important computational burden.\n\nThe optimization problem that yields the Kantorovich-Wasserstein distance can be solved with different methods.\nNowadays, the most popular methods are based on (i) the Sinkhorn's algorithm \\cite{Cuturi2013,Solomon2015,Altschuler2017}, \nwhich solves (heuristically) a regularized version of the basic optimal transport problem,\nand (ii) Linear Programming-based algorithms \\cite{Flood1953,Goldberg1989,Orlin}, which exactly solve the basic optimal transport problem\nby formulating and solving an equivalent uncapacitated minimum cost flow problem. For a nice overview of both computational approaches,\nwe refer the reader to Chapters 2 and 3 in \\cite{Peyre2018}, and the references therein contained.\n\nIn this paper, we propose a Linear Programming-based method to speed up the computation of Kantorovich-Wasserstein distances of order 2,\nwhich exploits the structure of the ground distance to formulate an uncapacitated minimum cost flow problem.\nThe flow problem is then solved with a state-of-the-art implementation of the well-known Network Simplex algorithm \\cite{Kovacs2015}.\n\nOur approach is along the line of research initiated in \\cite{LingOkada2007}, where the authors proposed a very efficient method\nto compute Kantorovich-Wasserstein distances of order 1 (i.e., the so--called {\\it Earth Mover Distance}), \nwhenever the ground distance between a pair of points is the $\\ell_1$ norm.\nIn \\cite{LingOkada2007}, the structure of the $\\ell_1$ ground distance and of regular $d$-dimensional histograms is exploited to \ndefine a very small flow network. More recently, this approach has been successfully generalized in \\cite{Bassetti2018} to the case of \n$\\ell_\\infty$ and $\\ell_2$ norms, providing both exact and approximations algorithms, which are able to compute distances\nbetween pairs of $512 \\times 512$ gray scale images. The idea of speeding up the computation of Kantorovich-Wasserstein distances by defining a minimum\ncost flow on smaller structured flow networks is also used in \\cite{Pele2009}, where a truncated distance is used as ground distance in place of a $\\ell_p$ norm.\n\nThe outline of this paper is as follows. Section 2 reviews the basic notion of discrete optimal transport and fixes the notation.\nSection 3 contains our main contribution, that is, Theorem 1 and Corollary 2, which permits to speed-up the computation\nof Kantorovich-Wasserstein distances of order 2 under quite general assumptions.\nSection 4 presents numerical results of our approaches, compared with the Sinkhorn's algorithm as implemented in \\cite{Cuturi2013} \nand a standard Linear Programming formulation on a complete bipartite graph \\cite{Rubner98}. \nFinally, Section 5 concludes the paper.\n\n\n\\section{Discrete Optimal Transport: an Overview}\nLet $X$ and $Y$ be two discrete spaces. \nGiven two probability vectors $\\mu$ and $\\nu$ defined on $X$ and $Y$, respectively, and a cost $c : X \\times Y \\to \\mathbb{R}_+$, \nthe {\\it Kantorovich-Rubinshtein functional} between $\\mu$ and $\\nu$ is defined as\n\\begin{equation}\n\\label{eq:kantorovich}\n\t\t \\mathcal{W}_c(\\mu,\\nu)= \\inf_{ \\pi \\in \\Pi(\\mu,\\nu)} \\sum_{ (x,y) \\in X\\times Y} c(x,y) \\pi(x,y)\n\\end{equation}\nwhere $\\Pi(\\mu,\\nu)$ is the set of all the probability measures on $X \\times Y$ with marginals $\\mu$ and $\\nu$, i.e. \nthe probability measures $\\pi$ such that $\\sum_{y \\in Y} \\pi(x,y)=\\mu(x)$ and $\\sum_{x \\in X} \\pi(x,y)=\\nu(y),$\nfor every $(x,y)$ in $X \\times Y$.\n Such probability measures are sometimes called transport plans or couplings for $\\mu$ and $\\nu$. \nAn important special case is when $X=Y$ and the cost function $c$ is a distance on $X$. In this case \n$ \\mathcal{W}_c$ is a distance on the simplex of probability vectors on $X$, also known as {\\it Kantorovich-Wasserstein distance} of order $1$. \n\nWe remark that {\\bf the Kantorovich-Wasserstein distance of order $p$} can be defined, more in general, \nfor arbitrary probability measures on a metric space $(X,\\delta)$ by \n\\begin{equation}\\label{wpgeneral}\n W_p(\\mu,\\nu):=\\left(\\inf_{ \\pi \\in \\Pi(\\mu,\\nu)} \\int_{ X\\times X} \\delta^p(x,y) \\pi(dx dy)\\right)^{\\min(1\/p,1)}\n\\end{equation}\nwhere now $\\Pi(\\mu,\\nu)$ is the set of all probability measures on the Borel sets of $X \\times X$ that have marginals $\\mu$ and $\\nu$, see, e.g., \\cite{AGS}.\nThe infimum in \\eqref{wpgeneral} is attained, and any probability $\\pi$ which realizes the minimum is called an {\\it optimal transport plan}.\n\nThe Kantorovich-Rubinshtein transport problem in the discrete setting can be seen as a special case of \nthe following Linear Programming problem, where we assume now that $\\mu$ and $\\nu$ are \ngeneric vectors of dimension $n$, with positive components, \n\\begin{align}\n\\label{p1:funobj} (P) \\quad \\min \\quad & \\sum_{x \\in X}\\sum_{y \\in Y} c(x,y) \\pi(x,y) \\\\\n\\mbox{s.t.} \\quad \n\\label{p1:supply} & \\sum_{y \\in Y} \\pi(x,y) \\leq \\mu(x) & \\forall x \\in X \\\\\n\\label{p1:demand} & \\sum_{x \\in X} \\pi(x,y) \\geq \\nu(y) & \\forall y \\in Y \\\\\n\\label{p1:posvar} & \\pi(x,y) \\geq 0.\n\\end{align}\n\\noindent If $\\sum_{x } \\mu(x) = \\sum_{y} \\nu(y)$ we have the so-called {\\it balanced} transportation problem, \notherwise the transportation problem is said to be {\\it unbalanced} \\cite{Liero2018,Chizat2016}.\nFor balanced optimal transport problems, constraints \\eqref{p1:supply} and \\eqref{p1:demand} must be satisfied with equality, and the problem \nreduces to the Kantorovich transport problem (up to normalization of the vectors $\\mu$ and $\\nu$). \n\n\nProblem (P) is related to the so-called {\\it Earth Mover's distance}.\nIn this case, $X,Y \\subset \\mathbb{R}^d$, $x$ and $y$ are the centers of two data clusters, and\n$\\mu(x)$ and $\\nu(y)$ give the number of points in the respective cluster. \nFinally, $c(x,y)$ is some measure of dissimilarity between the two clusters $x$ and $y$.\nOnce the optimal transport $\\pi^*$ is determined, the Earth Mover's distance\nbetween $\\mu$ and $\\nu$ is defined as (e.g., see \\cite{Rubner98})\n\\[\nEMD(\\mu,\\nu)= \\frac{\\sum_{x \\in X}\\sum_{y \\in Y} c(x,y) \\pi^*(x,y)}{\\sum_{x \\in X}\\sum_{y \\in Y} \\pi^*(x,y)}.\n\\]\n\nProblem (P) can be formulated as an uncapacitated minimum cost flow problem on a bipartite graph defined as follows \\cite{Ahuja}.\nThe bipartite graph has two partitions of nodes: the first partition has a node for each point $x$ of $X$, and the second partition has a node for each point $y$ of $Y$.\nEach node $x$ of the first partition has a supply of mass equal to $\\mu(x)$,\neach node of the second partition has a demand of $\\nu(y)$ units of mass.\nThe bipartite graph has an (uncapacitated) arc for each element in the Cartesian product $X \\times Y$ having cost equal to $c(x,y)$.\nThe minimum cost flow problem defined on this graph yields the optimal transport plan $\\pi^*(x,y)$, which indeed is an optimal solution of problem \\eqref{p1:funobj}--\\eqref{p1:posvar}.\nFor instance, in case of a regular 2D dimensional histogram of size $N \\times N$, that is, having $n=N^2$ bins, \nwe get a bipartite graph with $2N^2$ nodes and $N^4$ arcs (or $2n$ nodes and $n^2$ arcs). Figure \\ref{fig1}--\\subref{fig1a} shows an example for a $3 \\times 3$ histogram,\nand Figure \\ref{fig1}--\\subref{fig1b} gives the corresponding complete bipartite graph.\n\n\\begin{figure*}[t!]\n \\centering\n \\subfigure[]{\\label{fig1a}\\includegraphics[height=6cm]{.\/fig1a.PNG}}\\qquad \\qquad\n \\subfigure[]{\\label{fig1b}\\includegraphics[height=6cm]{.\/fig1b.PNG}}\\qquad \\qquad\n \\subfigure[]{\\label{fig1c}\\includegraphics[height=6cm]{.\/fig1c.PNG}}\n \\caption{\\subref{fig1a} Two given 2-dimensional histograms of size $N\\times N$, with $N=3$; \\subref{fig1b} Complete bipartite graph with $N^4$ arcs; \\subref{fig1c}: 3-partite graph with $(d+1)N^3$ arcs.\\label{fig1}}\n\\end{figure*}\n\nIn this paper, we focus on the case $p=2$ in equation \\eqref{wpgeneral} and the ground distance function $\\delta$ is the Euclidean norm $\\ell_2$, that is\nthe Kantorovich-Wasserstein distance of order $2$, which is denoted by $W_2$. We provide, in the next section,\nan equivalent formulation on a smaller $(d+1)$-partite graph.\n\n\n\t\n\\section{Formulation on $(d+1)$-partite Graphs}\t\nFor the sake of clarity, but without loss of generality, we present first our construction considering 2-dimensional histograms and the $\\ell_2$ Euclidean ground distance.\nThen, we discuss how our construction can be generalized to any pair of $d$-dimensional histograms.\n\nLet us consider the following flow problem: let $\\mu$ and $\\nu$ be two probability measures over a $N \\times N$ regular grid denoted by $G$.\nIn the following paragraphs, we use the notation sketched in Figure \\ref{fig2}. In addition, we define the set $U:=\\{1,\\dots,N\\}$.\n\n\\begin{figure}[t!]\n\\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={right,center},capbesidewidth=0.55\\textwidth}}]{figure}[\\FBwidth]\n{\\caption{Basic notation used in Section 3: in order to send a unit of flow from point $(a,j)$ to point $(i,b)$, we either send\n a unit of flow directly along arc $((a,j),(i,b))$ of cost $c((a,j),(i,b))=(a-i)^2 + (j-b)^2$, or, we first send a unit of flow\n from $(a,j)$ to $(i,j)$, and then from $(i,j)$ to $(i,b)$, having total cost $c((a,j),(i,j)) + c((i,j),(i,b)) = (a-i)^2 + (j-j)^2 + (i-i)^2 + (j-b)^2 = (a-i)^2 + (j-b)^2 = c((a,j),(i,b))$. Indeed, the cost of the two different path is exactly the same.}\n \\label{fig2}}\n{\\includegraphics[width=0.35\\textwidth]{.\/images\/fig2.PNG}}\n\\end{figure}\n\nSince we are considering the $\\ell_2$ norm as ground distance, we minimize the functional\t\n\\begin{equation}\n\\label{formulationflow}\n{R}:(F_1,F_2)\\rightarrow \\sum_{i,j=1}^N \\left[\\sum_{a=1}^N (a-i)^2f^{(1)}_{a,i,j}+ \\sum_{b=1}^N (j-b)^2f^{(2)}_{i,j,b}\\right]\n\\end{equation}\t\t\namong all $F_i = \\{f^{(i)}_{a,b,c}\\}$, with $a,b,c \\in \\{1,...,N\\}$ real numbers (i.e., flow variables) satisfying the following constraints\n\\begin{eqnarray}\n\\label{condizionecompmu}\t\\sum_{i=1}^N f^{(1)}_{a,i,j}&=&\\mu_{a,j}, \\qquad\\qquad \\forall a,j \\in U \\times U \\\\\n\\label{condizionecompnu}\t\\sum_{j=1}^N f^{(2)}_{i,j,b}&=&\\nu_{i,b}, \\qquad\\qquad \\forall i,b \\in U \\times U \\\\\n\\label{incollamento} \t\\sum_{a}f^{(1)}_{a,i,j}&=&\\sum_{b}f^{(2)}_{i,j,b}, \\qquad\\forall i,j \\in U \\times U, a \\in U, b \\in U.\n\\end{eqnarray}\n\\noindent Constraints \\eqref{condizionecompmu} impose that the mass $\\mu_{a,j}$ at the point $(a,j)$ is moved to the points $(k,j)_{k=1,...,N}$.\nConstraints \\eqref{condizionecompnu} force the point $(i,b)$ to receive from the points $(i,l)_{l=1,...,N}$ a total mass of $\\nu_{i,b}$.\nConstraints \\eqref{incollamento} require that all the mass that goes from the points $(a,j)_{a=1,...,N}$ to the point $(i,j)$ is moved to the points $(i,b)_{b=1,...,N}$.\nWe call a pair $(F_1,F_2)$ satisfying the constraints \\eqref{condizionecompmu}--\\eqref{incollamento} a {\\it feasible flow} between $\\mu$ and $\\nu$. \nWe denote by $\\mathcal{F}(\\mu,\\nu)$ the set of all feasible flows between $\\mu$ and $\\nu$.\n\t\t\t\nIndeed, we can formulate the minimization problem defined by \\eqref{formulationflow}--\\eqref{incollamento} \nas an uncapacitated minimum cost flow problem on a tripartite graph $T=(V,A)$. \nThe set of nodes of $T$ is $V:=V^{(1)}\\cup V^{(2)} \\cup V^{(3)}$, where $V^{(1)}, V^{(2)}$ and $V^{(3)}$ are the nodes corresponding to three $N\\times N$ regular grids. \nWe denote by $(i,j)^{(l)}$ the node of coordinates $(i,j)$ in the grid $V^{(l)}$. \nWe define the two disjoint set of arcs between the successive pairs of node partitions as\n\\begin{eqnarray}\n\tA^{(1)}&:=& \\{ ((a,j)^{(1)},(i,j)^{(2)}) \\mid i,a,j \\in U \\}, \\\\\n\tA^{(2)}&:=& \\{ ((i,j)^{(2)},(i,b)^{(3)}) \\mid i,b,j \\in U \\},\n\\end{eqnarray}\n\\noindent and, hence, the arcs of $T$ are $A:=A^{(1)} \\cup A^{(2)}$.\nNote that in this case the graph $T$ has $3N^2$ nodes and $2N^3$ arcs.\nWhenever $(F_1,F_2)$ is a feasible flow between $\\mu$ and $\\nu$, we can think of the values $f^{(1)}_{a,i,j}$ as the quantity of \nmass that travels from $(a,j)$ to $(i,j)$ or, equivalently, that moves along the arc $((a,j),(i,j))$ of the tripartite graph, \nwhile the values $f^{(2)}_{i,j,b}$ are the mass moving along the arc $((i,j),(i,b))$\n(e.g., see Figures \\ref{fig1}--\\subref{fig1c} and \\ref{fig2}).\n\t\nNow we can give an idea of the roles of the sets $V^{(1)}$, $V^{(2)}$ and $V^{(3)}$: $V^{(1)}$ is the node set where is drawn the initial distribution $\\mu$, \nwhile on $V^{(3)}$ it is drawn the final configuration of the mass $\\nu$. The node set $V^{(2)}$ is an auxiliary grid that hosts an intermediate \nconfiguration between $\\mu$ and $\\nu$. \n\nWe are now ready to state our main contribution.\t\n\\begin{theorem}\n\t\\label{teoremaequivalenza}\n\tFor each measure $\\pi$ on $G\\times G$ that transports $\\mu$ into $\\nu$, we can find a feasible flow $(F_1,F_2)$ such that\n\t\\begin{equation}\n\t\\label{result1}\n\t{R}(F_1,F_2)=\\sum_{((a,j),(i,b))} ((a-i)^2+(b-j)^2)\\pi_{(a,j),(i,b))}.\n\t\\end{equation}\n\\end{theorem}\t\n\\begin{proof}\\textit{(Sketch).} \n\tWe will only show how to build a feasible flow starting from a transport plan, \n\tthe inverse building uses a more technical lemma (the so--called {\\it gluing lemma} \\cite{AGS,Villani2008}) and can be found in the Additional Material.\t\t\t\t\n\tLet $\\pi$ be a transport plan, if we write explicitly the ground distance $\\ell_2((a,j),(i,b))$ we find that\n\t\\begin{eqnarray*}\n\t\t\\sum_{((a,j),(i,b))} \\ell_2((a,j),(i,b))\\pi_{((a,j),(i,b))}\\hspace{-0.3cm}&=&\\hspace{-0.3cm}\\sum_{((a,j),(i,b))} ((a-i)^2+(j-b)^2)\\pi_{((a,j),(i,b))}\\\\\n\t\t\\hspace{-0.3cm}&=&\\hspace{-0.3cm} \\sum_{j,i} \\left[\\sum_{a,b}(a-i)^2\\pi_{((a,j),(i,b))} + \\sum_{a,b} (j-b)^2 \\pi_{((a,j),(i,b))} \\right].\n\t\\end{eqnarray*}\n\tIf we set $f^{(1)}_{a,i,j}=\\sum_b\\pi_{((a,j),(i,b))}$ and $f^{(2)}_{i,j,b}=\\sum_a \\pi_{((a,j),(i,b))}$ we find\n\t\\begin{equation*}\n\t\t\\sum_{((a,j),(i,b))} \\ell_2((a,j),(i,b))\\pi_{((a,j),(i,b))}=\\sum_{i,j}^n \\left[\\sum_a^n (a-i)^2f^{(1)}_{a,i,j}+ \\sum_b^n (j-b)^2 f^{(2)}_{i,j,b}\\right].\n\t\\end{equation*}\n\tIn order to conclude we have to prove that those $f^{(1)}_{a,i,j}$ and $f^{(2)}_{i,j,b}$ satisfy the constraints \\eqref{condizionecompmu}--\\eqref{incollamento}.\t\t\n\t\n\tBy definition we have \n\t\\begin{equation*}\n\t\t\\sum_i f^{(1)}_{a,i,j}=\\sum_i \\sum_b\\pi_{((a,j),(i,b))}=\\mu_{a,j},\n\t\\end{equation*}\n\tthus proving (\\ref{condizionecompmu}); similarly, it is possible to check constraint \\eqref{condizionecompnu}.\n\tThe constraint \\eqref{incollamento} also follows easily since\n\t\\begin{equation*}\n\t\t\\sum_{a}f^{(1)}_{a,i,j}= \\sum_a \\sum_b\\pi_{((a,j),(i,b))} = \\sum_{b}f^{(2)}_{i,j,b}.\n\t\\end{equation*}\n\\end{proof}\n\nAs a straightforward, yet fundamental, consequence we have the following result.\n\t\n\\begin{corollary} \nIf we set $c((a,j),(i,b))=(a-i)^2+(j-b)^2$ then, for any discrete measures $\\mu$ and $\\nu$, we have that \n\\begin{equation}\nW^2_2(\\mu,\\nu)=\\min_{\\mathcal{F}(\\mu,\\nu)}R(F_1,F_2).\n\\end{equation}\n\\end{corollary}\n\nIndeed, we can compute the Kantorovich-Wasserstein distance of order 2 between a pair of discrete measures $\\mu, \\nu$, by solving an \nuncapacitated minimum cost flow problem on the given tripartite graph $T:=(V^{(1)} \\cup V^{(2)} \\cup V^{(3)}, A^{(1)} \\cup A^{(2)})$.\n\nWe remark that our approach is very general and it can be directly extended to deal with the following generalizations.\n\n\\paragraph{More general cost functions.} The structure that we have exploited of the Euclidean distance $\\ell_2$ is present in any\ncost function $c: G \\times G \\rightarrow [0,\\infty]$ that is separable, i.e., has the form\n\t\\begin{equation*}\n\t\tc(x,y)= c^{(1)}(x_1,y_1) + c^{(2)}(x_2,y_2),\n\t\\end{equation*}\n\twhere both $c^{(1)}$ and $c^{(2)}$ are positive real valued functions defined over $G$. \n\tWe remark that the whole class of costs $c_p(x,y)=(x_1-y_1)^p+(x_2-y_2)^p$ is of that kind, \n\tso we can compute any of the Kantorovich-Wasserstein distances related to each $c_p$.\n\t\n\\paragraph{Higher dimensional grids.} Our approach can handle discrete measures in spaces of any dimension $d$, that is, for instance, any $d$-dimensional histogram. \nIn dimension $d=2$, we get a tripartite graph because we decomposed the transport along the two main directions.\nIf we have a problem in dimension $d$, we need a $(d+1)$-plet of grids connected by arcs oriented as the $d$ fundamental directions, yielding a $(d+1)$-partite graph.\nAs the dimension $d$ grows, our approach gets faster and more memory efficient than the standard formulation given on a bipartite graph.\n\nIn the Additional Material, we present a generalization of Theorem 1 to any dimension $d$ and to {\\it separable} cost functions $c(x,y)$.\n\n\n\\section{Computational Results}\nIn this section, we report the results obtained on two different set of instances. \nThe goal of our experiments is to show how our approach scales with the size of the histogram $N$ and with the dimension of the histogram $d$.\nAs cost distance $c(x,y)$, with $x,y \\in \\mathbb{R}^d$, we use the squared $\\ell_2$ norm.\nAs problem instances, we use the gray scale images (i.e., 2-dimensional histograms) proposed by the DOTMark benchmark \\cite{Dotmark}, \nand a set of $d$-dimensional histograms obtained by bio medical data measured by flow cytometer \\cite{Bernas2008}.\n\n\n\\paragraph{Implementation details.}\nWe run our experiments using the Network Simplex as implemented in the Lemon C++ graph library\\footnote{\\url{http:\/\/lemon.cs.elte.hu} (last visited on October, 26th, 2018)},\nsince it provides the fastest implementation of the Network Simplex algorithm to solve uncapacitated minimum cost flow problems \\cite{Kovacs2015}.\nWe did try other state-of-the-art implementations of combinatorial algorithm for solving min cost flow problems, but the Network Simplex of\nthe Lemon graph library was the fastest by a large margin.\nThe tests are executed on a gaming laptop with Windows 10 (64 bit), equipped with an Intel i7-6700HQ CPU and 16 GB of Ram. \nThe code was compiled with MS Visual Studio 2017, using the ANSI standard C++17. The code execution is single threaded.\nThe Matlab implementation of the Sinkhorn's algorithm \\cite{Cuturi2013} runs in parallel on the CPU cores, but we do not use any GPU in our test.\nThe C++ and Matlab code we used for this paper is freely available at \\url{http:\/\/stegua.github.io\/dpartion-nips2018}.\n\n\\paragraph{Results for the DOTmark benchmark.} \nThe DOTmark benchmark contains 10 classes of gray scale images related to randomly generated images, classical images, \nand real data from microscopy images of mitochondria \\cite{Dotmark}. In each class there are 10 different images.\nEvery image is given in the data set at the following pixel resolutions: $32\\times32$, $64\\times64$, $128\\times128$, $256\\times256$, and $512\\times512$.\nThe images in Figure \\ref{fig:dot} are respectively the {\\it ClassicImages}, {\\it Microscopy}, and {\\it Shapes} images (one class for each row), shown at highest resolution.\n\n\\begin{figure}[!t]\n\\centering\n{\\renewcommand{\\arraystretch}{0.7}\n\\setlength{\\tabcolsep}{0.1em}\n\\begin{tabular}{cccccccccc}\n \\includegraphics[width=0.095\\linewidth]{images\/classic512_1001} & \\includegraphics[width=0.095\\linewidth]{images\/classic512_1002} &\n \\includegraphics[width=0.095\\linewidth]{images\/classic512_1003} & \\includegraphics[width=0.095\\linewidth]{images\/classic512_1004} &\n \\includegraphics[width=0.095\\linewidth]{images\/classic512_1005} &\n \\includegraphics[width=0.095\\linewidth]{images\/classic512_1006} & \\includegraphics[width=0.095\\linewidth]{images\/classic512_1007} &\n \\includegraphics[width=0.095\\linewidth]{images\/classic512_1008} & \\includegraphics[width=0.095\\linewidth]{images\/classic512_1009} &\n \\includegraphics[width=0.095\\linewidth]{images\/classic512_1010} \\\\ \n \\includegraphics[width=0.095\\linewidth]{images\/micro512_1001} & \\includegraphics[width=0.095\\linewidth]{images\/micro512_1002} &\n \\includegraphics[width=0.095\\linewidth]{images\/micro512_1003} & \\includegraphics[width=0.095\\linewidth]{images\/micro512_1004} &\n \\includegraphics[width=0.095\\linewidth]{images\/micro512_1005} &\n \\includegraphics[width=0.095\\linewidth]{images\/micro512_1006} & \\includegraphics[width=0.095\\linewidth]{images\/micro512_1007} &\n \\includegraphics[width=0.095\\linewidth]{images\/micro512_1008} & \\includegraphics[width=0.095\\linewidth]{images\/micro512_1009} &\n \\includegraphics[width=0.095\\linewidth]{images\/micro512_1010} \\\\\n \\includegraphics[width=0.095\\linewidth]{images\/shapes512_1001} & \\includegraphics[width=0.095\\linewidth]{images\/shapes512_1002} &\n \\includegraphics[width=0.095\\linewidth]{images\/shapes512_1003} & \\includegraphics[width=0.095\\linewidth]{images\/shapes512_1004} &\n \\includegraphics[width=0.095\\linewidth]{images\/shapes512_1005} &\n \\includegraphics[width=0.095\\linewidth]{images\/shapes512_1006} & \\includegraphics[width=0.095\\linewidth]{images\/shapes512_1007} &\n \\includegraphics[width=0.095\\linewidth]{images\/shapes512_1008} & \\includegraphics[width=0.095\\linewidth]{images\/shapes512_1009} &\n \\includegraphics[width=0.095\\linewidth]{images\/shapes512_1010} \\\\\n \\end{tabular}}\n\\caption{DOTmark benchmark: Classic, Microscopy, and Shapes images. \\label{fig:dot}}\n\\end{figure}\n\nIn our test, we first compared five approaches to compute the Kantorovich-Wasserstein distances on images of size $32\\times32$:\n\\begin{enumerate}\n\\item {\\bf EMD}: The implementation of Transportation Simplex provided by \\cite{Rubner98}, known in the literature as EMD code, \nthat is an exact general method to solve optimal transport problem. We used the implementation in the programming language C, as provided by the authors,\nand compiled with all the compiler optimization flags active.\n\n\\item {\\bf Sinkhorn}: The Matlab implementation of the Sinkhorn's algorithm\\footnote{\\url{http:\/\/marcocuturi.net\/SI.html} (last visited on October, 26th, 2018)} \n\\cite{Cuturi2013}, that is an approximate\napproach whose performance in terms of speed and numerical accuracy depends on a parameter $\\lambda$: for smaller values of $\\lambda$, the algorithm\nis faster, but the solution value has a large gap with respect to the optimal value of the transportation problem; \nfor larger values of $\\lambda$, the algorithm is more accurate (i.e., smaller gap), but it becomes slower.\nUnfortunately, for very large value of $\\lambda$ the method becomes numerically unstable.\nThe best value of $\\lambda$ is very problem dependent. In our tests, we used $\\lambda=1$ and $\\lambda = 1.5$. The second value, $\\lambda=1.5$, \nis the largest value we found for which the algorithm computes the distances for all the instances considered without facing numerical issues.\n\n\\item {\\bf Improved Sinkhorn}: We implemented in Matlab an improved version of the Sinkhorn's algorithm, \nspecialized to compute distances over regular 2-dimensional grids \\cite{Solomon2015,Solomon2018}.\nThe main idea is to improve the matrix-vector operations that are the true computational bottleneck of Sinkhorn's algorithm, by exploiting the structure of the cost matrix.\nIndeed, there is a parallelism with our approach to the method presented in \\cite{Solomon2015}, since\nboth exploits the geometric cost structure. In \\cite{Solomon2015}, the authors proposes a general method that exploits a heat kernel to speed up\nthe matrix-vector products.\nWhen the discrete measures are defined over a regular 2-dimensional grid, the cost matrix used by the Sinkhorn's algorithm can be obtained using a Kronecker\nproduct of two smaller matrices. Hence, instead of performing a matrix-vector product using a matrix of dimension $N \\times N$, we perform\ntwo matrix-matrix products over matrices of dimension $\\sqrt{N}\\times \\sqrt{N}$, yielding a significant runtime improvement.\nIn addition, since the smaller matrices are Toeplitz matrices, they can be embedded into circulant matrices, and, as consequence, it is possible\nto employ a Fast Fourier Transform approach to further speed up the computation. Unfortunately, the Fast Fourier Transform makes the approach\nstill more numerical unstable, and we did not used it in our final implementation.\n\n\\item {\\bf Bipartite}: The bipartite formulation presented in Figure \\ref{fig1}--\\subref{fig1b}, which is the same as \\cite{Rubner98}, but it is solved\nwith the Network Simplex implemented in the Lemon Graph library \\cite{Kovacs2015}.\n\n\\item {\\bf $3$-partite}: The $3$-partite formulation proposed in this paper, which for 2-dimensional histograms is represented in \\ref{fig1}--\\subref{fig1c}.\nAgain, we use the Network Simplex of the Lemon Graph Library to solve the corresponding uncapacitated minimum cost flow problem.\n\\end{enumerate}\n\nTables \\ref{tab:1}(a) and \\ref{tab:1}(b) report the averages of our computational results over different classes of images of the DOTMark benchmark. \nEach class of gray scale image contains 10 instances, and we compute the distance between every possible pair of images within the same class:\nthe first image plays the role of the source distribution $\\mu$, and the second image gives the target distribution $\\nu$. \nConsidering all pairs within a class, it gives 45 instances for each class.\nWe report the means and the standard deviations (between brackets) of the runtime, measured in seconds.\nTable \\ref{tab:1}(a) shows in the second column the runtime for EMD \\cite{Rubner98}. The third and fourth columns gives the runtime and the optimality gap\nfor the Sinkhorn's algorithm with $\\lambda=1$; the 6-$th$ and 7-$th$ columns for $\\lambda=1.5$.\nThe percentage gap is computed as $\\mbox{Gap}=\\frac{UB-opt}{opt}\\cdot 100$, where $UB$ is the upper bound computed by the Sinkhorn's algorithm, \nand $opt$ is the optimal value computed by EMD. The last two columns report the runtime for the bipartite and $3$-partite approaches presented in this paper.\n\nTable \\ref{tab:1}(b) compares our $3$-partite formulation with the Improved Sinkhorn's algorithm \\cite{Solomon2015,Solomon2018}, reporting the same statistics of the previous table.\nIn this case, we run the Improved Sinkhorn using three values of the parameter $\\lambda$, that are, 1.0, 1.25, and 1.5. While the Improved Sinkhorn is \nindeed much faster that the general algorithm as presented in \\cite{Cuturi2013}, it does suffer of the same numerical stability issues, and,\nit can yield very poor percentage gap to the optimal solution, as it happens for the GRFrough and the WhiteNoise classes, where the optimality gaps\nare on average 31.0\\% and 39.2\\%, respectively.\n\nAs shown in Tables \\ref{tab:1}(a) and \\ref{tab:1}(b), the $3$-partite approach is clearly faster than any of the alternatives considered here, despite being an exact method.\nIn addition, we remark that, even on the bipartite formulation, the Network Simplex implementation of the Lemon Graph library is order of magnitude faster than EMD,\nand hence it should be the best choice in this particular type of instances. We remark that it might be unfair to compare an algorithm implemented in C++ with\nan algorithm implemented in Matlab, but still, the true comparison is on the solution quality more than on the runtime. \nMoreover, when implemented on modern GPU that can fully exploit parallel matrix-vector operations, the Sinkhorn's algorithm can run much faster,\nbut they cannot improve the optimality gap.\n\n\\begin{table}[t!]\n\\caption{Comparison of different approaches on $32 \\times 32$ images. The runtime (in seconds) is given as ``Mean (StdDev)''.\nThe gap to the optimal value {\\it opt} is computed as $\\frac{UB-opt}{opt}\\cdot 100$, where $UB$ is the upper bound computed by Sinkhorn's algorithm.\nEach row reports the averages over 45 instances.\n\\label{tab:1}}\n\\centering\n{\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{lccrcrcc}\n & \\multicolumn{1}{c}{EMD \\cite{Rubner98}} & \\multicolumn{4}{c}{Sinkhorn \\cite{Cuturi2013}} & \\multicolumn{1}{c}{Bipartite} & \\multicolumn{1}{c}{$3$-partite} \\\\\n\t & & \\multicolumn{2}{c}{$\\lambda=1$} & \\multicolumn{2}{c}{$\\lambda=1.5$} & & \\\\\nImage Class & Runtime & Runtime & Gap & Runtime & Gap & Runtime & Runtime \\\\\n\\hline\nClassic & 24.0 (3.3) & 6.0 (0.5) & 17.3\\% & 8.9 (0.7) & 9.1\\% & 0.54 (0.05) & 0.07 (0.01)\\\\\nMicroscopy & 35.0 (3.3) & 3.5 (1.0) & 2.4\\% & 5.3 (1.4) & 1.2\\% & 0.55 (0.03) & 0.08 (0.01)\\\\\nShapes & 25.2 (5.3) & 1.6 (1.1) & 5.6\\% & 2.5 (1.6) & 3.0\\% & 0.50 (0.07) & 0.05 (0.01)\\\\\n \\hline\\noalign{\\smallskip}\n \\multicolumn{8}{c}{(a)} \\\\\n \\multicolumn{8}{c}{} \\\\\n \\end{tabular}}\n\\centering\n{\\renewcommand{\\arraystretch}{1.2}\n\\setlength{\\tabcolsep}{5pt}\n\\begin{tabular}{lcrcrcrc}\n & \\multicolumn{6}{c}{Improved Sinkhorn \\cite{Solomon2015,Solomon2018}} & \\multicolumn{1}{c}{$3$-partite} \\\\\n\t & \\multicolumn{2}{c}{$\\lambda=1$} & \\multicolumn{2}{c}{$\\lambda=1.25$} & \\multicolumn{2}{c}{$\\lambda=1.5$} & \\\\\nImage Class & Runtime & Gap & Runtime & Gap & Runtime & Gap & Runtime \\\\\n\\hline\n CauchyDensity\t&\t0.22\t(0.15)\t&\t2.8\\%\t&\t0.33\t(0.23)\t&\t2.0\\%\t&\t0.41\t(0.28)\t&\t1.5\\%\t&\t0.07\t(0.01)\t\\\\\n Classic\t\t&\t0.20\t(0.01)\t&\t17.3\\%\t&\t0.31\t(0.02)\t&\t12.4\\%\t&\t0.39\t(0.03)\t&\t9.1\\%\t&\t0.07\t(0.01)\t\\\\\n GRFmoderate\t&\t0.19\t(0.01)\t&\t12.6\\%\t&\t0.29\t(0.02)\t&\t9.0\\%\t&\t0.37\t(0.03)\t&\t6.6\\%\t&\t0.07\t(0.01)\t\\\\\n GRFrough\t\t&\t0.19\t(0.01)\t&\t58.7\\%\t&\t0.29\t(0.01)\t&\t42.1\\%\t&\t0.38\t(0.02)\t&\t31.0\\%\t&\t0.05\t(0.01)\t\\\\\n GRFsmooth\t\t&\t0.20\t(0.02)\t&\t4.3\\%\t&\t0.30\t(0.04)\t&\t3.1\\%\t&\t0.38\t(0.04)\t&\t2.2\\%\t&\t0.08\t(0.01)\t\\\\\n LogGRF\t\t\t&\t0.22\t(0.05)\t&\t1.3\\%\t&\t0.32\t(0.08)\t&\t0.9\\%\t&\t0.40\t(0.13)\t&\t0.7\\%\t&\t0.08\t(0.01)\t\\\\\n LogitGRF\t\t&\t0.22\t(0.02)\t&\t4.7\\%\t&\t0.33\t(0.03)\t&\t3.3\\%\t&\t0.42\t(0.04)\t&\t2.5\\%\t&\t0.07\t(0.02)\t\\\\\n Microscopy \t&\t0.18\t(0.03)\t&\t2.4\\%\t&\t0.27\t(0.04)\t&\t1.7\\%\t&\t0.34\t(0.05)\t&\t1.2\\%\t&\t0.08\t(0.02)\t\\\\\n Shapes\t\t\t&\t0.11\t(0.04)\t&\t5.6\\%\t&\t0.16\t(0.06)\t&\t4.0\\%\t&\t0.20\t(0.07)\t&\t3.0\\%\t&\t0.05\t(0.01)\t\\\\\n WhiteNoise\t\t&\t0.18\t(0.01)\t&\t76.3\\%\t&\t0.28\t(0.01)\t&\t53.8\\%\t&\t0.37\t(0.02)\t&\t39.2\\%\t&\t0.04\t(0.00)\t\\\\\n\\hline\\noalign{\\smallskip}\n \\multicolumn{8}{c}{(b)} \\\\\n \\end{tabular}}\n\\end{table}\n\nIn order to evaluate how our approach scale with the size of the images, we run additional tests using images of size $64\\times64$ and $128\\times128$.\nTable \\ref{tab:2} reports the results for the bipartite and $3$-partite approaches for increasing size of the 2-dimensional histograms.\nThe table report for each of the two approaches, the number of vertices $|V|$ and of arcs $|A|$, and the means and standard deviations of the runtime.\nAs before, each row gives the averages over 45 instances. Table \\ref{tab:2} shows that the $3$-partite approach is clearly better (i) in terms of memory,\nsince the 3-partite graph has a fraction of the number of arcs, and (ii) of runtime, since it is at least an order of magnitude faster in computation time.\nIndeed, the 3-partite formulation is better essentially because it exploits the structure of the ground distance $c(x,y)$ used, that is, the squared $\\ell_2$ norm.\n\\begin{table}[t!]\n\\caption{Comparison of the bipartite and the $3$-partite approaches on 2-dimensional histograms.\\label{tab:2}}\n\\centering\n{\\renewcommand{\\arraystretch}{1.2}\n\\setlength{\\tabcolsep}{0.5em}\n\\begin{tabular}{clrrrrrr}\n & & \\multicolumn{3}{c}{Bipartite} & \\multicolumn{3}{c}{$3$-partite} \\\\\nSize & Image Class & $|V|$ & $|A|$ & Runtime & $|V|$ & $|A|$ & Runtime \\\\\n\\hline\n$64 \\times 64$ &Classic & 8\\,193& 16\\,777\\,216 & 16.3 (3.6) & 12\\,288 & 524\\,288 & 2.2 (0.2) \\\\\n&Microscopy & & & 11.7 (1.4) & & & 1.0 (0.2) \\\\\n&Shape & & & 13.0 (3.9) & & & 1.1 (0.3) \\\\\n\\hline\\noalign{\\smallskip}\n$128\\times128$ & Classic & 32\\,768 & 268\\,435\\,456& 1\\,368 (545) & 49\\,152 & 4\\,194\\,304& 36.2 (5.4) \\\\\n&Microscopy & & & 959 (181) & & & 23.0 (4.8) \\\\\n&Shape & & & 983 (230) & & & 17.8 (5.2) \\\\\n \\hline\\noalign{\\smallskip}\n \\end{tabular}}\n\\end{table}\n\n\n\\paragraph{Flow Cytometry biomedical data.}\nFlow cytometry is a laser-based biophysical technology used to study human health disorders. Flow cytometry experiments produce huge set of data, which are very hard to analyze with standard statistics methods and algorithms \\cite{Bernas2008}. Currently, such data is used to study the correlations of only two factors (e.g., biomarkers) at the time, by visualizing 2-dimensional histograms and by measuring the (dis-)similarity between pairs of histograms \\cite{Orlova2016}. However, during a flow cytometry experiment up to hundreds of factors (biomarkers) are measured and stored in digital format. \nHence, we can use such data to build $d$-dimensional histograms that consider up to $d$ biomarkers at the time, and then comparing the similarity among different\nindividuals by measuring the distance between the corresponding histograms.\nIn this work, we used the flow cytometry data related to {\\it Acute Myeloid Leukemia (AML)}, available at \\url{http:\/\/flowrepository.org\/id\/FR-FCM-ZZYA}, \nwhich contains cytometry data for 359 patients, classified as ``normal'' or affected by AML. \nThis dataset has been used by the bioinformatics community to run clustering algorithms,\nwhich should predict whether a new patient is affected by AML \\cite{Aghaeepour2013}.\n\nTable \\ref{tab:3} reports the results of computing the distance between pairs of $d$-dimensional histograms, with $d$ ranging in the set $\\{2,3,4\\}$,\nobtained using the AML biomedical data. Again, the first $d$-dimensional histogram plays the role of the source distribution $\\mu$, while the second\nhistogram gives the target distribution $\\nu$.\nFor simplicity, we considered regular histograms of size $n=N^d$ (i.e., $n$ is the total number of bins), using $N=16$ and $N=32$. \nTable \\ref{tab:3} compares the results obtained by the bipartite\nand $(d+1)$-partite approach, in terms of graph size and runtime. Again, the $(d+1)$-partite approach, by exploiting the structure of the ground distance,\noutperforms the standard formulation of the optimal transport problem. We remark that for $N=32$ and $d=3$, we pass for going out-of-memory\nwith the bipartite formulation, to compute the distance in around 5 seconds with the $4$-partite formulation.\n\n\\begin{table}\n\\caption{Comparison between the bipartite and the $(d+1)$-partite approaches on Flow Cytometry data.}\n\t\\label{tab:3}\n\\centering\n{\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{llrrrrrrr}\n & & & \\multicolumn{3}{c}{Bipartite Graph} & \\multicolumn{3}{c}{$(d+1)$-partite Graph} \\\\\nN & $d$ & $n$ & $|V|$ & $|A|$ & Runtime & $|V|$ & $|A|$ & Runtime \\\\\n\\hline\n$16$ & 2 & 256 & 512 & 65\\,536 & 0.024 (0.01) & 768 & 8\\,192 & 0.003 (0.00) \\\\\n& 3 & 4\\,096 & 8\\,192 & 16\\,777\\,216& 38.2 (14.0) & 16\\,384 & 196\\,608& 0.12 (0.02) \\\\\n& 4 & 65\\,536 & \\multicolumn{3}{c}{{\\it out-of-memory}} & 327\\,680 & 4\\,194\\,304& 4.8 (0.84) \\\\\n \\hline\\noalign{\\smallskip}\n$32$ & 2 & 1\\,024 & 2\\,048 & 1\\,048\\,756 & 0.71 (0.14) & 3072 & 65\\,536& 0.04 (0.01) \\\\\n& 3 & 32\\,768& \\multicolumn{3}{c}{{\\it out-of-memory}} & 131\\,072 & 3\\,145\\,728& 5.23 (0.69) \\\\\n \\hline\\noalign{\\smallskip}\n \\end{tabular}}\n\\end{table}\n\n\n\n\n\n\\section{Conclusions}\nIn this paper, we have presented a new network flow formulation on $(d+1)$-partite graphs that can speed up the optimal solution of transportation problems\nwhenever the ground cost function $c(x,y)$ (see objective function \\eqref{p1:funobj}) has a separable structure along the main $d$ directions, such as, for instance, \nthe squared $\\ell_2$ norm used in the computation of the Kantorovich-Wasserstein distance of order 2.\n\nOur computational results on two different datasets show how our approach scales with the size of the histograms $N$ and with the dimension of the histograms $d$.\nIndeed, by exploiting the cost structure, the proposed approach is better in term of memory consumption, since it has only $dn^{\\frac{d+1}{d}}$ arcs instead of $n^2$.\nIn addition, it is much faster since it has to solve an uncapacitated minimum cost flow problem on a much smaller flow network.\n\n\\subsubsection*{Acknowledgments}\nWe are deeply indebted to Giuseppe Savar\\'e, for introducing us to optimal transport and for many stimulating discussions and suggestions. \nWe thanks Mattia Tani for a useful discussion concerning the Improved Sinkhorn's algorithm.\n\nThis research was partially supported by the Italian Ministry of Education, University and Research (MIUR): \nDipartimenti di Eccellenza Program (2018--2022) - Dept. of Mathematics ``F. Casorati'', University of Pavia.\n\nThe last author's research is partially supported by ``PRIN 2015. 2015SNS29B-002. Modern Bayesian nonparametric methods''.\n\n\\section*{Additional Material}\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\nThe problem of the restoration in dense or hot matter of the chiral\nsymmetry of the strong interactions, which is spontaneously violated in\nthe QCD vacuum has been extensively addressed. The interest has largely\nfocused on the quark condensate, considered as the order parameter.\nFor independent particles the evolution of the quark condensate with\ndensity or temperature is governed by the sigma commutator of the\nparticles present in the system with the simple following expression:\n\\begin{equation}\n\\frac{<\\overline{q} q(\\rho)>}{<\\overline{q} q(0)>} = 1 -\n\\sum_n\\frac{\\rho_n^s\\;\\Sigma_n}{f_{\\pi}^2m_{\\pi}^2}\n\\label{sigma}\n\\end{equation}\nwhere the sum extends over the species present in the medium, $\\rho^s$ is\ntheir scalar density and $\\Sigma$ their sigma commutator.\n Pions play a crucial role in the restoration process especially in the heat\nbath where they enter as the lightest particles created\nby the thermal fluctuations. In the nuclear medium the main ingredients are the\nnucleons, with some corrections from the exchanged pions. At normal nuclear\n density the magnitude of the condensate has dropped by\nabout 1\/3, a large amount of restoration. It is essentially the effect of\nthe nucleons adding their effects independently, the corrections due to\nthe interaction being small.\nSuch a large amount of restoration raises the question about\nmanifestations directly linked to the symmetry. If there is no\nspontaneous violation of the symmetry, {\\it i.e.}, if it is realized in the\nWigner mode, the hadron masses vanish or there \nexist parity doublets, each hadronic state being degenerate with its\nchiral partner. It is therefore legitimate to believe that the large\namount of restoration at normal density manifests itself either by a decrease\n of the hadron masses, or by a mixing between opposite parities. A link\nbetween the evolution of the hadron masses and the amount of restoration\nhas been suggested~\\cite{BR}. But it cannot be a straightforward one. Indeed\n the density or temperature evolution of the masses\ncannot have a direct relation to that of the condensate\n, as follows from the works of several authors~\\cite{LS,EI,BIR}.\n On the other hand\nthe significance of chiral symmetry restoration for the parity mixing was\nfirst established by Dey et al.~\\cite{IOF} for the thermal case. They showed \nthat in a pion gas a mixing occurs between the vector and axial\ncorrelators. It arises from the\nemission or absorption of s-wave thermal pions, which changes the parity\nof the system. The mixing goes along with a quenching effect of the correlators,\n which, to first order in the pion density, \nequals 4\/3 of the quenching of the quark condensate. These points were also\n made by Steele et al.~\\cite{ZAH}. The extension\n of the formalism of Dey et al.\nto finite densities has been attempted by Krippa~\\cite{KRI}.\n\nThe aim of this work is the discussion of the implications of chiral symmetry\nrestoration in the nuclear medium, in\na world restricted here to nucleons and pions. The only transitions\nallowed in the nucleus are then nuclear transitions or pion production. We\ngive the explicit expressions of the axial and vector\ncurrent in a formalism based on chiral lagrangians. We will show that the\n nuclear pions renormalize the coupling\n constants of the axial current and that the renormalization\ncan be expressed in terms of the pion scalar density. This last quantity also\nenters in the quark condensate evolution. However the\ncomplexity of the nuclear interactions bars a simple link between this\n evolution, which is an average concept, and the\nrenormalizations. For instance, for the axial coupling constant $g_A$\n the detailed spatial structure of the pion scalar density is needed. \n\nOur article is organized as follows. In section 1 we derive the expressions\nof the axial and vector currents from the chiral lagrangians. In section 2\nwe use these expressions to study the renormalization of the pion decay\nconstant in the hot pion gas and in the nuclear medium. The thermal case is\nonly introduced as an illustration of the method since the results are already\nknown. In section 3 we apply the same technique to the axial coupling constant.\n To account for the nucleon-nucleon correlations we express the renormalization\nin the traditional treatment by the meson exchange currents. We give an estimate\nof the quenching of the axial coupling constant. We also discuss the\nrenormalization of the Kroll-Ruderman matrix element of pion photoproduction. \n\n\\bigskip\n\\section{The Lagrangian and the currents} \nOur starting point is the chiral Lagrangian in the form introduced by\nWeinberg. We use, as in our previous work of ref.~\\cite{DCE}, the version of\n Lynn~\\cite{LYN}, which allows one to obtain the nucleon sigma commutator\n in the tree approximation. The Lagrangian writes:\n\\begin{eqnarray}\n{\\cal L}& = & -\\frac{1}{2} m_{\\pi}^2 \\frac{\\displaystyle{\\hbox{\\boldmath$\n\\phi $\\unboldmath}}^2}{\\displaystyle 1 + {\\hbox{\\boldmath$\\phi $\\unboldmath}}\n ^2\/4f_{\\pi}^2} + \\frac{1}{2}\\frac{\\displaystyle \\partial_{\\mu}\n{\\hbox{\\boldmath$\\phi $\\unboldmath}}\\cdot \n\\partial^{\\mu}{\\hbox{\\boldmath$\\phi $\\unboldmath}}}\n{\\displaystyle(1 + {\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2)^2}\n \\nonumber \\\\\n& & + 2\\sigma_N \\overline{\\psi}\\psi\n\\frac{\\displaystyle{\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2}\n{\\displaystyle 1 + {\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2} +\n\\overline{\\psi}(i\\gamma_{\\mu}\\partial^{\\mu}-M)\\psi \\nonumber \\\\\n& & - \\frac{1}{4f_{\\pi}^2}\n\\frac{\\displaystyle\\overline{\\psi}\\gamma_{\\mu}{\\hbox{\\boldmath$(\\tau \\times\n\\phi)$\\unboldmath}} \\cdot \\partial^{\\mu}{\\hbox{\\boldmath$\\phi $\\unboldmath}}\n\\psi}{\\displaystyle 1 +\n{\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2}\n+\n\\frac{g_A}{2f_{\\pi}}\n\\frac{\\displaystyle\\overline{\\psi}\\gamma_{\\mu}\\gamma_5\n{\\hbox{\\boldmath$\\tau $\\unboldmath}}\\cdot\\partial^{\\mu}\n{\\hbox{\\boldmath$\\phi $\\unboldmath}}\\psi}{\\displaystyle 1\n+{\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2} \\; .\n\\label{lag}\n\\end{eqnarray}\n\n\n We have to specify the quantity $\\sigma_N$\nassociated with the nucleon density in eq.~(\\ref{lag}). The free nucleon\nsigma commutator $\\Sigma_N$ cannot be entirely attributed to the pion cloud.\nWe define $\\sigma_N$ to be the difference between the total and pionic \ncontributions:\n\\begin{equation}\n\\Sigma_N = \\sigma_N + \\frac{1}{2} m_{\\pi}^2 \\int\n d{\\hbox{\\boldmath$x $\\unboldmath}}\\langle N \\vert \n{\\hbox{\\boldmath$\\phi^2(x) $\\unboldmath}}\\vert N \\rangle .\n\\label{sig}\n\\end{equation}\n\nFor instance in a description of the nucleon in terms of\nvalence quarks and pions, the pionic contribution is approximatively\n1\/2 to 2\/3 of the total value~\\cite{JCT,BMG}. \n\nFrom the Lagrangian of eq.~(\\ref{lag}) we derive the expressions of the axial\n and isovector vector currents:\n\\begin{eqnarray}\n{\\hbox{\\boldmath${\\cal A} $\\unboldmath}}_{\\mu} & = & \nf_{\\pi}\\frac\n{\\displaystyle\\partial_{\\mu}{\\hbox{\\boldmath$\\phi $\\unboldmath}}}\n{\\displaystyle 1 +\n{\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2}\n-\\frac{1}{2f_{\\pi}}\\frac{\\displaystyle\\big[({\\hbox{\\boldmath$\\phi\n \\times $\\unboldmath}}\\partial_{\\mu}{\\hbox{\\boldmath$\\phi)\\times\\phi\n $\\unboldmath}}\\big]}{\\displaystyle (1 +\n{\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2)^2} \\nonumber \\\\ \n & & + \\frac{g_A}{2}\\overline{\\psi}\\gamma_{\\mu}\\gamma_5\n{\\hbox{\\boldmath$\\tau $\\unboldmath}}\\psi\n+\\frac{g_A}{4f_{\\pi}^2}\\frac{\\displaystyle\\overline{\\psi}\\gamma_{\\mu}\\gamma_5\n\\big[{\\hbox{\\boldmath$(\\tau\\times\\phi)\\times\\phi $\\unboldmath}}\\big]\\psi}\n{\\displaystyle 1 +\n{\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2}\n-\\frac{1}{2f_{\\pi}}\\frac{\\displaystyle\\overline{\\psi}\\gamma_{\\mu}\n{\\hbox{\\boldmath$(\\tau\\times\\phi) $\\unboldmath}}\\psi}\n{\\displaystyle 1 +\n{\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2}\n\\label{acur} \\\\\n& & \\nonumber \\\\\n{\\hbox{\\boldmath${\\cal V} $\\unboldmath}}_{\\mu} & = &\n\\frac{\\displaystyle({\\hbox{\\boldmath$\\phi\n \\times $\\unboldmath}}\\partial_{\\mu}{\\hbox{\\boldmath$\\phi)\n $\\unboldmath}}}{\\displaystyle( 1 +\n{\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2)^2} \\nonumber \\\\\n& & +\\frac{1}{2}\\overline{\\psi}\\gamma_{\\mu}\n{\\hbox{\\boldmath$\\tau $\\unboldmath}}\\psi\n+\\frac{1}{4f_{\\pi}^2}\\frac{\\displaystyle\\overline{\\psi}\\gamma_{\\mu}\n\\big[{\\hbox{\\boldmath$(\\tau\\times\\phi)\\times\\phi $\\unboldmath}}\\big]\\psi}\n{\\displaystyle 1 +\n{\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2}\n-\\frac{g_A}{2f_{\\pi}}\\frac{\\displaystyle\\overline{\\psi}\\gamma_{\\mu}\\gamma_5\n{\\hbox{\\boldmath$(\\tau\\times\\phi) $\\unboldmath}}\\psi}\n{\\displaystyle 1 +\n{\\hbox{\\boldmath$\\phi $\\unboldmath}}^2\/4f_{\\pi}^2}\\; .\n\\label{vcur}\n\\end{eqnarray}\n\nThe conservation law of the vector current can be shown, using the equations of\nmotion for the nucleon and the pion fields.\nThe divergence of the axial current instead satisfies the following\nrelation:\n\\begin{equation}\n\\partial^{\\mu}{\\hbox{\\boldmath${\\cal A}$\\unboldmath}}_{\\mu} = -\nf_{\\pi}m_{\\pi}^2\\;\\frac{{\\hbox{\\boldmath$\\phi$\\unboldmath}}}\n{\\displaystyle 1 + {\\hbox{\\boldmath$\\phi$\\unboldmath}}^2\/4f_{\\pi}^2}\n\\;(1-\\sigma_N\\frac{\\overline{\\psi}\\psi}{f_{\\pi}^2m_{\\pi}^2}) \\; .\n\\label{div}\n\\end{equation}\n\n\nSome comments on the expressions~(\\ref{acur}) and~(\\ref{vcur}) are in order.\n Let us first discuss the free\ncase. We recognize in some of the terms the usual expressions for the\nvector or axial current coupled to a free nucleon or pion. In addition the axial\ncurrent can create one or more pions, either in free space (first\nterms of eq.~(\\ref{acur})), or when it acts on the nucleon \nvia a term (last one of eq.~(\\ref{acur})) which is the equivalent\n for the axial current of the Weinberg-Tomozawa\nterm of $\\pi$-N scattering. Similarly the vector current acting on the\nnucleon can create one (or more) pion via the Kroll-Ruderman term, {\\it i.e.}\nthe contact piece of \nphotoproduction (last term of eq.~(\\ref{vcur})).\n\nLet us now turn to the case of a hadronic medium. \nThe expressions~(\\ref{acur}) and (\\ref{vcur}) illustrate in a striking fashion\nthe way in which the axial and vector current mixing occurs.\nIndeed, in the heat bath any of the pions can be a thermal one. As an example,\n consider the Kroll-Ruderman\n term of the vector current, ignoring at this stage the denominator. \nThe creation or annihilation of a thermal pion of momentum $q$ in this term\n takes care of the pion field, leaving a factor $e^{\\pm iqx}$ and we are left\nwith a current of opposite parity, to be taken at the momentum transfer $k\\pm q$\nwhere $k$ is the photon momentum, as in the formalism of ref.~\\cite{ZAH}.\n Similarly the pion production or annihilation by the Weinberg term of\nthe axial current introduces the vector current nuclear matrix element. \nTo the extent that the Weinberg term is mediated by the rho meson\nand the Kroll-Ruderman one by the $A_1$ meson, these expressions include the\neffects, at low momenta, of the $\\rho-A_1$ mixing. \n\nIt is interesting to observe on expressions~(\\ref{acur}) and (\\ref{vcur}) that\nthe Kroll-Ruderman term itself can be obtained from the fourth term of the\naxial current by suppression of one of the pion fields (representing creation\nor annihilation of a thermal pion). Thus the three terms containing $g_A$ in\neqs~(\\ref{acur}) and (\\ref{vcur}) are linked together by suppression or\naddition of one pion field. The same is true for the three purely pionic terms\nand for the three terms in $\\gamma_{\\mu}$ as well. Thus a grouping three by\n three of the various terms naturally emerges from our expressions.\n \nIn the nuclear medium the virtual pions can be seen as a pion bath and\nsimilar considerations about the mixing might apply. However the pions do not\ncome from an external reservoir but fully belong to the nucleus. Strictly\nspeaking there is no mixing. However the mixing terms of the currents can pick\na pion from the cloud of a nucleon introducing a similarity with the heat bath\nas displayed in fig.~\\ref{fig-figc} in the case of the Kroll-Ruderman term.\n The corresponding process is the\nexcitation of high lying nuclear states (2p-2h). In the case of the third\nisospin component of the current, it is part of the well known\nquasi-deuteron photoabsorption cross-section. Another\nexample is the influence of the Weinberg-Tomozawa term on the time part of\nthe axial current, which enters via the Pauli correlations and sizeably\nincreases the time-like axial coupling constant~\\cite{KDR}. The present \napproach puts these effects, where the mixing terms of the currents pick a pion \nfrom the cloud, in a perspective linked to chiral symmetry. \n\n\nThe mixing goes along with a renormalization of certain coupling constants,\n such as the axial one, that will now be discussed.\n \n\\bigskip\n\\section{The pion decay constant}\n We start with the case of the hot pion gas. This is meant as an illustration\nof our method as no new result is reached. It serves to introduce quantities\nsuch as the residue $\\gamma$ ({\\it i.e.}, the wave function renormalization) \nthat will be used later. \nThe production of a pion by the axial current is governed by the first two\nterms of the expression~(\\ref{acur}). Limiting the expansion to first order in\nthe squared pion field we obtain: \n\\begin{equation}\n{\\hbox{\\boldmath${\\cal A}$\\unboldmath}}_{\\mu} = \n f_{\\pi}\\partial_{\\mu}{\\hbox{\\boldmath$\\phi$\\unboldmath}}\\;(1 -\n\\frac{7}{12}\\frac{{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2}{f_{\\pi}^2}) \\; .\n\\end{equation}\nThe pion field is expanded in terms of creation and annihilation operators\n$B$ and $B^{\\dag}$ for a quasi-pion in the medium:\n\\begin{equation}\n{\\hbox{\\boldmath$\\phi$\\unboldmath}}(x) =\n\\gamma^{1\/2}\\int\\frac{d{\\hbox{\\boldmath$k$\\unboldmath}}}{(2\\pi)^3}\n\\frac{1}{(2\\omega_k^*)^{1\/2}}\n\\big( {\\hbox{\\boldmath$B_k$\\unboldmath}} +\n {\\hbox{\\boldmath$B_{-k}$\\unboldmath}}^{\\dag}\\big)\n e^{i({\\hbox{\\boldmath$k\\cdot x$\\unboldmath}}-\\omega_k^* t)}\\; ,\n\\label{pifi} \n\\end{equation}\nwhere $\\omega_k^*$ is the energy of a quasi-pion of momentum {\\bf k},\n$\\omega_k^* = \\sqrt{\\displaystyle {\\hbox{\\boldmath$k$\\unboldmath}}^2 \n+ m_{\\pi}^{*2}}$ with $m_{\\pi}^*$ the effective pion mass.\n The quantity $\\gamma$ is the residue of the pion pole. Since the derivative of\nthe pion field gives no contribution when it acts on the pions of the bath, \nthe matrix element for production of a quasi-pion by the axial current reduces\nto: \n\\begin{equation}\n\\langle 0\\vert{\\hbox{\\boldmath${\\cal A}$\\unboldmath}}_{\\mu}(0)\\vert\n \\tilde{\\pi}\\rangle\n=\\frac{\\gamma^{1\/2}}{(2\\omega_k^*)^{1\/2}} if_{\\pi} k_{\\mu}\\big(1\n-\\frac{7}{12}\\langle\\displaystyle\\frac{{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2}\n{f_{\\pi}^2}\\rangle\\big) = \\frac{1}{(2\\omega_k^*)^{1\/2}} if_{\\pi}^* k_{\\mu}\\;\n\\end{equation}\nwhere the second equation defines the renormalized pion decay constant\n$f_{\\pi}^*$.\n In a pion gas the residue $\\gamma$ has been derived by Chanfray et\nal.~\\cite{CEW}. To first order in the quantity \n${\\hbox{\\boldmath$\\phi$\\unboldmath}}^2$, equivalently\nthe pion density, it writes, in the Weinberg representation: \n\\begin{equation} \n\\gamma = \\big(1 -\n\\frac{1}{2}\\langle\\frac{{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2}\n{f_{\\pi}^2}\\rangle\\big)^{-1}\\; .\n\\label{res}\n\\end{equation}\n The renormalized pion decay constant then reads:\n\\begin{equation}\nf_{\\pi}^* = f_{\\pi} \\gamma^{1\/2}\\big(1-\\frac{7}{12}\\langle\\displaystyle\n\\frac{{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2}{f_{\\pi}^2}\\rangle\\big)\n\\approx f_{\\pi} \\big(1-\\frac{1}{3}\\langle\\displaystyle\n\\frac{{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2}{f_{\\pi}^2}\\rangle\\big) \\; .\n\\label{fpist}\n\\end{equation}\nOn the other hand, the temperature evolution of the condensate in a hot pion\ngas is, to first order in the quantity ${\\hbox{\\boldmath$\\phi$\\unboldmath}}^2$,\n as given in ref.~\\cite{CEW}:\n\\begin{equation}\n\\frac{<\\overline{q} q>_T}{<\\overline{q} q>_0} = 1 - \\frac{1}{2}\\langle\n\\displaystyle\\frac{{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2}\n{f_{\\pi}^2}\\rangle_T \\;.\n\\label{condT}\n\\end{equation}\nThus to first order in $\\langle{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2\\rangle$ \nthe renormalization of $f_\\pi$ follows the evolution of the\ncondensate but with the coefficient 2\/3, in agreement with chiral perturbation\nresults and other works~\\cite{IOF,CEW,GL,GeL}. Note that this renormalization\n applies to both space and time components of the axial current. This agrees\nwith the findings of ref~\\cite{EEK} where it is shown that to order $T^2$ \nLorentz invariance is preserved.\n\n We now turn to the dense medium. Formally we can follow the same\nprocedure. The presence in the nuclear medium of a pion scalar density, in the\nform of an expectation value of the quantity \n${\\hbox{\\boldmath$\\phi$\\unboldmath}}^2$, renormalizes the pion\ndecay constant. Formally the expression is the same as previously, \n$f_{\\pi}^* = f_{\\pi} \\gamma^{1\/2}\\big(1-\\frac{7}{12}\\langle\\displaystyle\n\\frac{{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2}{f_{\\pi}^2}\\rangle\\big)$.\nIf we treat the nuclear medium as a pion gas, the residue $\\gamma$ entirely\narises from $\\pi$-$\\pi$ interactions and is the same as given previously in\neq.~(\\ref{res}). In this simplified treatment $f_{\\pi}^*$ is given by \neq.~(\\ref{fpist}). It is linked to the pion scalar density, {\\it i.e.}\n the expectation value ${\\hbox{\\boldmath$\\phi$\\unboldmath}}^2$.\n Even with the simple form~(\\ref{fpist}), the\nrenormalization of $f_{\\pi}$ does not follow 2\/3 of the condensate one. The\nreason is that the condensate evolution in the nuclear medium is governed by\nthe full nucleon sigma commutator $\\Sigma_N$, which is not entirely due to\n${\\hbox{\\boldmath$\\phi$\\unboldmath}}^2$. There exists also the non pionic\n contribution embodied in\n$\\sigma_N$, as discussed previously. Thus the two renormalizations do not\nfollow each other. This result is general and applies as well to the axial\ncoupling constant $g_A$. \n\nThis is not the only restriction which prevents a simple link to the condensate\nin the nuclear medium. The residue $\\gamma$ itself is not entirely due to\n$\\pi$-$\\pi$ scattering. There exist other sources for the energy dependence of\nthe s-wave $\\pi$-N interaction, such as the $\\Delta$ excitation. The medium\nrenormalization of $f_{\\pi}$ cannot be written in the simple \nform~(\\ref{fpist}). This illustrates the complexity of the dense medium as\ncompared to the hot pion gas. A more phenomenological approach has been\nfollowed by Chanfray et al.~\\cite{CEK} who linked the \nin-medium pion decay constant through the nuclear\nGell-Mann-Oakes-Renner relation to the evolution of the pion mass, itself\nobtained empirically from the s-wave pion-nucleus optical potential. \n\n\\bigskip\n\\section{The axial coupling constant}\nWe now turn to the axial coupling constant. Its renormalization\n is governed by the fourth term of eq.~(\\ref{acur}).\nAfter rearrangement with the Gamow-Teller current (third term), we get:\n\\begin{equation}\n\\frac{1}{2}g_A\\overline{\\psi}\\gamma_{\\mu}\\gamma_5\\big({\\hbox{\\boldmath$\\tau$\n\\unboldmath}} +\n\\frac{1}{2f_{\\pi}^2}\\frac{{\\hbox{\\boldmath$\\phi\\tau \\cdot\\phi -\\tau\\phi^2$\n\\unboldmath}}}\n{\\displaystyle 1 + {\\hbox{\\boldmath$\\phi$\\unboldmath}}^2\/4f_{\\pi}^2}\\big)\\psi\n=\\frac{1}{2}g_A\\overline{\\psi}\\gamma_{\\mu}\\gamma_5{\\hbox{\\boldmath$\\tau$\n\\unboldmath}}\\psi\\big(1-\\frac{1}{3}\n\\langle\\frac{{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2\/f_{\\pi}^2}\n{\\displaystyle 1 + {\\hbox{\\boldmath$\\phi$\\unboldmath}}^2\/4f_{\\pi}^2}\n\\rangle_T\\big)\\; ,\n\\label{axialr}\n\\end{equation}\nwhere on the right hand side the average is taken over the heat bath.\n On the other hand the condensate is obtained from the chiral symmetry breaking\nLagrangian ${\\cal L}_{sb} = -\\frac{1}{2}m_{\\pi}^2{\\hbox{\\boldmath$\n\\phi $\\unboldmath}}^2\/( 1 + {\\hbox{\\boldmath$\\phi $\\unboldmath}}\n ^2\/4f_{\\pi}^2)$.\n Therefore the condensate evolution follows~\\cite{CEW}:\n\\begin{equation}\n\\frac{<\\overline{q} q>_{T,\\rho}}{<\\overline{q} q>_0} - 1 = \n-\\frac{1}{2}\\langle\\frac{\\displaystyle{\\hbox{\\boldmath$\n\\phi $\\unboldmath}}^2\/f_{\\pi}^2}{\\displaystyle 1 + {\\hbox{\\boldmath$\\phi $\\unboldmath}}\n ^2\/4f_{\\pi}^2}\\rangle_{T,\\rho} \\; . \n\\label{condF}\n\\end{equation}\n Hence the axial coupling constant renormalized by\n the pion loops (fig.~\\ref{fig-figb}a) can be written: \n\\begin{equation}\ng_A^*\/g_A = \\big(1 - \\frac{2}{3}\\frac{<\\overline{q} q>_{T}}\n{<\\overline{q} q>_0}\\big) \\; .\n\\label{gar}\n\\end{equation}\nThus with this chiral Lagrangian, in a hot medium the axial coupling constant\nfollows, to all orders in the pion density, 2\/3 of\nthe quark condensate evolution (as long as it is pion dominated).\n The factor 2\/3 is easily understood here: only two charges out of three\n contribute to the renormalization while all three charge\nstates participate in the condensate evolution. The quenching of $g_A$ is in\nagreement with the universal behaviour of ref.~\\cite{IOF} and with the former\nresult of ref.~\\cite{EK}.\n We have checked the expected independence\n of our results on the particular representation of the non-linear Lagrangian.\n\nWe now turn to the case of finite density. The starting expression is the same\nas the left hand side of eq.~(\\ref{axialr}). In the nuclear medium the pions\n originate from the other nucleons so that the nucleon-nucleon correlations\n cannot be ignored. Here it is useful to make the link between this\nrenormalization and the traditional picture of meson exchange currents. We keep\nonly the two-body terms which are the dominant ones and work to lowest order in\nthe pion field. \nThe corresponding graph is that of fig.~\\ref{fig-figb}b. This type of exchange\ngraph with two pions is not usually considered in nuclear physics. It is\ndictated to us only by these chiral symmetry considerations. \nWe have to express the triangular graph of the figure as an effective two-body\n operator to be\nevaluated between correlated two-nucleon wave functions. A simplification\noccurs in the static approximation where the pions do not transfer energy to\nthe nucleon line. We are left with an integral over the squared pion\npropagator with leads to a simple form in $x$-space for the two-body operator:\n\n\\begin{equation} \n O_{12} = - \\frac{1}{6f_{\\pi}^2} g_A (\\gamma_{\\mu} \\gamma_5)_1\n{\\hbox{\\boldmath$ \\big(\\tau_1 -\\frac{i}{2}(\\tau_1\\times\\tau_2)\\big)\n$\\unboldmath}}\n\\varphi^2{\\hbox{\\boldmath$ (x_1,x_2)$\\unboldmath}}\\; , \n\\label{Otb}\n\\end{equation}\nwhere $\\varphi{\\hbox{\\boldmath$(x_1,x_2)$\\unboldmath}}$\n is the Yukawa field, taken at the point \n\\boldmath$x_1$\\unboldmath, emitted by the nucleon located at the point \n\\boldmath$x_2$\\unboldmath\\ and we have made explicit the dependence in\nthe isospin operator \\boldmath$\\tau_2$\\unboldmath\\ of the emitting nucleon.\n The operator $O_{12}$ has a direct and an exchange contribution. The latter\ncontribution vanishes for the second piece of the two-body operator\n in the limit of zero momentum current. We will furthermore ignore the exchange\nterm of the first piece ({\\it i.e.} in \\boldmath$\\tau_1$\\unboldmath)\n and consider only the short range correlations. We now focus on the direct\n terms and specialize to the charged currents. The isospin factors\n in eq.~(\\ref{Otb}) reduce to the\n expression $3\\tau_1^{\\pm} - 2\\tau_1^{\\pm}\\tau_2^{\\pm}\\tau_2^{\\mp} =\n 2\\tau_1^{\\pm}(1\\mp\\tau_2^0\/2)$. The resulting contributions depend on the\nrelative number of protons and neutrons. In symmetric nuclear matter where they\nare equal, the factor, once summed over all the pion emitters (the\nnucleons with index 2), gives $2\\tau_1^{\\pm}$ multiplied by the nuclear density\n$\\rho$. In the neutron gas instead, depending whether we consider neutron decay\n(the $+$ component) or proton decay (the $-$ one), we would get a factor\n $3\\tau_1^+$ or $1\\tau_1^-$ multiplied by the neutron density $\\rho_n$.\nWe can summarize these results by introducing an effective density, which\ndepends on the charge of the current and on the neutron excess number:\n\\begin{equation}\n \\rho_{eff}^+ = \\frac{3N+Z}{2A}\\rho \\qquad \n\\rho_{eff}^- = \\frac{3Z+N}{2A}\\rho \\, \n\\label{rhoe}\n\\end{equation} \nfrom which we recover the previous results.\nSandwiching the whole\noperator $O_{12}$ between two-nucleon wave functions, we thus obtain:\n \n\\begin{equation}\n\\delta g_A^{ex}\/g_A = -\\frac{1}{3f_{\\pi}^2}\\int\nd{\\hbox{\\boldmath$x_2$\\unboldmath}}\\rho_{eff}^{\\pm}(x_2)[1+\nG({\\hbox{\\boldmath$x_1,x_2$\n\\unboldmath}})] \\varphi^2\n({\\hbox{\\boldmath$x_1,x_2$\\unboldmath}})\\; ,\n\\label{gad}\n\\end{equation}\n where $ G({\\hbox{\\boldmath$x_1,x_2$\\unboldmath}})$ is the short range\n nucleon-nucleon\n correlation function. In symmetric nuclear matter $\\rho_{eff} = \\rho$ whereas\na similar formula would hold in the neutron gas, with the obvious replacement\n of $\\rho$ by the neutron density $\\rho_n$ and of the factor 1\/3 in front by \n 1\/2 and 1\/6 for neutron and proton decay respectively. \n As is apparent on the expression~(\\ref{gad}) it is not\n the full pion field squared which acts in the renormalization\nof $g_A$, but only the part which extends beyond the range of the correlation\nhole. No such distinction occurred for the pion decay constant since the pion\n produced by the axial current can be anywhere in the nucleus. Thus the\nuniversality of the quenching which exists in the heat bath is lost. \n\nIn order to obtain an estimate for $g_A^*$ in symmetric matter, we assume a\n total exclusion of\nother nucleons in a sphere of radius $r_0 = 0.6 fm$ . In\norder to facilitate the comparison of the quenching effect of $g_A$ to\nthat of the condensate which is governed by the nucleon sigma term, we\nintroduce a quantity $(\\Sigma_N)_{eff}$:\n\\begin{equation}\n(\\Sigma_N)_{eff} = \\frac{1}{2}m_\\pi^2\\int d{\\hbox{\\boldmath$x$\\unboldmath}}\n\\theta(x-r_0) {\\hbox{\\boldmath$\\varphi^2(x)$\\unboldmath}} \\; .\n\\label{sigeff}\n\\end{equation}\nNumerically for point-like pion emitters, we find an effective value\n $(\\Sigma_N)_{eff}\\approx 21 MeV$. It is interesting to compare this value\nwith a model calculation in the quark picture. We have used the results of\nWakamatsu~\\cite{WA} in a chiral soliton model.\nThe quantity\n${\\hbox{\\boldmath$\\varphi$\\unboldmath}}^2({\\hbox{\\boldmath$x$\\unboldmath}})$ \nis replaced by the sea quark density distribution according to:\n$\\frac{1}{2}m_\\pi^2 {\\hbox{\\boldmath$\\varphi^2(x)$\\unboldmath}} \\to\n 2m_q\\overline{q} q{\\hbox{\\boldmath$(x)$\\unboldmath}} $.\nThis gives a very similar value $(\\Sigma_N)_{eff}\\approx 19 MeV$.\n \nHowever these numbers \ndo not include the Pauli blocking effect which removes\nthe occupied states in the process of pion emission. This effect\nhas been calculated in refs.~\\cite{ERC,CE} but for the whole space integral\n({\\it i.e.} without a cut-off) of the quantity\n ${\\hbox{\\boldmath$\\phi$\\unboldmath}}^2$.\n Expressed in terms of a modification of the sigma commutator it amounts\nto a reduction $(\\Delta\\Sigma_N)_{Pauli} = -2.6 MeV $.\n The blocking effect, which is moderate, should \n be even less pronounced with the cut-off. We ignore it in the following. \n\nComing back to the renormalized axial coupling constant, we have:\n\\begin{equation}\ng_A^*\/g_A = 1 - \\frac{2}{3}\\frac{\\rho(\\Sigma_N)_{eff}}{f_{\\pi}^2m_{\\pi}^2}\\; .\n\\label{gaq}\n\\end{equation} \nThis represents a 10\\% quenching at normal nuclear density in symmetric matter\n(15\\% for neutron decay in a neutron gas of the same density),\n while the condensate\nhas dropped by 35\\%. Notice that the evolution of $g_A$ is sizeably slower. \nThis quenching applies to all the components, space or time, of the axial\ncurrent. Other renormalization effects have to be added. They are known to\n act differently on the\ndifferent components. For instance the Weinberg-Tomozawa term acts on the\ntime component alone, producing a sizeable enhancement~\\cite{KDR}. In the case\nof the space component the nucleon polarization under the influence of\nthe pion field $N \\to \\Delta$ leads to the Lorentz-Lorenz\n quenching~\\cite{EFT}. In the latter case the two\nrenormalizations go in the same direction of a quenching. The extra reduction\n that we have introduced in this work could help to explain\n the large amount of quenching observed in\nGamow-Teller transitions. To get an idea, we fictitiously translate the\nreduction by chiral symmetry into an equivalent Lorentz-Lorenz effect. We\nintroduce an effective Landau-Migdal parameter $\\delta g_{N\\Delta}'$, to be\nadded to the genuine one, so as to reproduce the 10\\% quenching. This\ncorresponds to an increase $\\delta g_{N\\Delta}' \\approx 0.16$, a significant\nincrease. Indeed the quenching of the Gamow-Teller sum rule requires, if all\nattributed to the Lorentz-Lorenz effect, \n$g_{N\\Delta}'$ to be as big as 0.6 - 0.7 while the favoured theoretical value \n is around 0.4~\\cite{DIC}. Hence the chiral induced quenching would help to fill\n the gap. \n\n\nClosely related to the Gamow-Teller transition is the pion photoproduction at\nthreshold through the Kroll-Ruderman term. To lowest order the nuclear\ntransition is governed by the axial current. We want now to discuss how it is\nrenormalized in the medium following chiral symmetry requirements. Expanding to\nfirst order in ${\\hbox{\\boldmath$\\phi$\\unboldmath}}^2\/f_{\\pi}^2$\n and applying Wick theorem, the relevant current writes:\n\\begin{equation}\n({\\hbox{\\boldmath${\\cal V}$\\unboldmath}}_{\\mu})_{KR} =\n-\\frac{g_A}{2f_{\\pi}}\\overline{\\psi}\\gamma_{\\mu}\\gamma_5\n({\\hbox{\\boldmath$\\tau\\times\\phi$\\unboldmath}})\n\\psi\\big(1 - \\frac{5}{12}\\langle\\displaystyle\\frac{\n{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2}{f_{\\pi}^2}\\rangle\\big) \\; .\n\\label{KR}\n\\end{equation}\nFor the production of a quasi-pion in the medium, the renormalization $r_{KR}$\n of the amplitude involves again the residue $\\gamma$:\n\\begin{equation}\nr_{KR} = \\gamma^{1\/2}\\big(1 - \\frac{5}{12}\\langle\\displaystyle\\frac{\n{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2}{f_{\\pi}^2}\\rangle\\big) \\; .\n\\end{equation}\n \nFor illustrating the complexity of the situation we first assume that the\n residue is entirely given by $\\pi-\\pi$ scattering and\ntake the value of eq.~(\\ref{res}). Moreover we ignore the correlation\n complications. We obtain then: \n\\begin{equation}\nr_{KR} = \\big(1 - \\frac{1}{6}\\langle\\displaystyle\\frac{\n{\\hbox{\\boldmath$\\phi$\\unboldmath}}^2}{f_{\\pi}^2}\\rangle\\big) \\; ,\n\\label{rKR}\n\\end{equation}\nwhich is 1\/3 of the variation of the condensate, in contradistinction to the\naxial transitions where the factor is 2\/3. This result does not contradict the\ngeneral expressions of Dey et al.~\\cite{IOF} as the Kroll-Ruderman term\nrepresents already a mixing of the axial current into the vector one. This\nreduction factor could apply to other mixing amplitudes, but we have not\nestablished it. \nThe evolution as 1\/3 of the condensate one would apply in the hot pion gas\nsituation. In the nuclear medium all the complications mentioned previously\noccur: the role of the correlations, the link between the condensate evolution\nand the expectation value of ${\\hbox{\\boldmath$\\phi$\\unboldmath}}^2$ and\n the problem with the residue $\\gamma$.\nThis case cumulates all of the difficulties of the dense medium. In all\ninstances the overall renormalization of the Kroll-Ruderman matrix element\nin the nuclear medium should be small.\n\n\\section{Conclusion}\nIn conclusion we have investigated the behaviour of the nuclear medium in\nrelation with chiral symmetry restoration. We have\nfocused\non the extension of the parity mixing concept between the axial and vector\ncorrelators, which exists in the hot pion gas. In the nuclear medium there is\n no mixing {\\it stricto sensu}. Indeed the pions, which induce the \nmixing, are not part of an\nexternal system, as in the thermal case, but they belong to the virtual pion\ncloud which is an integral part of the nucleus. We have shown that nevertheless \ncertain consequences of the mixing survive. The nucleus behaves in certain\nrespects as a pion reservoir. The virtual pion emitted by a nucleon acts, as\nillustrated in fig.~\\ref{fig-figa}, on the\nremainder of the nucleus, {\\it i.e.} on the system of (A-1) nucleons, as the\npion of the heat bath. The mixing which does not exist at the level of the\nwhole nucleus is present only at the sublevel of the (A-1) nucleon system. This \ntranslates by the fact that, in the ``mixing'' cross-sections (such as the\nquasi-deuteron photoabsorption one), at least one nucleon has to be ejected: \nthe emitter or absorber of the\n pion. As for the heat bath, this pion reservoir produces a quenching\nof the axial coupling constants. Since the pion originates from a neighbouring\nnucleon this renormalization is nothing else than a meson exchange\ncontribution. It involves the exchange of two pions and has not been so far \nconsidered,\n to our knowledge. We have expressed the renormalizations in terms of the\npion scalar density. The same quantity also enters in the quark condensate\nevolution. One can therefore think of a link between the two quantities, as\noccurs in the heat bath where the link is simple. There is however an important\ndifference of the nuclear medium with respect to the heat bath: the\nrenormalizations are not described by a universal quenching factor expressed in\nterms of the average squared pion field. The nucleonic observables such as\n$g_A$ are renormalized differently due the sensitivity to nucleon-nucleon\nshort range correlations. In this case only that part of the pionic field\n which is beyond\nthe correlation hole enters in the renormalization.\nThis prevents the link to the condensate evolution which instead\ninvolves the average scalar density. This is an illustration of\nthe point made by T. Ericson~\\cite{TE} about the possible importance of\n the spatial fluctuations of the condensate. We have\ngiven an estimate for the quenching of the axial coupling constant arising\nfrom the requirements of chiral symmetry. Although it is not very large (about\n10\\%), this additional quenching is significant and may help explain the large observed\nquenching of the Gamow-Teller sum rule.\nWe have also discussed the photoproduction amplitude arising from the\nKroll-Ruderman term. It represents a mixing term of the axial current\ninto the vector one. We have shown that its evolution is slower than the\naxial coupling constant one.\n\nThis work can be extended to enlarge the space. The first step is to\ninclude the Delta excitation. Another extension concerns the explicit\nintroduction of the rho and the $A_1$ mesons, which the mixing of the axial and\nvector correlators allows to be excited either by the vector or by the\naxial current.\n\n\n\n\n\\bigskip\nWe thank Prof. T. Ericson for useful comments. We are very grateful to\n Prof. M. Wakamatsu for\ncommunication of his detailed results on the quark scalar density.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgrnc b/data_all_eng_slimpj/shuffled/split2/finalzzgrnc new file mode 100644 index 0000000000000000000000000000000000000000..6d47c36a2ea97eaa1f6e2dc2dff2e3a0cbf1ef0c --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzgrnc @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction}\n\\subsection{Gradient Descent at the Edge of Stability}\nAlmost all neural networks are trained using a variant of gradient descent, most commonly stochastic gradient descent (SGD) or ADAM \\citep{kingma2015adam}. When deciding on an initial learning rate, many practitioners rely on intuition drawn from classical optimization. In particular, the following classical lemma, known as the \"descent lemma,\" provides a common heuristic for choosing a learning rate in terms of the sharpness of the loss function:\n\\begin{definition}\n Given a loss function $L(\\theta)$, the sharpness is defined to be $S(\\theta) := \\lambda_{max}(\\nabla^2 L(\\theta))$. When this eigenvalue is unique, the associated eigenvector is denoted by $u(\\theta)$.\n\\end{definition}\n\\begin{lemma}[Descent Lemma]\n Assume that $S(\\theta) \\le \\ell$ for all $\\theta$. If $\\theta_{t+1} = \\theta_t - \\eta \\nabla L(\\theta_t)$,\n \\begin{align*}\n L(\\theta_{t+1}) \\le L(\\theta_t) - \\frac{\\eta \\qty(2 - \\eta \\ell)}{2} \\norm{\\nabla L(\\theta_t)}^2.\n \\end{align*}\n\\end{lemma}\nHere, the loss decrease is proportional to the squared gradient, and is controlled by the quadratic $\\eta (2 - \\eta \\ell)$ in $\\eta$. This function is maximized at $\\eta = 1\/\\ell$, a popular learning rate criterion. For any $\\eta < 2\/\\ell$, the descent lemma guarantees that the loss will decrease. As a result, learning rates below $2\/\\ell$ are considered \"stable\" while those above $2\/\\ell$ are considered \"unstable.\" For quadratic loss functions, e.g. from linear regression, this is tight. Any learning rate above $2\/\\ell$ provably leads to exponentially increasing loss.\n\nHowever, it has recently been observed that in neural networks, the descent lemma is not predictive of the optimization dynamics. Recently, \\citet{cohen2021eos} observed two interesting phenomena:\n\n\\paragraph{Progressive Sharpening} Throughout most of the optimization trajectory, the gradient of the loss is negatively aligned with the gradient of sharpness, i.e. $\\nabla L(\\theta) \\cdot \\nabla S(\\theta) < 0.$ As a result, for any reasonable learning rate $\\eta$, the sharpness increases throughout training until it reaches $S(\\theta) = 2\/\\eta$.\n\n\\paragraph{Edge of Stability} Once the sharpness reaches $2\/\\eta$, it ceases to increase and remains around $2\/\\eta$ for the rest of training. Despite the fact that the descent lemma no longer guarantees the loss decreases, the loss still continues to rapidly decrease, albeit non-monotonically (see \\Cref{fig:basicEOS}).\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=\\textwidth]{Experiments\/basic_eos.pdf}\n \\caption{\\textbf{Progressive Sharpening and Edge of Stability:} We train an MLP on CIFAR10 with learning rate $\\eta = 2\/100$. It reaches instability after around $2200$ training steps after which the sharpness hovers at $2\/\\eta = 100$, which is denoted by the horizontal dashed line.}\n \\label{fig:basicEOS}\n\\end{figure}\n\n\\subsection{Self-stabilization: The Implicit Bias of Instability}\n\nIn this work we explain the second stage, \"edge of stability.\" We identify a new implicit bias of gradient descent which we call \\textit{self-stabilization}. Self-stabilization is the mechanism by which the sharpness remains bounded around $2\/\\eta$, despite the continued force of progressive sharpening, and by which the gradient descent dynamics do not diverge, despite instability. Unlike progressive sharpening, which is only observed for neural network loss functions \\citep{cohen2021eos}, self stabilization is a general property of gradient descent.\n\n\nTraditional non-convex optimization analyses involve Taylor expanding the loss function to second order around $\\theta$ to prove loss decrease when $\\eta \\le 2\/S(\\theta)$. When this is violated, the iterates diverge exponentially in the top eigenvector direction, $u$, thus leaving the region in which the loss function is locally quadratic. Understanding the dynamics thus necessitates a \\emph{cubic} Taylor expansion.\n\nOur key insight is that the missing term in the Taylor expansion of the gradient after diverging in the $u$ direction is $\\nabla^3 L(\\theta)(u,u)$, which is conveniently equal to the gradient of the sharpness at $\\theta$:\n\\begin{lemma}[Self-Stabilization Property]\\label{lem:nabla_eig}\n If the top eigenvalue of $\\nabla^2 L(\\theta)$ is unique, then the sharpness $S(\\theta)$ is differentiable at $\\theta$ and $\\nabla S(\\theta) = \\nabla^3 L(\\theta)(u(\\theta),u(\\theta))$.\n\\end{lemma}\nAs the iterates move in the negative gradient direction, this term has the effect of \\emph{decreasing the sharpness}. The story of self-stabilization is thus that as the iterates diverge in the $u$ direction, the strength of this movement in the $-\\nabla S(\\theta)$ direction grows until it forces the sharpness below $2\/\\eta$, at which point the iterates in the $u$ direction shrink and the dynamics re-enter the quadratic regime. \n\nThis negative feedback loop prevents both the sharpness $S(\\theta)$ and the movement in the top eigenvector direction, $u$, from growing out of control. As a consequence, we show that gradient descent \\emph{implicitly} solves the \\emph{constrained minimization problem}:\n\\begin{align}\n \\min_\\theta L(\\theta) \\qqtext{such that} S(\\theta) \\le 2\/\\eta.\n\\end{align}\nSpecifically, if the stable is defined by $\\mathcal{M} := \\qty{\\theta ~:~ S(\\theta) \\le 2\/\\eta \\text{ and } \\nabla L(\\theta) \\cdot u(\\theta) = 0}$\\footnote{The condition that $\\nabla L(\\theta) \\cdot u(\\theta) = 0$ is necessary to ensure the stability of the constrained trajectory.} then the gradient descent trajectory $\\{\\theta_t\\}$ tracks the following projected gradient descent trajectory $\\{\\theta^\\dagger_t\\}$ which solves the constrained problem \\citep{barber2017pgd}:\n\\begin{align}\\label{eq:constrained_update}\n \\theta^\\dagger_{t+1} = \\mathrm{proj}_\\mathcal{M}\\qty(\\theta^\\dagger_t - \\eta \\nabla L(\\theta^\\dagger_t)) \\qq{where} \\mathrm{proj}_\\mathcal{M}\\qty(\\theta) := \\argmin_{\\theta' \\in \\mathcal{M}} \\norm{\\theta - \\theta'}.\n\\end{align}\n\nOur main contributions are as follows. First, we explain self-stabilization as a generic property of gradient descent for a large class of loss functions, and provide precise predictions for the loss, sharpness, and deviation from the constrained trajectory $\\{\\theta^\\dagger_t\\}$ throughout training (\\Cref{sec:heuristic}). Next, we prove that under mild conditions on the loss function (which we verify empirically for standard architectures and datasets), our predictions track the true gradient descent dynamics up to higher order error terms (\\Cref{sec:theory}). Finally, we verify our predictions by replicating the experiments in \\citet{cohen2021eos} and show that they model the true gradient descent dynamics (\\Cref{sec:experiments}). \n\n\n\n\n\n\n\n\\section{Related Work}\n\n\\citet{cohen2021eos} conducted an extensive empirical study showing progressive sharpening and edge of stability in a wide range of models. Prior work~\\citep{wu2018selectminimizer, Xing2018AWW} had also observed that for neural networks full-batch gradient descent reaches instability and the loss is not monotonically decreasing. \\citet{Lewkowycz2020} observed that when the initial sharpness is larger than $2\/\\eta$, gradient descent \"catapults\" into a stable region and converges.\n\nRecent works have sought to provide a theoretical analysis for the edge of stability phenomenon. \\citet{Ma2022TheMS} analyzes edge of stability when the loss satisfies a \"subquadratic growth\" assumption. \\citet{Ahn2022UnderstandingTU} argues that unstable convergence is possible when there exists a \"forward invariant subset\" near the set of minimizers. \\citet{Arora2022EoS} analyzes progressive sharpening and the edge of stability phenomenon for normalized gradient descent close to the manifold of global minimizers. \\citet{Lyu2022Normalization} uses the edge of stability phenomenon to analyze the effect of normalization layers on sharpness for scale-invariant loss functions.\n\\citet{Chen2022EoS} show global convergence despite instability for certain 2D toy problems and in a 1-neuron student-teacher setting. The concurrent work \\citet{Li2022EoS} proves progressive sharpening for a two-layer network and analyzes the edge of stability dynamics through four stages similar to ours using the norm of the output layer as a proxy for sharpness. \n\nBeyond the edge of stability phenomenon itself, prior work has also shown that SGD with large step size or small batch size will lead to a decrease in sharpness \\citep{keskar2017, Jastrzebski2017ThreeFI, jastrzebski2018on, Jastrzebski2020The}. \\citet{gilmer2022} also describes connections between edge of stability, learning rate warm-up, and gradient clipping.\n\nAt a high level, our proof relies on the idea that oscillations in an unstable direction prescribed by the quadratic approximation of the loss cause a longer term effect arising from the third-order Taylor expansion of the dynamics. This overall idea has also been used to analyze the implicit regularization of SGD~\\citep{BlancGVV20, damian2021label, li2022what}. In those settings, oscillations come from the stochasticity, while in our setting the oscillations stem from instability.\n\n\\section{Setup}\\label{sec:setup}\n\nWe denote the loss function by $L \\in C^3(\\mathbb{R}^d)$. Let $\\theta \\in \\mathbb{R}^d$ follow gradient descent with learning rate $\\eta$, i.e. $\\theta_{t+1} := \\theta_t - \\eta \\nabla L(\\theta_t)$. Recall that \n\\begin{align*}\n \\mathcal{M} := \\qty{\\theta ~:~ S(\\theta) \\le 2\/\\eta \\text{ and } \\nabla L(\\theta) \\cdot u(\\theta) = 0}\n\\end{align*}\nis the set of stable points and $\\mathrm{proj}_\\mathcal{M} := \\argmin_{\\theta' \\in \\mathcal{M}} \\norm{\\theta - \\theta'}$ is the orthogonal projection onto $\\mathcal{M}$. For notational simplicity, we will shift time so that $\\theta_0$ is the first point such that $S(\\mathrm{proj}_\\mathcal{M}(\\theta)) = 2\/\\eta$. \n\nAs in \\eqref{eq:constrained_update}, the constrained trajectory $\\theta^\\dagger$ is defined by \n\\begin{align*}\n \\theta^\\dagger_0 := \\mathrm{proj}_\\mathcal{M}(\\theta_0) \\qand \\theta^\\dagger_{t+1} := \\mathrm{proj}_{\\mathcal{M}}(\\theta^\\dagger_t - \\eta \\nabla L(\\theta^\\dagger_t)).\n\\end{align*}\n\n\nOur key assumption is the existence of progressive sharpening along the constrained trajectory, which is captured by the progressive sharpening coefficient $\\alpha(\\theta)$:\n\\begin{definition}[Progressive Sharpening Coefficient]\\label{def:alpha}\n We define $\\alpha(\\theta) := -\\nabla L(\\theta) \\cdot \\nabla S(\\theta)$.\n\\end{definition}\n\n\\begin{assumption}[Existence of Progressive Sharpening]\\label{assumption:progressive_sharpening}\n $\\alpha(\\theta^\\dagger_t) > 0$.\n\\end{assumption}\n\n\nWe focus on the regime in which there is a single unstable eigenvalue, and we leave understanding multiple unstable eigenvalues to future work. We thus make the following assumption on $\\nabla^2 L(\\theta^\\dagger_t)$:\n\n\\begin{assumption}[Eigengap]\\label{assumption:eigval_gap}\n For some absolute constant $c < 2$ we have $\\lambda_2(\\nabla^2 L(\\theta^\\dagger_t)) < c\/\\eta$.\n\\end{assumption}\n\n\\section{The Self-stabilization Property of Gradient Descent}\\label{sec:heuristic}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/stages1x4.pdf}\n \\caption{The four stages of edge of stability (see \\Cref{sec:four_stages}), demonstrated on a simple loss function (see \\Cref{sec:toy_model}).}\n \n \\label{fig:stages}\n\\end{figure}\n\nIn this section, we derive a set of equations that predict the displacement between the gradient descent trajectory $\\{\\theta_t\\}$ and the constrained trajectory $\\{\\theta^\\dagger_t\\}$. Viewed as a dynamical system, these equations give rise to a negative feedback loop, which prevents both the sharpness and the displacement in the unstable direction from diverging. These equations also allow us to predict the values of the sharpness and the loss throughout the gradient descent trajectory.\n\n\n\n\n\n\n\\subsection{The Four Stages of Edge of Stability: A Heuristic Derivation}\\label{sec:four_stages}\n\n\nThe analysis in this section proceeds by a cubic Taylor expansion around a fixed reference point $\\theta^\\star := \\theta^\\dagger_0$.\\footnote{Beginning in \\Cref{sec:theory}, the reference points for our Taylor expansions change at every step to minimize errors. However, fixing the reference point in this section simplifies the analysis, better illustrates the negative feedback loop, and motivates the definition of the constrained trajectory.}\nFor notational simplicity, we will define the following quantities at $\\theta^\\star$:\n\\begin{alignat*}{5} \n \\nabla L &:= \\nabla L(\\theta^\\star),&\\qquad \\nabla^2 L &:= \\nabla^2 L(\\theta^\\star),&\\qquad u &:= u(\\theta^\\star) \\\\\n \\nabla S &:= \\nabla S(\\theta^\\star),&\\qquad \\alpha &:= \\alpha(\\theta^\\star),&\\qquad \\beta &:= \\norm{\\nabla S}^2,\n\\end{alignat*}\nwhere $\\alpha = -\\nabla L \\cdot \\nabla S > 0$ is the progressive sharpening coefficient at $\\theta^\\star$. For simplicity, in \\Cref{sec:heuristic} we assume that $\\nabla S \\perp u$ and $\\nabla L,\\nabla S \\in \\mathrm{ker}(\\nabla^2 L)$, and ignore higher order error terms.\\footnote{We give an explicit example of a loss function satisfying these assumptions in \\Cref{sec:toy_model}.} Our main argument in \\Cref{sec:theory} does not require these assumptions and tracks all error terms explicitly.\n\nWe would like to track the movement in the unstable direction $u$ and the direction of changing sharpness $\\nabla S$, and thus define \n\\begin{align*}\n x_t := u \\cdot (\\theta_t - \\theta^\\star) \\qand y_t := \\nabla S \\cdot (\\theta_t - \\theta^\\star).\n\\end{align*}\nNote that $y_t$ is approximately to the change in sharpness from $\\theta^\\star$ to $\\theta_t$, since Taylor expanding the sharpness yields\n\\begin{align*}\n S(\\theta_t) \\approx S(\\theta^\\star) + \\nabla S \\cdot (\\theta_t - \\theta^\\star) = 2\/\\eta + y_t.\n\\end{align*}\n\nThe mechanism for edge of stability can be described in 4 stages (see \\Cref{fig:stages}):\n\\paragraph{Stage 1: Progressive Sharpening} While $x,y$ are small, $\\nabla L(\\theta_t) \\approx \\nabla L$. In addition, because $\\nabla L \\cdot \\nabla S < 0$, gradient descent naturally increases the sharpness at every step. In particular,\n\\begin{align*}\n y_{t+1} - y_t = \\nabla S \\cdot(\\theta_{t+1} - \\theta_t) \\approx - \\eta \\nabla L \\cdot \\nabla S = \\eta \\alpha.\n\\end{align*}\nThe sharpness therefore increases linearly with rate $\\eta \\alpha$.\n\\paragraph{Stage 2: Blowup} As $x_t$ measures the deviation from $\\theta^\\star$ in the $u$ direction, the dynamics of $x_t$ can be modeled by gradient descent on a quadratic with sharpness $S(\\theta_t) \\approx 2\/\\eta + y_t$. In particular, the rule for gradient descent on a quadratic gives\\footnote{A rigorous derivation of this update in terms of $S(\\theta_t)$ instad of $S(\\theta^\\star)$ requires a third-order Taylor expansion around $\\theta^\\star$; see \\Cref{sec:proofs} for more details.}\n\\begin{align*}\n x_{t+1} = x_t - \\eta u \\cdot \\nabla L(\\theta_t) \\approx x_t - \\eta S(\\theta_t) x_t \\approx x_t - \\eta \\qty[2\/\\eta + y_t] x_t = -(1 + \\eta y_t) x_t.\n\\end{align*}\nWhen the sharpness exceeds $2\/\\eta$, i.e. when $y_t > 0$, $\\abs{x_t}$ begins to grow exponentially.\n\n\\paragraph{Stage 3: Self-Stabilization} Once the movement in the $u$ direction is sufficiently large, the loss is no longer locally quadratic. Understanding the dynamics necessitates a third order Taylor expansion. The missing cubic term in the Taylor expansion of $\\nabla L(\\theta_t)$ is\n\\begin{align*}\n \\frac12\\nabla^3 L(\\theta - \\theta^\\star,\\theta - \\theta^\\star) \\approx \\nabla^3 L(u,u)\\frac{x_t^2}{2} = \\nabla S\\frac{x_t^2}{2}\n\\end{align*}\nby \\Cref{lem:nabla_eig}. This biases the optimization trajectory in the $-\\nabla S$ direction, which decreases sharpness.\nRecalling $\\beta = \\norm{\\nabla S}^2$, the new update for $y$ becomes:\n\\begin{align*}\n y_{t+1} -y_t &= \\eta \\alpha + \\nabla S \\cdot \\qty(-\\eta\\nabla^3L(u, u)\\frac{x_t^2}{2}) = \\eta\\qty(\\alpha - \\beta \\frac{x_t^2}{2})\n\\end{align*}\nTherefore once $x_t > \\sqrt{2\\alpha\/\\beta}$, the sharpness begins to decrease and continues to do so until until the sharpness goes below $2\/\\eta$ and the dynamics return to stability.\n\n\\paragraph{Stage 4: Return to Stability} At this point $\\abs{x_t}$ is still large from stages 1 and 2. However, the self-stabilization of stage 3 eventually drives the sharpness below $2\/\\eta$ so that $y_t < 0$. Because the rule for gradient descent on a quadratic with sharpness $S(\\theta_t) = 2\/\\eta + y_t$ is still\n\\begin{align*}\n x_{t+1} \\approx -(1 + \\eta y_t) x_t,\n\\end{align*}\n$\\abs{x_t}$ begins to shrink exponentially and the process returns to stage 1.\n\nCombining the update for $x_t, y_t$ in all four stages, we obtain the following simplified dynamics:\n\\begin{align}\\label{eq:simple_update}\n x_{t+1} \\approx -(1+\\eta y_t)x_t \\qand y_{t+1} \\approx y_t + \\eta\\qty(\\alpha - \\beta \\frac{x_t^2}{2})\n\\end{align}\nwhere we recall $\\alpha = -\\nabla L \\cdot \\nabla S$ is the progressive sharpening coefficient and $\\beta = \\norm{\\nabla S}^2$.\n\n\\subsection{Analyzing the simplified dynamics}\\label{sec:ode_potential}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/bean_plot_no_legend.pdf}\n \\caption{\\textbf{The effect of $\\mathbf{X(0)}$ (left):} We plot the evolution of the ODE in \\cref{eq:bean_ode} with $\\alpha = \\beta = 1$ for varying $X(0)$. Observe that smaller $X(0)$'s correspond to larger curves. \\textbf{The four stages of edge of stability (right):} We show how the four stages of edge of stability described in \\Cref{sec:four_stages} and \\Cref{fig:stages} correspond to different parts of the curve generated by the ODE in \\cref{eq:bean_ode}.}\n \\label{fig:bean_plots}\n\\end{figure}\nWe now analyze the dynamics in \\cref{eq:simple_update}. First, note that $x_t$ changes sign at every iteration, and that, $x_{t+1} \\approx -x_t$ due to the instability in the $u$ direction. While \\cref{eq:simple_update} cannot be directly modeled by an ODE due to these rapid oscillations, we can instead model $\\abs{x_t}, y_t$, whose update is controlled by $\\eta$.\nAs a consequence, we can couple the dynamics of $\\abs{x_t},y_t$ to the following ODE $X(t),Y(t)$:\n\\begin{align}\\label{eq:bean_ode}\n X'(t) = X(t) Y(t) \\qand Y'(t) = \\alpha - \\beta \\frac{X(t)^2}{2}.\n\\end{align}\nThis system has the unique fixed point $(X,Y) = (\\delta,0)$ where $\\delta := \\sqrt{2\\alpha\/\\beta}$. We also note that this ODE can be written as a Lotka-Volterra predator-prey model after a change of variables, which is a classical example of a negative feedback loop. In particular, the following quantity is conserved:\n\\begin{lemma}\n Let $h(z) := z - \\log z - 1$. Then the quantity \n \\begin{align*}\n g(X(t),Y(t)) := h\\qty(\\frac{\\beta X(t)^2}{2\\alpha}) + \\frac{Y(t)^2}{\\alpha}\n \\end{align*} is conserved.\n\\end{lemma}\n\\begin{proof} \n\\begin{align*}\n \\frac{d}{dt} g(X(t),Y(t))\n = \\frac{\\beta X(t)^2 Y(t)}{\\alpha} - 2 Y(t) + \\frac{2}{\\alpha} Y(t) \\qty[\\alpha - \\beta \\frac{X(t)^2}{2}] = 0. \\qquad\\qedhere\n\\end{align*}\n\\end{proof}\nAs a result we can use the conservation of $g$ to explicitly bound the size of the trajectory:\n\\begin{corollary}\n For all $t$, $X(0) \\le X(t) \\lesssim \\delta\\sqrt{\\log(\\delta\/X(0))}$ and $\\abs{Y(t)} \\lesssim \\sqrt{\\alpha \\log(\\delta\/X(0))}$.\n\\end{corollary}\nThe fluctuations in sharpness are $\\tilde O(\\sqrt{\\alpha})$, while the fluctuations in the unstable direction are $\\tilde O(\\delta)$. Moreover, the \\emph{normalized} displacement in the $\\nabla S$ direction, i.e. $\\frac{\\nabla S}{\\norm{\\nabla S}} \\cdot (\\theta - \\theta^\\star)$ is also bounded by $\\tilde O(\\delta)$, so the entire process remains bounded by $\\tilde O(\\delta)$. Note that the fluctuations increase as the progressive sharpening constant $\\alpha$ grows, and decrease as the self-stabilization force $\\beta$ grows.\n\n\n\\subsection[Relationship with the constrained trajectory]{Relationship with the constrained trajectory $\\theta^\\dagger_t$}\n\\Cref{eq:simple_update} completely determines the displacement $\\theta_t - \\theta^\\star$ in the $u,\\nabla S$ directions and \\Cref{sec:ode_potential} shows that these dynamics remain bounded by $\\tilde O(\\delta)$ where $\\delta = \\sqrt{2\\alpha\/\\beta}$. However, progress is still made in all other directions. Indeed, $\\theta_t$ evolves in these orthogonal directions by $-\\eta P_{u,\\nabla S}^\\perp \\nabla L$ at every step where $P_{u,\\nabla S}^\\perp$ is the projection onto this orthogonal subspace. This can be interpreted as first taking a gradient step of $-\\eta \\nabla L$ and then projecting out the $\\nabla S$ direction to ensure the sharpness does not change. \\Cref{lem:dagger_step}, given in the Appendix, shows that this is precisely the update for $\\theta^\\dagger_t$ (\\cref{eq:constrained_update}) up to higher order terms. The preceding derivation thus implies that $\\|\\theta_t - \\theta^\\dagger_t\\| \\le \\tilde O(\\delta)$ and that this $\\tilde O(\\delta)$ error term is controlled by the self-stabilizing dynamics in \\cref{eq:simple_update}.\n\n\n\\section{The Predicted Dynamics and Theoretical Results}\\label{sec:theory}\nWe now present the equations governing edge of stability for general loss functions.\n\\subsection{Notation}\nOur general approach Taylor expands the gradient of each iterate $\\theta_t$ around the corresponding iterate $\\theta^\\dagger_t$ of the constrained trajectory. We define the following Taylor expansion quantities at $\\theta^\\dagger_t$:\n\\begin{definition}[Taylor Expansion Quantities at $\\theta^\\dagger_t$]~\n\\begin{gather*}\n \\nabla L_t := \\nabla L(\\theta^\\dagger_t) \\qc \\nabla^2 L_t := \\nabla^2 L(\\theta^\\dagger_t) \\qc \\nabla^3 L_t := \\nabla^3 L(\\theta^\\dagger_t) \\\\ \\nabla S_t := \\nabla S(\\theta^\\dagger_t) \\qc\\qquad u_t := u(\\theta^\\dagger_t).\n\\end{gather*}\nFurthermore, for any vector-valued function $v(\\theta)$, we define $v_t^\\perp := P_{u_t}^\\perp v(\\theta^\\dagger_t)$ where $P_{u_t}^\\perp$ is the projection onto the orthogonal complement of $u_t$.\n\\end{definition}\n\nWe also define the following quantities which govern the dynamics near $\\theta^\\star_t$.\n\\begin{definition}\\label{def:alpha_beta_epsilon} Define\n\\begin{align*}\n \\alpha_t := -\\nabla L_t \\cdot \\nabla S_t \\qc \\beta_t := \\norm{\\nabla S_t^\\perp}^2 \\qc \\delta_t := \\sqrt{\\frac{2\\alpha_t}{\\beta_t}} \\qand \\delta := \\sup_t \\delta_t.\n\\end{align*}\nFurthermore, we define \n\\begin{align*}\n \\beta_{s \\to t} := \\nabla S_{t+1}^\\perp\\qty[\\prod_{k = t}^{s+1} (I - \\eta \\nabla^2L_k)P_{u_k}^\\perp] \\nabla S_s^\\perp.\n\\end{align*}\n\\end{definition}\nRecall that $\\alpha_t$ is the progressive sharpening force, $\\beta_t$ is the strength of the stabilization force, and $\\delta_t$ controls the size of the deviations from $\\theta^\\dagger_t$ and was the fixed point in the $x$ direction in \\Cref{sec:ode_potential}. The scalars $\\beta_{s \\to t}$ capture the effect of the interactions between $\\nabla S$ and the Hessian.\n\n\n\n\\subsection{The equations governing edge of stability}\nWe now introduce the equations governing edge of stability. We track the following quantities:\n\\begin{definition}\\label{def:vxy}\n Define $v_t := \\theta_t - \\theta^\\dagger_t$, $x_t := u_t \\cdot v_t$, $y_t := \\nabla S_t^\\perp\\cdot v_t$.\n\\end{definition}\nOur predicted dynamics directly predict the displacement $v_t$ and the full definition is deferred to \\Cref{sec: define predicted dynamics}. However, they have a relatively simple form in the $u_t,\\nabla S_t^\\perp$ directions:\n\n\\begin{lemma}[Predicted Dynamics for $x,y$]\\label{lemma:predicted_x_y}\n Let $\\accentset{\\star}{v}_t$ denote our predicted dynamics (defined in \\Cref{sec: define predicted dynamics}). Letting $\\accentset{\\star}{x}_t = u_t \\cdot \\accentset{\\star}{v}_t$ and $\\accentset{\\star}{y}_t = \\nabla S_t^\\perp \\cdot \\accentset{\\star}{v}_t$, we have\n \\begin{align}\\label{eq:predicted_x_y_only}\n \\accentset{\\star}{x}_{t+1} = - (1 + \\eta \\accentset{\\star}{y}_t)\\accentset{\\star}{x}_t \\qand \\accentset{\\star}{y}_{t+1} = \\eta\\sum_{s = 0}^t \\beta_{s \\to t}\\qty[\\frac{\\delta_s^2 - {\\accentset{\\star}{x}_s}^2}{2}].\n\\end{align}\n\\end{lemma}\nNote that when $\\beta_{s \\to t}$ are constant, our update reduces to the simple case discussed in \\Cref{sec:heuristic}, which we analyze fully. When $x_t$ is large, \\cref{eq:predicted_x_y_only} demonstrates that there is a self-stabilization force which acts to decrease $y_t$; however, unlike in \\Cref{sec:heuristic}, the strength of this force changes with $t$.\n\n\\subsection{Coupling Theorem}\nWe now show that, under a mild set of assumptions which we verify to hold empirically in \\Cref{sec:verify_assumptions}, the true dynamics are accurately governed by the predicted dynamics. This lets us use the predicted dynamics to predict the loss, sharpness, and the distance to the constrained trajectory $\\theta^\\dagger_t$.\n\nOur errors depend on the unitless quantity $\\epsilon$, which we verify is small in \\Cref{sec:verify_assumptions}.\n\\begin{definition}\n Let $\\epsilon_t := \\eta \\sqrt{\\alpha_t}$ and $\\epsilon := \\sup_t \\epsilon_t$.\n\\end{definition}\nTo control Taylor expansion errors, we require upper bounds on $\\nabla^3 L$ and its Lipschitz constant:\\footnote{For simplicity of exposition, we make these bounds on $\\nabla^3 L$ globally, however our proof only requires them in a small neighborhood of the constrained trajectory $\\theta^\\dagger$.}\n\\begin{assumption}\\label{assume:rho4}\n Let $\\rho_3$, $\\rho_4$ to be the minimum constants such that for all $\\theta$, $\\norm{\\nabla^3 L(\\theta)}_{op} \\le \\rho_3$ and $\\nabla^3 L$ is $\\rho_4$-Lipschitz with respect to $\\norm{\\cdot}_{op}$. Then we assume that $\\rho_4 = O(\\eta \\rho_3^2)$.\n\\end{assumption}\nNext, we require the following generalization of \\Cref{assumption:progressive_sharpening}: \n\\begin{assumption}\\label{assume:prog_general}\n \n For all $t$,\n\\begin{align*}\n \\frac{-\\nabla L_t \\cdot \\nabla S_t}{\\norm{\\nabla L_t}\\norm{\\nabla S_t^\\perp}} = \\Theta(1) \\qand \\norm{\\nabla S_t^\\perp} = \\Theta(\\rho_3).\n\\end{align*}\n\\end{assumption}\n\nFinally, we require a set of ``non-worst-case\" assumptions, which are that the quantities $\\nabla^2 L, \\nabla^3 L,$ and $\\lambda_{min}(\\nabla^2 L)$ are nicely behaved in the directions orthogonal to $u_t$, which generalizes the eigengap assumption. We verify the assumptions on $\\nabla^2 L$ and $\\nabla^3 L$ empirically in \\Cref{sec:verify_assumptions}.\n\\begin{assumption}\\label{assume:non_worst} For all $t$ and $v,w \\perp u_t$,\n\\begin{align*}\\frac{\\norm{\\nabla^3L_t(v, w)}}{\\norm{\\nabla^3 L_t}_{op}\\norm{v}\\norm{w}} \\qc \\frac{|\\nabla^2 L_t( \\accentset{\\star}{v}^\\perp_t,\\accentset{\\star}{v}^\\perp_t)|}{\\norm{\\nabla^2 L_t}\\norm{\\accentset{\\star}{v}^\\perp_t}^2} \\qand \\frac{\\abs{\\lambda_{min}(\\nabla^2 L_t)}}{\\|\\nabla^2 L_t\\|_2} \\le O(\\epsilon).\n\\end{align*}\n\\end{assumption}\n\n \n\n\nWith these assumptions in place, we can state our main theorem which guarantees $\\accentset{\\star}{x},\\accentset{\\star}{y},\\accentset{\\star}{v}$ predict the loss, sharpness, and deviation from the constrained trajectory up to higher order terms:\n\\begin{theorem}\\label{thm:coupling}\n Let $\\mathscr{T} := O(\\epsilon^{-1})$ and assume that $\\min_{t \\le \\mathscr{T}} \\abs{\\accentset{\\star}{x}_t}\\ge c_1\\delta.$ Then for any $t \\le \\mathscr{T}$, we have\n \\begin{align*}\n L(\\theta_t) &= L(\\theta^\\dagger_t) + \\accentset{\\star}{x}_t^2\/\\eta + O\\qty(\\epsilon \\delta^2\/\\eta) \\tag{Loss} \\\\\n S(\\theta_t) &= 2\/\\eta + \\accentset{\\star}{y}_t + \\qty(S_t \\cdot u_t) \\accentset{\\star}{x}_t + O\\qty(\\epsilon^2\/\\eta) \\tag{Sharpness} \\\\\n \\theta_t &= \\theta^\\dagger_t + \\accentset{\\star}{v}_t + O\\qty(\\epsilon\\delta) \\tag{Deviation from $\\theta^\\dagger$}\n \\end{align*}\n\\end{theorem}\nThe sharpness is controlled by the slowly evolving quantity $\\accentset{\\star}{y}_t$ and the period-2 oscillations of $(\\nabla S \\cdot U) \\accentset{\\star}{x}_t$. This combination of gradual and rapid periodic behavior was observed by \\citet{cohen2021eos} and appears in our experiments. \\Cref{thm:coupling} also shows that the loss at $\\theta_t$ spikes whenever $\\accentset{\\star}{x}_t$ is large. On the other hand, when $\\accentset{\\star}{x}_t$ is small, $L(\\theta_t)$ approaches the loss of the constrained trajectory.\n\n\n\n\n\\section{Experiments}\\label{sec:experiments}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{Experiments\/loss_sharpness.pdf}\n \\caption{We empirically demonstrate that the predicted dynamics given by \\cref{eq:predicted_x_y_only} track the true dynamics of gradient descent at the edge of stability. For each learning rate, the top row is a zoomed in version of the bottom row which isolates one cycle and is reflected by the dashed rectangle in the bottom row. Reported sharpnesses are two-step averages for visual clarity. For additional experimental details, see \\Cref{sec:experiments} and \\Cref{sec:experimental_details}.}\n \n \\label{fig:loss_sharpness}\n\\end{figure}\n\nWe verify that the predicted dynamics defined in \\cref{eq:predicted_x_y_only} accurately capture the dynamics of gradient descent at the edge of stability by replicating the experiments in \\citep{cohen2021eos} and tracking the deviation of gradient descent from the constrained trajectory. In \\Cref{fig:loss_sharpness}, we evaluate our theory on a 3-layer MLP and a 3-layer CNN trained with mean squared error (MSE) on a 5k subset of CIFAR10 and a 2-layer Transformer \\citep{Vaswani2017AttentionIA} trained with MSE on SST2 \\citet{socher-etal-2013-recursive}. We provide additional experiments varying the learning rate and loss function in \\Cref{sec:additional_experiments}, which use the generalized predicted dynamics described in \\Cref{sec:generalized_discussion}. For additional details, see \\Cref{sec:experimental_details}.\n\n\\Cref{fig:loss_sharpness} confirms that the predicted dynamics \\cref{eq:predicted_x_y_only} accurately predict the loss, sharpness, and distance from the constrained trajectory. In addition, while the gradient flow trajectory diverges from the gradient descent trajectory at a linear rate, the gradient descent trajectory and the constrained trajectories remain close \\emph{throughout training}. In particular, the dynamics converge to the fixed point $(\\abs{x_t},y_t) = (\\delta_t,0)$ described in \\Cref{sec:ode_potential} and $\\|\\theta_t - \\theta^\\dagger_t\\| \\to \\delta_t$. This confirms our claim that gradient descent implicitly follows the constrained trajectory~\\cref{eq:constrained_update}.\n\nIn \\Cref{sec:theory}, various assumptions on the model were made to obtain the edge of stability behavior. In \\Cref{sec:verify_assumptions}, we numerically verify these assumptions to ensure the validity of our theory.\n\n\n\n\n\\section{Discussion}\\label{sec:discussion}\n\n\\subsection{Takeaways from the Predicted Dynamics}\n\nThe predicted dynamics enable many interesting observations about the edge of stability dynamics. First, the loss and sharpness only depend on the quantities $(\\accentset{\\star}{x}_t, \\accentset{\\star}{y}_t)$, which are governed by the 2D dynamical system with time-dependent coefficients \\cref{eq:predicted_x_y_only}. When $\\alpha_t, \\beta_{s \\to t}$ are constant, we showed that this system cycles and has a conserved potential. In general, understanding the edge of stability dynamics only requires analyzing the 2D system \\cref{eq:predicted_x_y_only}, which is generally well behaved (\\Cref{fig:loss_sharpness}).\n\nIn the limit, we expect $\\accentset{\\star}{x}_t, \\accentset{\\star}{y}_t$ to approach $(\\pm \\delta_t, 0)$, the fixed point of the system \\cref{eq:predicted_x_y_only}.\nIn fact, \\Cref{fig:loss_sharpness} shows that after a few cycles, $(\\accentset{\\star}{x}_t, \\accentset{\\star}{y}_t)$ indeed converges to this fixed point. We are able to accurately predict its location, as well as the loss increase from the constrained trajectory due to $\\accentset{\\star}{x}_t \\neq 0$.\n\n\\subsection{Generalized Predicted Dynamics}\\label{sec:generalized_discussion}\nIn order for our cubic Taylor expansions to track the true gradients, we need a bound on the fourth derivative of the loss (\\Cref{assume:rho4}). This is usually sufficient to capture the dynamics at the edge of stability as demonstrated by \\Cref{fig:loss_sharpness} and \\Cref{sec:verify_assumptions}. However, this condition was violated in some of our experiments, especially when using logistic loss. To overcome this challenge, we developed a generalized form of the predicted dynamics whose definition we defer to \\Cref{sec:generalized_dynamics}. These generalized predictions are qualitatively similar to those given by the predicted dynamics in \\Cref{sec:theory}; however, they precisely track the dynamics of gradient descent in a wider range of settings. See \\Cref{sec:additional_experiments} for empirical verification of the generalized predicted dynamics.\n\n\n\n\\subsection{Implications for Neural Network Training}\n\\paragraph{Non-Monotonic Loss Decrease} A central phenomenon at edge of stability is that despite non-monotonic fluctuations of the loss, the loss still decreases over long time scales. Our theory provides a clear explanation for this decrease. We show that the gradient descent trajectory remains close to the constrained trajectory (\\Cref{sec:heuristic,sec:theory}). Since the constrained trajectory is \\emph{stable}, it satisfies a descent lemma (\\Cref{lem:dagger_descent}), and has monotonically decreasing loss. Over short time periods, the loss is dominated by the rapid fluctuations of $x_t$ described in \\Cref{sec:heuristic}. Over longer time periods, the loss decrease of the constrained trajectory due to the descent lemma overpowers the bounded fluctuations of $x_t$, leading to an overall loss decrease. This is reflected in our experiments in \\Cref{sec:experiments}.\n\n\n\\paragraph{Generalization \\& the Role of Large Learning Rates} Prior work has shown that in neural networks, both decreasing sharpness of the learned solution~\\citep{keskar2017, dziugaite2017, neyshabur2017, Jiang2020Fantastic} and increasing the learning rate~\\citep{smith2018dont, li2019large, Lewkowycz2020} are correlated with better generalization. Our analysis shows that gradient descent implicitly constrains the sharpness to stay near $2\/\\eta$, which suggests larger learning may improve generalization by reducing the sharpness. In \\Cref{fig:training_speed} we confirm that in a standard setting, full-batch gradient descent generalizes better with large learning rates.\n\n\\paragraph{Training Speed} Additional experiments in \\citep[Appendix F]{cohen2021eos} show that, despite the instability in the training process, larger learning rates lead to faster convergence. \nThis phenomenon is explained by our analysis. Gradient descent is coupled to the constrained trajectory which minimizes the loss while constraining movement in the $u_t,\\nabla S_t^\\perp$ directions. Since only two directions are ``off limits,\" the constrained trajectory can still move quickly in the orthogonal directions, using the large learning rate to accelerate convergence. We demonstrate this empirically in \\Cref{fig:training_speed}. \n\n\nWe defer additional discussion of our work, including the effect of multiple unstable eigenvalues and connections to Sharpness Aware Minimization~\\citep{foret2021sharpnessaware}, warm-up~\\citep{gilmer2022}, and scale-invariant loss functions~\\citep{Lyu2022Normalization} to~\\Cref{sec:additional_discussion}.\n\n\\subsection{Future Work}\n\nAn important direction for future work is understanding the dynamics when there are multiple unstable eigenvalues, which we briefly discuss in~\\Cref{sec:additional_discussion}. Another interesting direction is understanding the \\emph{global} convergence properties at the edge of stability, including convergence to a KKT point of the constrained update \\cref{eq:constrained_update}.\nNext, our analysis focused on the edge of stability dynamics but left open the question of why neural networks exhibit progressive sharpening.\nFinally, we would like to understand the role of self-stabilization in \\emph{stochastic}-gradient descent and how it interacts with the implicit biases of SGD \\citep{BlancGVV20, damian2021label, li2022what}.\n\n\\section*{Acknowledgements}\nAD acknowledges support from a NSF Graduate Research Fellowship. EN acknowledges support from a National Defense Science \\& Engineering Graduate Fellowship. JDL, AD, EN acknowledge support of the ARO under MURI Award W911NF-11-1-0304, the Sloan Research Fellowship, NSF CCF 2002272, NSF IIS 2107304, NSF CIF 2212262, ONR Young Investigator Award, and NSF-CAREER under award \\#2144994. \n\nThe authors would like to thank Jeremy Cohen, Kaifeng Lyu, and Lei Chen for helpful discussions throughout the course of this project.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nPrecision measurements of the Cosmic Microwave Background (CMB)\nradiation have, in recent years, enormously advanced our understanding\nof the origins, content and structure of our universe. Measurements\nof secondary anisotropies induced, not at the surface of last\nscattering ($z\\approx 1100$), but in the more recent universe ($z\\sim\n\\mathcal{O}(1)$) also provide detailed information about the late time\nevolution of the universe enabling, for example, measurements of the Hubble constant\nthrough the Sunyaev-Zel'dovich (SZ) effect (for a review see \\cite{2002ARA&A..40..643C}), and measurements of the\nproperties of dark energy through the late time Integrated Sachs-Wolfe\neffect (ISW) \\cite{Sachs:1967er,Giannantonio:2008zi}.\n\nThese precision measurements can also be used to test `new physics',\nincluding the existence of new light, weakly interacting particles\nif they influence the propagation of photons. In this article we\nfocus in particular on mini-charged particles (MCPs). MCPs are\nparticles with a small and not necessarily quantized charge. Such\nparticles arise naturally in extensions of the Standard Model which\ncontain at least one additional U(1) hidden sector gauge\ngroup~\\cite{Holdom:1985ag,Bruemmer:2009ky}. The gauge boson of this\nadditional U(1) is known as a hidden photon, and hidden sector\nparticles, charged under the hidden U(1) get an induced electric\ncharge proportional to the small mixing angle between the kinetic\nterms of the two photons. In string theory such hidden U(1)s and the\nrequired kinetic mixing are a generic\nfeature~\\cite{Abel:2006qt,Abel:2008ai,Dienes:1996zr,Lukas:1999nh,Lust:2003ky,Abel:2003ue,Blumenhagen:2005ga,Blumenhagen:2006ux,Mark}.\nHidden photons are not {\\it necessary} however to explain\nmini-charged particles, and explicit brane world scenarios have been\nconstructed \\cite{Batell:2005wa} where MCPs arise without the need\nfor hidden photons.\n\nThe existence of MCPs would lead to the decay of photons in\nthe presence of electric or magnetic fields \\cite{Gies:2006ca,Ahlers:2006iz}. This has lead to\nsearches for MCPs in high precision optical experiments (BFRT \\cite{PhysRevD.47.3707},\nPVLAS \\cite{zavattini:110406}, Q\\&A \\cite{Chen:2006cd}, BMV \\cite{Rizzo} and OSQAR~\\cite{Pugnat:2005nk}) where\n a laser beam is passed\nthrough a transverse magnetic field and the real and virtual production of MCPs leads to rotation and ellipticity\nof the polarization of the beam. This signal\ndiffers depending on whether or not the model under examination includes\nhidden photons. In addition the presence of hidden photons can lead\nto more exotic effects, such as light-shining-through-walls \\cite{Ahlers:2007rd,Ahlers:2007qf}.\n\nFor a wide range of parameters, however, more stringent constraints\non MCPs come from observations in astrophysics and cosmology\n\\cite{Davidson:1993sj,Davidson:2000hf}. In particular extensions of the Standard\nModel which include MCPs must be in agreement with the bounds of Big\nBang Nucleosynthesis (BBN), and must not lead to overly fast cooling\nof white dwarf and red giant stars. However it has been shown\n\\cite{Masso:2006gc} that in models containing more than one hidden\nphoton, where at least one of the hidden photons is massless, the\nbounds obtained in settings with high density\/temperature can be\nconsiderably relaxed. Most prominently this affects bounds from\nenergy loss considerations in stars. Therefore alternative\nconstraints obtained in low density\/temperature environments are of\nparticular interest~\\cite{Melchiorri:2007sq,Ahlers:2009kh}. For this\nwe turn to observations of the CMB; the light we observe from the\nCMB has traveled solely through low density\/temperature\nenvironments, and therefore constraints on MCPs derivable from the\nCMB also apply to those models which avoid the stellar evolution\nbounds. These constraints would be of direct relevance for upcoming\nlaboratory searches for MCPs which are typically performed under\nvacuum conditions. The light from the CMB passes through magnetic\nfields in the neighborhood of galaxy clusters, where real and\nvirtual MCPs are produced exactly as in laboratory experiments. The\nanisotropies induced by such interactions contribute to the\nstandard late time CMB anisotropies. In this article we show that\nobservations of these effects can be used to constrain new regions\nof the MCP parameter space.\n\n\nCluster magnetic fields are well understood on distance scales at\nwhich the SZ effect dominates over the ISW effect, but little is\nknown about magnetic fields in the\nlargest structures in the universe. Therefore in this article we\nmainly focus on an MCP contribution to the SZ effect. When photons from the CMB pass through galaxy clusters there is a\nsmall probability that they will interact with an energetic electron\nin the plasma of the intracluster medium. If this happens the photons\ncan be Thomson scattered up to higher energies, distorting the CMB\nspectrum. This is the Sunyaev-Zel'dovich (SZ) effect\n\\cite{Zeldovich:1969ff,Sunyaev:1972eq}. The temperature distortions\ninduced in the CMB have the form\n\\begin{equation}\n\\frac{\\Delta T}{T} =f\\left(\\frac{\\omega}{T_{CMB}}\\right)\\int\\frac{n_e\nT_e \\sigma_T}{m_e}\\;dl,\n\\end{equation}\nwhere $\\omega=2\\pi \\nu$ is the photon energy, $T_{CMB}$ the CMB temperature, $n_e$ the\nelectron number density in the plasma, $T_e$ the electron temperature,\n$\\sigma_T$ the Thomson scattering cross section and $m_e$ the mass of\nthe electron\\footnote{We are using units $k_B=\\hbar=c=1$.}.\nThe function $f(x)$ describes the frequency dependence\nof the SZ effect and in the non-relativistic and Rayleigh-Jeans\n($\\omega\\ll T$) limits $f(x)\\rightarrow -2$. The integral is along a\nline of sight through a cluster.\nA typical galaxy cluster is expected to induce temperature anisotropies of the\norder $10^{-4}$ in the CMB spectrum. Photons are lost in the long\nwavelength part of the CMB spectrum $\\nu \\lesssim 218 \\mbox{ GHz}$ and\nthere is an increase in the power of the spectrum at higher\nfrequencies. There are now a large number of high quality\nmeasurements of the SZ effect for a variety of clusters. The physics of the SZ\neffect is reviewed in \\cite{Birkinshaw199997} and the current\nobservations are reviewed in \\cite{2000cucg.confE..43C}.\n\n Constraints on MCPs from\nobservations of the\nCMB have also been derived from processes where two CMB photons\nannihilate into two MCPs \\cite{Melchiorri:2007sq} and in the region of\nparameter space where MCPs do not decouple from the acoustic\noscillations of the baryon-photon plasma at recombination \\cite{Dubovsky:2003yn}. Other cosmological\nprobes of MCPs have also been considered, including their effect on\nthe propagation of light from type Ia supernovae\n\\cite{Ahlers:2009kh}. Modifications of the SZ effect by chameleonic\naxion-like-particles have also been discussed \\cite{Davis:2009vk}.\n\n\nThe outline of this article is as follows. In Section \\ref{sec:optics} we\ndescribe the effect of MCPs on the propagation of photons through\nmagnetic fields. We solve the equations of motion and compute the\nsurvival probability for photons. In Section \\ref{sec:SZ} we show\nhow measurements of the SZ effect can be used to constrain MCPs, in\nSection \\ref{sec:coma} we give the constraints on MCP models that come from\n observations of the Coma cluster and in Section \\ref{sec:hyper} we\n discuss the relevance of these constraints to hyperweak gauge\n interactions in the LARGE volume scenario of string theory. Section \\ref{sec:ISW} describes how,\nin the future, observations of the ISW effect may also be used to\nconstrain MCPs, and we conclude in Section \\ref{sec:conc}.\n\n\n\\section{Optics with MCPs and hidden photons}\n\\label{sec:optics}\nPhoton propagation in the presence of mini-charged particles can be studied\nin the Holdom model \\cite{Holdom:1985ag} starting with the most general\nLagrangian\n\\begin{equation}\n{\\cal L} = -\\frac{1}{4}\\tilde F_{\\mu\\nu}\\tilde F^{\\mu\\nu}-\\frac{1}{4}\\tilde B_{\\mu\\nu}\\tilde B^{\\mu\\nu}\n-\\frac{\\sin\\chi}{2}\\tilde B_{\\mu\\nu}\\tilde F^{\\mu\\nu} +\n\\tilde e j_{\\rm em}^\\mu \\tilde A_\\mu + e_h j_{\\rm h}^\\mu \\tilde B_\\mu,\n\\end{equation}\ndescribing visible sector photons $\\tilde A_\\mu$, hidden sector\nphotons $\\tilde B_\\mu$, and visible and hidden sector matter fields,\nwritten here as the currents $j_{\\rm em}^\\mu$ and $j_{\\rm h}^\\mu$\nrespectively. The visible and hidden photons have field strength\ntensors $\\tilde F_{\\mu\\nu}$ and $\\tilde B_{\\mu\\nu}$ respectively, and $\\chi$ controls the\nstrength of the kinetic mixing between the photon fields.\nThe visible and hidden sector gauge couplings are $\\tilde e$ and $e_h$.\n\n\nThe following change of variables\n\\begin{equation}\n\\label{shift}\n\\tilde A = \\frac{1}{\\cos \\chi}A \\quad ; \\quad \\tilde B = B- \\tan \\chi A,\n\\end{equation}\ndiagonalizes the kinetic part of the Lagrangian, which can then be written as\n\\begin{equation}\n\\label{lag}\n{\\cal L} = -\\frac{1}{4} F_{\\mu\\nu}F^{\\mu\\nu}-\\frac{1}{4} B_{\\mu\\nu} B^{\\mu\\nu} +\ne j_{\\rm em}^\\mu A_\\mu + e_h j_{\\rm h}^\\mu B_\\mu + \\epsilon e j^{\\mu}_{\\rm h} A_\\mu,\n\\end{equation}\nwith $\\epsilon = (e_h\/e)\\tan \\chi \\quad$ and $\\quad\ne = \\tilde e\/\\cos \\chi$.\nThe last term of (\\ref{lag}) gives an effective charge under the visible sector gauge group to the hidden\nmatter.\nIf either $\\chi$ or $e_{\\rm h}$ are small the effective charge of the\nhidden matter has a naturally small value $|\\epsilon|\\approx e_{h}\\chi\/e\\ll 1$.\n\n\n\nIn the presence of a strong transverse magnetic field the hidden matter contributes to the refraction\nand absorption coefficients of photons and hidden photons\n\\cite{Gies:2006ca} through the complex refractive index,\n$\\epsilon^2e^2\\Delta N_i(\\epsilon e\n\\mathbf{B},m_{\\epsilon})=\\epsilon^2e^2(\\Delta\nn_i+i\\frac{1}{2\\omega}\\kappa_i)=n_i-1$, for a photon of frequency\n$\\omega$ and an MCP with mass $m_{\\epsilon}$. $i=\\perp,\\parallel$\nlabels the photons polarized parallel and perpendicular to the\ndirection of the magnetic field. The real parts, $\\Delta n_i$, are the\nrefractive indices and the imaginary parts, $\\kappa_i$, are\nthe absorption coefficients due to the production of MCPs. Full expressions for $\\Delta N_i$\nare given in Refs.~\\cite{Erber:1966vv,Tsai:1974fa,Gies:2006ca,Ahlers:2006iz}.\nThe equations of motion derived from the Lagrangian (\\ref{lag}) are\n\\begin{equation}\n\\label{eom}\n\\left[(\\omega^2+\\partial_z^2)\\left(\\begin{array}{cc}\n1 & 0\\\\\n0 & 1\n\\end{array}\\right)-\\left(\\begin{array}{cc}\n\\omega_{\\rm P}^2 + \\mu^2\\chi^2 & - \\mu^2\\chi\\\\\n- \\mu^2\\chi & \\mu^2\n\\end{array}\\right)\\right]\\left(\\begin{array}{c}\nA\\\\\nB\n\\end{array}\\right)=0,\n\\end{equation}\nwhere we have defined\n\\begin{equation}\n\\mu^2 = -2\\omega^2e_h^2\\Delta N_i\n\\end{equation}\nand $\\tan(\\chi)\\to \\chi$. $\\omega_{\\rm P}^2=4\\pi^2\\alpha n_e\/m_e$ is the plasma\nfrequency depending on the fine structure constant, the mass of an\nelectron, $m_e$, and the number density, $n_e$, of free electrons in a plasma.\nThe effective mass $\\mu$, normalized by the MCP mass, depends only on\n the polarization of the light and on the\nadiabatic parameter\n\\begin{equation}\n\\lambda = \\frac{3}{2}\\frac{\\omega}{m_\\epsilon}\\frac{\\epsilon e B}{m_\\epsilon^2}.\n\\end{equation}\nThe dependence of $\\mu^2\/m^2_\\epsilon$ on $\\lambda$ is shown in\nFigure ~\\ref{fig:mu2}.\n\n\\begin{figure}[tbp]\n\\centering\n{\n\\psfragscanon\n\\psfrag{a}[][l]{$\\lambda=\\frac{3}{2}\\frac{\\omega}{m_\\epsilon}\\frac{\\epsilon e B}{m_\\epsilon^2}$}\n\\psfrag{b}[][l][0.8]{}\n\\psfrag{c}[][l][0.8]{}\n\\psfrag{d}[][l][0.8]{\\vspace{1cm}\\hspace{1cm} $\\begin{array}{c}\\ |{\\rm Re} (\\mu^2)| $ - - - $ \\\\ |{\\rm Im} (\\mu^2)|\\ $-----$ \\end{array}$}\n\\includegraphics[width=8cm]{mu3.eps}\n}\n\\caption{\\small The absolute value of the real (solid) and imaginary (dashed) parts of the MCP-induced mass $\\mu^2$\nare shown for photon polarization parallel (black) and perpendicular (red) to the magnetic field direction.\nThe MCP particle is a Dirac spinor with mass $m_\\epsilon$ and electric charge $\\epsilon$. The scalar case is\nvery similar. They only depend on the adiabatic parameter $\\lambda$.\nThe imaginary part of $\\mu^2$ is always negative while the real part is negative\nfor $\\lambda \\lesssim 20$ and becomes positive for larger\nvalues.}\n\\label{fig:mu2}\n\\end{figure}\n\nSolving the equations of motion, the propagating eigenstates are\n\\begin{equation}\n\\label{prop}\nV_+(t,z) = \\left(\\begin{array}{c} 1 \\\\ - a \\end{array} \\right) e^{i (\\omega t - k_+ z)} \\quad ; \\quad\nV_-(t,z) = \\left(\\begin{array}{c} a \\\\ 1 \\end{array} \\right) e^{i (\\omega t - k_- z)},\n\\end{equation}\nwhere the momenta are\n\\begin{eqnarray}\nk_\\pm &=& \\sqrt{\\omega^2-m_\\pm^2 }\\simeq \\omega -\\frac{m_\\pm^2}{2\\omega}\\equiv \\omega - \\Delta_\\pm, \\\\\n\\label{mpm}\n2 m_\\pm^2 &=& \\omega_{\\rm P}^2+\\mu^2(1+\\chi^2)\\pm \\sqrt{(\\omega_{\\rm P}^2-\\mu^2(1-\\chi^2))^2+4\\mu^4\\chi^2},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{a}\na=\\frac{\\mu^2}{m_+^2-\\mu^2}\\chi.\n\\end{eqnarray}\nTherefore a state which is purely photon-like initially at $z=0$ evolves as\n\\begin{equation}\n\\label{Vz}\nV(t,z) = \\left(\\frac{1}{1+a^2}V_+(0,0)e^{i\\Delta_+ z} + \\frac{a}{1+a^2}V_-(0,0)e^{i\\Delta_- z}\\right)e^{i\\omega(t-z)}.\n\\end{equation}\nThe photon survival amplitude is the first component of this vector\nand, from this, the photon survival probability is\n\\begin{equation}\n\\label{genprob}\nP_{\\gamma\\to\\gamma}(z)=\\left|\\frac{1}{1+a^2}\\right|^2\\Bigl(e^{-2 {\\rm Im}\\Delta_+z} + |a|^4e^{-2 {\\rm Im}\\Delta_-z} +2{\\rm Re} \\{a^2 e^{i (\\Delta_--\\Delta^*_+)z}\\}\\Bigr)\n\\end{equation}\nThe last term inside the bracket in (\\ref{genprob}) is oscillatory and can be neglected when\nthe phase is large $\\phi = {\\rm Re}\\{\\Delta_--\\Delta^*_+\\} z \\gg 1$,\ncorresponding to situations where a large number of oscillations\noccur within the propagation length $z$.\n\nThe expression for the survival probability (\\ref{genprob}) breaks down\nwhen $a^2=-1$ or, equivalently, when\n\\begin{equation}\n\\label{resonance*}\n\\mu^2= \\frac{\\omega_{\\rm P}^2}{(1\\pm i \\chi)^2}.\n\\end{equation}\nThis is the point in phase space where the photon and hidden photon\nare exactly degenerate.\nAs the imaginary and real parts of $\\mu^2$ satisfy\n\\begin{equation}\n\\label{caca}\n{{\\rm Im}}\\{ \\mu^2\\} > {{\\rm Re}}\\{ \\mu^2\\} \\hspace{2cm} {\\rm for} \\hspace{1.5cm} {{\\rm Re}}\\{ \\mu^2\\}>0 \\ ,\n\\end{equation}\nthe condition (\\ref{resonance*}) can only be satisfied for values of $\\chi\\sim {\\cal O}(1)$.\nIn this paper we restrict ourselves to considering small values of\n$\\chi$, which are not only more realistic from the theoretical point of view but also are not excluded by laboratory experiments.\nTherefore we are always far from the resonance.\n\nIn the small mixing regime $\\chi \\ll 1$ a simpler formula for the photon survival probability (\\ref{genprob})\ncan be obtained.\nExpanding (\\ref{mpm}) and (\\ref{a}) around $\\chi=0$ we find\n\\begin{eqnarray}\nm_+^2 &=& \\omega_{\\rm P}^2\\(1+a\\chi\\) \\quad ; \\quad m_-^2 = \\mu^2\\(1-a\\chi\\) \\\\\na &=& \\frac{\\mu^2}{\\omega_{\\rm P}^2-\\mu^2}\\chi\n\\end{eqnarray}\nSo that the photon survival probability becomes\n\\begin{equation}\nP_{\\gamma\\to\\gamma}(z)=1-2{{\\rm Re}}\\{a^2\\}-\\chi\\frac{z\\omega_P^2}{\\omega}{{\\rm Im}}\\{a\\}+2{{\\rm Re}}\\{a^2e^{-iz(\\omega_P^2-\\mu^2)\/2\\omega}\\}+\\mathcal{O}(\\chi^4).\n\\label{smallchi}\n\\end{equation}\nThe last term in this expression is exponentially damped and when the\ndistances under consideration satisfy $z{{\\rm Im}}\\{\\mu^2\\}\\gg2\\omega$ it\ncan be safely neglected.\n\nAs mentioned in the introduction the inclusion of the hidden photon\nis not necessary for the existence of MCPs but it certainly\nprovides one of the few natural theoretical explanations of the small mini-charges.\nUse of the framework described above imposes no restriction\non the origin of MCPs because the hidden photon field can be consistently decoupled by\nformally sending $e_h\\to 0$ whilst keeping $\\epsilon$ constant. This can be seen\ndirectly from the Lagrangian (\\ref{lag}) where the only term that\nconnects the hidden photon to the other fields\n is proportional to $e_h$\\footnote{This decoupling can also be seen in the matrix form of\nthe equations of motion (\\ref{eom}).\nAs $\\mu^2$ equals $e_h^2$ multiplied by some function $f(\\epsilon)$ the $A-A$ matrix element\nis $\\propto \\chi^2 e_h^2 f(\\epsilon)=\\epsilon^2e^2 f(\\epsilon)$ and will stay constant in the decoupling limit,\nhowever the $A-B$ element is $\\propto \\chi e_h^2 f(\\epsilon)=e_h \\epsilon e f(\\epsilon)$ and vanishes as\n$e_h\\to 0$.}.\nNote that $m_+^2$ is a constant in this limit but $\\chi \\mu^2$ and\n$\\mu^2$ vanish and therefore the mixing parameter $a$ also\ntends to zero.\nWhen the hidden photon decouples the photon survival probability\nbecomes simply\n\\begin{equation}\nP=e^{-2 {\\rm Im}\\Delta_+z}.\n\\end{equation}\n\n\n\\section{Using the Sunyaev-Zel'dovich effect to constrain MCPs}\n\\label{sec:SZ}\nAs discussed in the introduction, magnetic fields exist in clusters of\ngalaxies. As light from the CMB traverses these fields the properties\nof this light can be affected by the real and virtual production of\nMCPs. Clusters of galaxies are some of the largest objects in the\n universe; a typical cluster contains $\\sim 10^3$ galaxies in a\n region $\\sim 2$ Mpc in radius. The magnetic fields of these\n clusters are relatively well understood \\cite{Carilli}, and it is\n common to model these magnetic fields as being made up of a large\n number of equally sized magnetic domains. Within each domain the\n magnetic field is constant, and the magnitude of the magnetic field\n strength is constant over the whole cluster, but within each domain\n the direction of the magnetic field vector is essentially a random\n variable. Photons passing\nthrough such clusters may interact with the cluster magnetic\nfield and convert into real or virtual MCPs. This loss of photons would\nlook like a contribution to the SZ effect of the form\n\\begin{equation}\n\\frac{\\Delta T}{T}=\\frac{1-e^{-x}}{x}\\frac{\\Delta I}{I_0},\n\\label{MCPSZ}\n\\end{equation}\nwhere $I_0$ is the photon flux\nfrom the CMB, $\\Delta I$ is the flux decrement due to MCPs and\n$x=\\omega\/T_{CMB}$, and $ T_{CMB}$ is the\ntemperature of the CMB radiation today.\nThe best constraints would come from a cluster for which both the SZ effect\nand the properties of its magnetic field have been directly measured.\nThis is uniquely\nthe case for the Coma cluster (Abell 1656) which lies at a redshift\n$z=0.023$. The properties of the Coma cluster will be discussed\nfurther in Section \\ref{sec:coma}.\n\n\nIn order to constrain the MCP contribution to the SZ effect we must\ncompute the flux deficit induced as the photons propagate through a large\nnumber of randomly oriented magnetic domains. As the CMB is a {\\it very} non\ncompact source, light from the CMB takes many\ndifferent paths through the random magnetic field of a\ngalaxy cluster and we need only compute the\neffects of MCP production averaged over this large class of paths.\n\nWe will assume that the size of a magnetic domain is sufficiently\nlarge that the final term in the photon survival probability\n(\\ref{smallchi}) can be neglected. This is consistent whenever\n${{\\rm Im}}\\{\\mu^2\\}\\gg2\\omega\/L$, where $L$ is the size of the\ndomain. Then the photon\nsurvival probability can be written as $P_i(z)\n=1-p_i-q_iz$ for $i=\\parallel,\\perp$, where $p=2{{\\rm Re}}\\{a^2\\}$,\n$q=\\chi\\omega^2_P{{\\rm Im}}\\{a\\}\/\\omega$. $a$ is given by (\\ref{a}). To evolve the system through the cluster we will need to\naverage over the two angles $\\theta_n$ and $\\psi_n$ that determine the\ndirection of the magnetic field in the n-th domain. The average over\n$\\psi_n$, the angle of inclination of the magnetic field to the\ndirection of motion of the photons, we will absorb into a\nconservative order of magnitude estimate for $B$. Letting $\\theta_n$ be the\norientation of the magnetic field in the plane perpendicular to the\ndirection of motion of the photons, the\nphoton components at the start of the $(n+1)$-th domain are related to those at the start of the $n$-th\ndomain by\\footnote{In principle the propagation is described by a $4\\times4$ matrix. However, after the initial damping only the $V_{+}$ eigenmodes\nsurvive and we can use effectively a $2\\times 2$ description.}\n\\begin{eqnarray}\n\\left(\\begin{array}{c}\nA_x\\\\\nA_y\n\\end{array}\\right)_{n+1}&=&\\left[ \\left(1-\\frac{\\delta_{n1}\\langle p\\rangle+\\langle\n q\\rangle L}{2}\\right)\\mathbb{1}\\right.\\nonumber\\\\\n& &+\\left.\\frac{\\delta_{n1}dp +dqL}{2}\\left(\\begin{array}{cc}\n\\cos 2\\theta_n & \\sin2\\theta_n\\\\\n\\sin2\\theta_n & -\\cos2\\theta_n\n\\end{array}\\right)\\right]\\left(\\begin{array}{c}\nA_x\\\\\nA_y\n\\end{array}\\right)_{n}\n\\end{eqnarray}\nwhere $\\langle q\\rangle =(q_{\\bot}+q_{\\parallel})\/2$ and\n$dq=(q_{\\bot}-q_{\\parallel})\/2$, with equivalent definitions for\n$\\langle p\\rangle$ and $dp$.\nAssuming that, on average, $\\cos^2\\theta_n=\\sin^2\\theta_n=1\/2$ and\n$\\cos\\theta_n=\\sin\\theta_n=0$ it can be shown that after\npassing through $N$ magnetic domains the average photon flux $I_{N}$, is\nrelated to the initial photon flux, $I_0$, by\n\\begin{equation}\nI_{N}=I_0\\Bigl(1-\\langle p\\rangle-N\\langle q\\rangle L +\\mathcal{O}(N^2\\langle q \\rangle^2)\\Bigr).\n\\label{flux}\n\\end{equation}\nCombining equations (\\ref{MCPSZ}) and (\\ref{flux}) we can constrain the CMB\ntemperature anisotropies induced in MCP models to be less than those\nof the SZ effect.\n\nIn the general case we compute the photon survival probabilities\n numerically, but there are two limits which can be\nunderstood analytically.\nIf the plasma frequency is sufficiently large $\\omega_{\\rm P}^2\\gg |\\mu^2|$, the\nimaginary part of the mass of the photon like state (+) is well\napproximated by ${{\\rm Im}}\\{m_+^2\\} =\\chi^2 {{\\rm Im}}\\{ \\mu^2\\} = -\\epsilon^2 e^2 \\kappa \\omega$.\nThis regime is equivalent to the decoupling of the hidden photon\ndiscussed in Section \\ref{sec:optics}\n($e_h\\to 0$, $\\mu^2\\to 0$ but $\\chi^2\\mu^2=\\mbox{constant}$).\nIn this regime the mixing angle $a$ is suppressed not only by $\\chi$\nbut also by the ratio $\\mu^2\/\\omega_{\\rm P}^2$ and the photon survival probability is simply\n\\begin{equation}\nP(\\gamma\\to \\gamma) = 1-z\\epsilon^2e^2\\kappa + {\\cal O}(a^2) .\n\\label{nohp}\n\\end{equation}\nSo $\\langle p\\rangle=0$ and $\\langle q\\rangle=\\epsilon^2e^2\\langle\\kappa\\rangle$ in (\\ref{flux}).\n\nIn the opposite limit, $|\\mu^2|\\gg \\omega_{\\rm P}^2$, the situation also simplifies, recovering the expressions derived\nin \\cite{Ahlers:2007rd}. The mixing angle is only suppressed by $\\chi$ and is\nreal in the limit $\\omega_{\\rm P}\\to 0$. The mass of the photon like state (+)\nstill has an imaginary part, but this is suppressed by the small quantity $\\omega_{\\rm P}^2\/|\\mu^2|$,\n\\begin{equation}\n {{\\rm Im}}\\, \\{m^2_+\\} \\simeq \\chi^2{{\\rm Im}}\\, \\{\\mu^2\\} \\frac{\\omega_{\\rm P}^4}{|\\mu^2|^2} = \\omega\\epsilon^2 e^2\\kappa \\frac{\\omega_{\\rm P}^4}{|\\mu^2|^2}.\n\\end{equation}\nIn this limit the photon survival probability is\n\\begin{equation}\nP(\\gamma\\to \\gamma) \\simeq 1-2\\chi^2-4\\chi^2\\frac{\\omega^2_P{{\\rm Re}}\\{\\mu^2\\}}{|\\mu^2|^2}-\\chi^2\\frac{z\\omega^2_P{{\\rm Im}}\\{\\mu^2\\}}{\\omega|\\mu^2|^2}+\\mathcal{O}\\left(\\frac{\\omega_P^4}{|\\mu^2|^2}\\right).\n\\label{Pnoomega}\n\\end{equation}\nIn the limit $\\omega_{\\rm P}\\to 0$ the photon survival probability becomes constant\n\\begin{equation}\n\\lim_{\\omega_{\\rm P}\\to 0}P(\\gamma\\to \\gamma) \\simeq 1-2\\chi^2 .\n\\label{justhp}\n\\end{equation}\nSo $\\langle p\\rangle =2\\chi^2$ and $\\langle q\\rangle =0$ in (\\ref{flux}).\nThis can be understood by considering equation\n(\\ref{Vz}). In the $\\omega_{\\rm P}\\to 0$ limit, $a\\to -\\chi$ and an initial photon state\nis mainly $V_+$ with a very small (proportional to $\\chi^2$) component of $V_-$.\nIn this case, the $V_-$ state is the original hidden photon $\\tilde{B}_\\mu$ which,\nby definition, is the state that couples to the hidden sector\nparticles.\nTherefore only the $V_-$ component of the initial state can be damped by pair production of\nMCPs and after traveling distances $z\\gtrsim 2\\omega\\ln (\\chi)\/{{\\rm Im}}\\{\\mu^2\\}$ the final state\nis $V_+\/(1+\\chi^2)$. Squaring the amplitude of this gives the photon\nsurvival probability (\\ref{justhp}).\n\n\n\n\\subsection{Constraints from the Coma cluster}\n\\label{sec:coma}\n The most detailed information\nabout the strength and structure of the magnetic fields in clusters of\ngalaxies typically comes from measurements of Faraday rotation of\nlight at radio frequencies. The presence of a magnetic field in an\nionized plasma, such as the intracluster medium, sets a preferred direction for the gyration of\nelectrons, leading to a difference in the index of refraction for left\nand right circularly polarized radiation as it passes through the\nplasma. This is equivalent to a rotation of\nthe plane of polarization of linearly polarized light, known as the\nFaraday rotation, which depends on the thermal electron density and\nthe magnetic field strength. By taking Faraday rotation measurements\nof an entire cluster, not only the magnetic field strength, but also\nits coherence length, can be estimated. It has been shown\n\\cite{Gies:2006ca,Ahlers:2006iz}, however, that the\ninteractions of photons and mini-charged particles in a transverse magnetic field also lead to the rotation of polarization, and\nthese effects are expected to be strongest at low frequencies.\nTherefore it is unclear whether the magnetic field strengths calculated from Faraday rotation\nmeasurements can be reliably used in the computation of CMB\ntemperature anisotropies due to MCPs.\n\nFaraday rotation is not, however, the only way to estimate the\nmagnetic field strength of a cluster. For the Coma cluster a hard\nX-ray flux has been measured exceeding the thermal emission. This has\nits origin in the inverse Compton scattering of photons by relativistic\nelectrons which are accelerated by the magnetic field of the galaxy cluster. This inference of the magnetic field is not based on subtle\npolarization measurements and therefore we expect it to be essentially free of\ncontamination by MCPs. Hard X-ray\nobservations of the Coma cluster imply a magnetic field\nstrength of $0.16\\times 10^{-10}\\mbox{ T}$ \\cite{1999A&A...344..409E,1999ApJ...520..529S}, roughly an order of magnitude\nsmaller than that inferred from Faraday rotation measurements \\cite{1990ApJ...355...29K}.\nWhilst, within the Standard Model, it is thought that these two sets of observations can be\nreconciled by more realistic cluster models \\cite{Brunetti}. The\nsize of a magnetic domain can be estimated from images of the Faraday\nrotation of the Coma cluster, $L\\approx 10 \\mbox{ kpc}$,\n\\cite{1990ApJ...355...29K} and the size of these correlations\nwill not be affected by mixing with MCPs.\n\n\nWe note that it would be possible to derive constraints on MCP models by requiring that\nthe\nMCP induced rotation of linearly polarized light at radio frequencies\nbe less than the measured Faraday rotation \\cite{1990ApJ...355...29K}. However these\nconstraints are subsumed by the constraints coming from measurements of\nthe SZ effect, which we focus on for the remainder of this article.\n\nTo compute the contribution of MCPs to the SZ effect we also need to\nknow the electron density in the intracluster plasma. This can be inferred, for the Coma cluster, from soft X-ray\nmeasurements taken during the ROSAT all sky survey\n\\cite{1992A&A...259L..31B}. From these observations it is inferred that $n_e=2.89\\times 10^{-3}\\mbox{ cm}^{-3}$\nin the core of the cluster and $n_e\\approx 1 \\times\n10^{-3}\\mbox{ cm}^{-3}$ away from the core. This corresponds to a\nplasma frequency of $\\omega_P\\approx 10^{-12} \\mbox{ eV}$.\n\nThe SZ effect of the Coma cluster has been measured precisely in\na number of frequency bands \\cite{1538-4357-598-2-L75}, as shown in\nTable \\ref{table:obs}.\n\\begin{table}[ht]\n\n\\centering \n\\begin{tabular}{|c| c| c| c|} \n\\hline\nInstrument & & Temperature Decrement &Observational Frequency\\\\[0.5ex] \n\\hline \nOVRO & \\cite{1538-4357-449-1-L5} &$\\Delta T = -520\\pm 83 \\mbox{\n $\\mu$K}$ &32.0 GHz \\\\\nWMAP & \\cite{0067-0049-148-1-97} & $\\Delta T = -240\\pm 180 \\mbox{\n $\\mu$K}$ & 60.8 GHz\\\\\nWMAP & \\cite{0067-0049-148-1-97} & $\\Delta T = -340\\pm 180\n\\mbox{ $\\mu$K}$ & 93.5 GHz\\\\\nMITO & \\cite{1538-4357-574-2-L119} &$\\Delta T = -184\\pm 39\n\\mbox{ $\\mu$K}$ & 143 GHz\\\\\nMITO & \\cite{1538-4357-574-2-L119} & $\\Delta T = -32\\pm 79\n\\mbox{ $\\mu$K}$ & 214 GHz\\\\\nMITO & \\cite{1538-4357-574-2-L119} & $\\Delta T = 172\\pm 36\n\\mbox{ $\\mu$K}$ & 272 GHz\\\\ [1ex]\n\\hline \n\\end{tabular}\n\\caption{SZ observations of the Coma Cluster}\n\\label{table:obs} \n\\end{table}\t\nFor each observation we constrain the temperature decrement due to\nproduction of MCPs with the largest measured temperature decrement\nwithin $2\\sigma$ error bars. All the observations detailed above\ngive constraints on the parameters of the MCP model of the same order\nof magnitude, with those by MITO at 214 GHz, being the most\nconstraining. The numerical bounds quoted below come from this measurement.\n\nTo calculate the constraints on MCPs from the SZ measurements of the\nComa cluster we assume that in most of the volume of the\nComa cluster $B\\approx 1\\times 10^{-11}\\mbox{ T}$, which estimate\nincludes an averaging over the angles $\\psi_n$, and that the size\nof a magnetic\ndomain is $L\\approx 10\\mbox{ kpc}$. The cluster magnetic fields\nextend out to a radius of at least 1 Mpc, so we assume that light from\nthe CMB traverses approximately 100 domains. We take the average plasma\nfrequency to be $\\omega_P=10^{-12}\\mbox{ eV}$. The MCP induced\ntemperature anisotropies (\\ref{MCPSZ}) at $\\nu \\sim 214\\mbox{ GHz}$ are,\nfrom (\\ref{flux}), constrained to be\n\\begin{equation}\n\\frac{\\Delta T}{T}=0.25\\Bigl(\\langle p\\rangle+NL\\langle q\\rangle\\Bigr) < 7.0 \\times 10^{-5}.\n\\end{equation}\nOur assumption that the last term in (\\ref{smallchi}) can be safely\nneglected when calculating the survival probability in each domain of\nthe Coma cluster means that our constraints are valid for\n${{\\rm Im}}\\{\\mu^2\\}\\gg2\\omega\/L$, which for observations at 214 GHz implies\n${{\\rm Im}}\\{\\mu^2\\}\\gg1.1\\times 10^{-30}\\mbox{ eV}^2$.\n\n\\begin{figure}[tbp]\n{ \n\\psfragscanon\n\\psfrag{a}[][l]{$m_\\epsilon$ [eV]}\n\\psfrag{b}[][l]{$\\epsilon$}\n\\psfrag{ss1}[][l][0.7]{$e_h=10^{-5}$}\n\\psfrag{ss2}[][l][0.7]{$e_h=10^{-4}$}\n\\psfrag{ss3}[][l][0.7]{$e_h=10^{-3}$}\n\\psfrag{ss4}[][l][0.7]{$e_h=3\\times 10^{-3}$}\n\\psfrag{ss5}[][l][0.7]{$e_h=10^{-2}$}\n\\psfrag{ss6}[][l][0.7]{$e_h=0.1$}\n \\includegraphics[height=11cm]{ISW} }\n\\caption{The constraints of the SZ effect on MCPs. The solid black\n line shows the upper bound in the mini-charge, MCP mass phase space on models which contain MCPs but no\n hidden photons (we consider a Dirac fermion, but scalars are very similar). The gray region shows the excluded region for\n models which include hidden photons. The different plots show\n different values of the hidden sector gauge coupling, given in units\n of the electric charge.}\n\\label{fig:caca}\n \\end{figure}\n\nThe constraints this imposes on the charge and mass of MCPs are shown\nin Figure \\ref{fig:caca}. The upper bound on models without hidden\nphotons is shown as a solid line. The gray regions are those\nexcluded for models which include both hidden sector photons\nand MCPs. The upper bound on these regions comes from requiring\nthat $\\chi\\equiv\\epsilon e \/e_h < 1$. The excluded region in the $(m_{\\epsilon},\\epsilon)$\nplane for models with hidden photons varies significantly\nwith $e_h$. For small values of the hidden sector gauge coupling the\nconstraints are indistinguishable from those of the pure MCP case. This is the $|\\mu^2|\\ll\\omega_{\\rm P}^2$\nlimit discussed in the previous section. For larger gauge couplings the\nconstraints on the mini-charge are much weaker because in this region\nof the parameter space only the hidden photon component of the initial\nstate is damped by the production of MCPs. This corresponds to\nthe $|\\mu|^2\\gg \\omega_P^2$ limit.\n\nIn certain regions of the parameter space it is possible to understand\nthese limits analytically. When the adiabatic parameter is large\n$\\lambda\\gg1$ an analytic expression for the complex refractive index exists.\n For light at 214 GHz passing through the magnetic fields of the Coma\ncluster large adiabatic parameter corresponds to\n\\begin{equation}\n\\epsilon^{1\/3} \\gg 1.1\\times 10^4\n\\left(\\frac{m_{\\epsilon}}{\\mbox{eV}}\\right).\n\\end{equation}\nand this corresponds to the region on the left of\nFigure \\ref{fig:caca} where the constraints are independent of the\nMCP mass.\n\nWhen the adiabatic parameter is large and $|\\mu|^2\\ll\n\\omega_P^2$, corresponding to $ e_h\\epsilon^{1\/3}\\ll 7\\times\n10^{-8}$, the hidden photons effectively decouple. The\nphoton survival probability is given by (\\ref{nohp}) and measurements of the SZ effect in the\nComa cluster constrain\n\\begin{equation}\n\\epsilon < 4\\times 10^{-10}.\n\\end{equation}\nIn the alternative limit $|\\mu|^2\\gg \\omega_P^2$ the photon survival\nprobability is given by (\\ref{justhp}) and measurements of the SZ effect constrain\n\\begin{equation}\n\\chi<1\\times 10^{-2},\n\\end{equation}\nor equivalently\n\\begin{equation}\n\\epsilon<3\\times 10^{-2}e_h.\n\\end{equation}\n\\subsection{Hyperweak hidden gauge couplings}\\label{sec:hyper}\nAbove we have seen that our bounds are strongest for mini-charged particles without hidden photons and for very small hidden sector gauge couplings.\nAlthough the first case is interesting in itself, let us also briefly motivate the case of small hidden sector gauge couplings.\nIndeed we will see below that our bounds start to penetrate a theoretically very interesting region.\n\nHyperweak gauge interactions~\\cite{Burgess:2008ri} are a typical feature in so-called LARGE volume scenarios in string theory.\nIn LARGE volume scenarios the string scale $M_{s}$ is related to the Planck scale $M_{P}$ (and the string coupling $g_{s}$) via\n\\begin{equation}\nM^2_{P}=\\frac{4\\pi}{g^2_{s}}{\\mathcal{V}}M^{2}_{s},\n\\end{equation}\nwith a LARGE volume ${\\mathcal{V}}=V_{6}M^{6}_{s}$ of the six compactified dimensions.\nDue to the LARGE volume the string scale can now be much lower than the Planck scale.\nFor example for a volume of the order of\n${\\mathcal{V}}\\sim 10^{12}$ one obtains a string scale $M_{s}\\sim 10^{11}\\,{\\rm GeV}$. This scale is particularly interesting in scenarios where\na supersymmetry breaking of size $\\sim M^{2}_{s}$ is mediated to the Standard Model by gravity resulting in masses for the superpartners of the order\nof $\\sim M^{2}_{s}\/M_{P}\\sim 1\\,{\\rm TeV}$.\nFor an even larger volume ${\\mathcal{V}}\\sim 10^{27}$ the string scale itself could be as low as a TeV.\n\nIn the LARGE volume scenarios gauge groups live on D7 branes with $7+1$ dimensions. The extra 4 space dimensions are removed by wrapping the brane\naround a cycle in the extra dimensions. The gauge coupling is then given by\n\\begin{equation}\ng^{2}=\\frac{2\\pi g_{s}}{{\\mathcal{V}}_{4}}\\sim \\frac{2\\pi g_{s}}{{\\mathcal{V}}^{\\frac{2}{3}}},\n\\end{equation}\nwhere ${\\mathcal{V}}_{4}=V_{4}M^{4}_{s}$ is the volume of the compactified 4 extra dimensions of the brane.\nIn the last step we have assumed that the brane extents through\nmost of the LARGE volume and the latter has roughly the same extent in all directions.\nIn this case the gauge coupling will be very small. (The Standard Model gauge groups live on smaller cycles or singularities and accordingly have\nlarger gauge couplings $\\sim 1$.)\nTypical values for these hyperweak gauge couplings are of the order\n\\begin{equation}\ne_{h}\\sim \\bigg\\{\\begin{array}{clcc}\n 10^{-4}& \\sim 10^{-3}\\,e & {\\rm for} & {\\mathcal{V}}\\sim 10^{12} \\\\\n 10^{-10} &\\sim 10^{-9}\\, e& {\\rm for} & {\\mathcal{V}}\\sim 10^{27}\n \\end{array}.\n\\end{equation}\nThis is exactly the regime where our bound is most constraining.\n\nFinally, let us also take a brief look at the kinetic mixing in these scenarios (cf. \\cite{Mark} for details).\nAn estimate of the kinetic mixing between a visible sector (non-hyperweak) U(1) and a hidden hyperweak U(1) gauge group\nyields,\n\\begin{equation}\n\\chi\\sim \\frac{e e_{h}}{6\\pi^2}.\n\\end{equation}\nAccordingly we find for the mini-charge\n\\begin{equation}\n|\\epsilon|=\\left|\\frac{e_{h}\\chi}{e}\\right|\\sim \\frac{e^{2}_{h}}{6\\pi^2}\\sim \\bigg\\{\\begin{array}{ccc}\n {\\rm few}\\times 10^{-10}&{\\rm for}&{\\mathcal{V}}\\sim 10^{12} \\\\\n {\\rm few}\\times 10^{-20}&{\\rm for}&{\\mathcal{V}}\\sim 10^{27}\n \n\\label{hyper} \\end{array}.\n\\end{equation}\n\nComparing with Fig.~\\ref{fig:caca} we see that our bound probes the region of interest for the scenario with a string scale $M_{s}\\sim 10^{11}\\,{\\rm GeV}$.\nWe have summarized this in Fig.~\\ref{fig:summary} where we have also\nincluded bounds from laboratory searches and low density\/temperature\nbounds from cosmology. The solid line shows the edge of the excluded\nregion for models with only mini-charged particles. If the strength of\nthe magnetic field and the plasma frequency are assumed to be the same in each magnetic domain\nin the cluster, for models with\nhidden photons and hyperweak hidden gauge couplings satisfying (\\ref{hyper}) there are `holes'\nin the excluded region. However if small fluctuations in the\nmagnetic field strength and plasma frequency are allowed, and such\nfluctuations would naturally be expected to occur in the galaxy\ncluster, these holes are closed and the entire region above the solid\nline is excluded.\n\n\\begin{figure}[tbp] \n\\centering\n{\n\\includegraphics[width=13cm,angle=0]{final_hyper.eps}\n}\n\\caption{Bounds on mini-charged particles for very weak hidden sector gauge couplings. They apply also to models with only mini-charged particles.\nThe solid black line shows the upper bound on the mini-charge obtained in this paper from the SZ effect.\nThe green area is a prediction in LARGE volume scenarios in string theory with a hyperweak U(1) and a string scale $M_{s}\\lesssim10^{11}\\,{\\rm GeV}$.\nFor comparison, we have also included bounds arising from accelerators\n\\cite{Davidson:1993sj,Davidson:2000hf}, Lamb shift\n\\cite{Gluck:2007ia}, positronium decays \\cite{Badertscher:2006fm}, tests\nof Coulomb's Law \\cite{Jaeckel:2009dh}, accelerator cavities \\cite{Gies:2006hv}, laser polarization experiments\n\\cite{Ahlers:2007qf}, the CMB~\\cite{Melchiorri:2007sq}\nand supernova dimming~\\cite{Ahlers:2009kh}.\nAll these bounds arise from physics occurring in low density\/temperature regions.}\n\\label{fig:summary}\n\\end{figure}\n\n\n\\section{The ISW effect: A future test}\n\\label{sec:ISW}\n On the largest scales on the sky ($\\theta>1^{\\circ}$) the dominant source of secondary\n anisotropies of the CMB is not the SZ effect but the Integrated\n Sachs-Wolfe effect (ISW) \\cite{Sachs:1967er}. The ISW effect occurs when\n gravitational potentials evolve with time, as then the blue-shifting\n of a photon falling into the gravitational well is not exactly\n canceled by the red-shifting of the photon climbing out of the\n potential well, and there is a net effect on the energy of a\n photon,\n\\begin{equation}\n\\frac{\\Delta T}{T}\\approx-2 \\int \\dot{\\Phi} d\\tau,\n\\end{equation}\nwhere $\\Phi$ is the gravitational potential along a line of sight, and a dot denotes differentiation with\nrespect to conformal time, $\\tau$.\n\nIf the universe is close to being flat most of\nthe late time ISW effect is caused by dark energy, and observations of\nthe\nISW effect have been used to probe its properties. However measuring the ISW effect is not easy as the signal\nis a fraction of the size of the primordial CMB anisotropies. It can\nbe extracted, however, by looking for correlations between the CMB\ntemperature fluctuations and tracers of the density of matter. These\ncorrelations have been detected using a variety of density probes and\nover a wide range of the electromagnetic spectrum \\cite{Giannantonio:2008zi}.\n\nThe ISW effect dominates on the very largest scales, those of\nsuperclusters of galaxies which can stretch over distances of tens of mega-parsecs.\nWe would expect mini-charged particles to be constrained by\nobservations of the ISW effect\nonly if there exist magnetic fields in galaxy superclusters. So far no\ndetailed study has been done of the magnetic fields on such scales, however there are some hints that magnetic fields exist in these\nenvironments from observations of the diffuse radio emission from\nlarge scale networks of galaxies \\cite{Bagchi:2002vf}. Also if the magnetic fields\nin galaxies are produced during structure formation (as opposed to\nhaving their origin in a primordial magnetic field) then simulations\nshow that magnetic fields should indeed exist in superclusters of\ngalaxies (e.g. \\cite{Ryu:1998up}).\n\nIt is to be hoped that as we continue to learn more abut the magnetic\nfields in super clusters of galaxies we will be able to use\nmeasurements of the ISW effect to constrain the properties of\nmini-charged particles. The magnetic fields in these structures are\nexpected, from simulations, to be of the same order of magnitude as\nthose in galaxy clusters and so we\nexpect to be able to significantly improve our bounds, both\nbecause the temperature fluctuations in the CMB due to the ISW effect\nare much smaller $\\Delta T\/T \\sim 10^{-6}$ than those of the SZ\neffect, and also because the distances traveled through the magnetic\nfields of superclusters of galaxies are greater than the equivalent\ndistances used in the SZ calculation.\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conc}\nMini-charged particles arise in a variety of extensions to the\nStandard Model, most naturally in models which also contain hidden photons. In such models the\npropagation of photons in a transverse magnetic field is affected by the real and\nvirtual production of MCPs.\nMCPs can be constrained by a wide variety of terrestrial,\nastrophysical and cosmological experiments. However some of these\nconstraints, those coming from processes occurring in dense environments such as the\ninterior of stars, do not constrain all available MCP models. Therefore it is necessary to have\nalternative probes of this region of parameter space which require only\nphysics in low density\/temperature environments. Such constraints are particularly\nrelevant for upcoming laboratory searches for MCPs.\n\nIn this article we have developed such a new test for MCPs from\nobservations of the Sunyaev-Zel'dovich effect. As photons from the\nCMB pass through the magnetic fields of galaxy clusters some photons\nare lost due to the production of MCPs and this decrement in the\nphoton flux appears as an additional contribution to the SZ\neffect. Insisting that this flux decrement is not larger than the\nobserved SZ effect constrains new regions of\nthe MCP parameter space. Our bounds are most constraining for models\nof MCPs without hidden sector photons, and for models with small\nhidden sector gauge coupling which are well motivated in the LARGE\nvolume string scenario.\n\nIf, in the future, new observations lead us to a better understanding\nof the magnetic fields in the largest scale structures in the\nuniverse, a similar test could be made with observations of the ISW\neffect, which have\nthe prospect to be even more constraining for MCP models.\n\n\\section*{Acknowledgments}\nWe would like to thank Mark Goodsell for valuable discussions and\ncollaboration on related subjects.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nRecommender systems play an important role in modern web services, like e-commerce and online advertising, which mitigate the information overload problem by suggesting users with personalized items according to their own interests. Thanks to the recent development of deep learning techniques, deep neural networks have been intensively applied to recommender systems \\cite{covington2016deep,hidasi2015session,ying2018graph}, which give rise to remarkable improvement for the recommendation quality. Knowing that the online platforms need to deal with a tremendous amount of items, deep recommender systems call for collaboration of two different models: the retrieval model (retriever) and the ranking model (ranker)~\\cite{covington2016deep}. Particularly, the retriever targets selecting a small set of candidate items at a context (i.e. all information except items, such as user, interaction history, time and location) from the whole items with high efficiency. Typically, such kind of models learns to represent contexts and items as dense embeddings, where context-aware preference towards the item can be efficiently estimated based on the embedding similarity, e.g., inner product and cosine similarity. In contrast, the ranker is used to identify the best items from the retrieval results with high precision. To this end, it usually leverages expressive yet time-consuming networks, especially those establishing the in-depth interactions between user and item (e.g., DIN~\\cite{zhou2018deep}, DIEN~\\cite{zhou2019deep} and DeepFM~\\cite{guo2017deepfm}), for the fine-grained prediction of context-aware preference towards the retrieved items. \n\n\\subsection{Existing Problems} \nDespite the collaborative nature of the retriever and ranker, the typical training workflow of the two models lacks effective cooperation, which severely harms the overall recommendation quality. Most of the time, the two models are independently trained and directly applied to the recommender systems~\\cite{zhu2018learning,li2019multi,cen2020controllable,zhou2018deep,zhou2019deep}. In other times, the ranker is successively trained based on retrieval results; whereas, the retriever remains independently trained~\\cite{covington2016deep}. Such training workflows are inferior due to the following reasons. Firstly, the independently-trained retriever is prone to low recall due to its limited expressiveness. Secondly, the independently-trained ranker may suffer from poor discriminative capability, considering that the retrieval results can be largely differentiated from the randomly sampled items. \n\n\\subsection{Our Solution} \nTo overcome the existing problem, we propose a novel framework for cooperatively training the retriever and ranker, a.k.a. \\textbf{CoRR} (\\textbf{Co}operative \\textbf{R}etriever and \\textbf{R}anker). In our framework, the retriever and ranker are simultaneously trained within a unified workflow, where both models can be mutually reinforced. \n\n$\\bullet$ \\textbf{Training of ranker}. The ranker is trained with sampled softmax~\\cite{bengio2008adaptive} on top of the hard negative items sampled from the retriever. Particularly, instead of working with the ``easy negatives'' randomly sampled from the whole items~\\cite{rendle2009bpr,zhou2018deep,guo2017deepfm}, the ranker is trained to discriminate the true positive from the hard negatives sampled from the retriever. {Such a change is beneficial to the overall recommendation quality when the retriever and ranker are assembled as a cascaded pipeline.} Moreover, with the improved accuracy of the retriever, negative samples drawn from the retriever will become increasingly harder, which may further contribute to the discriminative capability of the ranker. \n\n$\\bullet$ \\textbf{Training of retriever}. On one hand, the retriever is learned from recommendation data via sampled softmax and from the ranker via knowledge distillation~\\cite{hinton2015distilling}. In a specific context, a small set of items are sampled in the first place. Then, the ranker is required to predict context-aware fine-grained preferences towards the sampled items. The retriever is expected to generate the same ranking order for the sampled items as the ranker; to this end, the KL-divergence is minimized for the normalized predictions between the retriever and ranker~\\cite{hinton2015distilling}. Compared with the conventional training methods (typically on top of contrastive learning), knowledge distillation is preferable for two reasons. {Firstly, the high-preference items, whether interacted or not, will probably get highly-rated by the ranker; as a result, such items will virtually become positive samples, which substantially augments the recommendation data. Secondly, unlike the conventional methods where all the non-interacted items are equally treated as negative samples, the low-preference items will probably be low-rated by the ranker, thus being penalized more when the retriever is trained. By doing so, the false negative issues for the non-interacted items can be largely alleviated.}\n\nThe mutual reinforcement of the retriever and ranker will finally result in a highly competitive recommendation accuracy. To facilitate the optimized conduct of the proposed framework, a couple of technical designs are introduced for more effective sampling and knowledge distillation, respectively. Firstly, knowing that directly sampling from the retriever can be extremely time-consuming when dealing with a large itemset, we propose a \\textbf{scalable and adaptive sampling strategy}, where the items favored by the retriever can be efficiently sampled with constant time complexity. Secondly, the direct knowledge distillation over the sampled items is biased and prone to inferior performances. To mitigate this problem, an \\textbf{asymptotic-unbiased approximation of KL divergence} is introduced for the compensation of the bias resulting from the sampled items. On top of this operation, the retriever will become more consistent with the ranker in terms of the global ranking order. \n\nWe conduct comprehensive experimental studies over four benchmark datasets for the evaluation of CoRR. According to the experiment results, the overall recommendation quality can be substantially improved by CoRR in comparison with the existing training methods. More detailed analysis further verifies CoRR's standalone effectiveness to the retriever and ranker, and its general effectiveness as a model-agnostic framework. Finally, the contribution of our work is summarized with the following points. \n\n\\begin{itemize}[leftmargin=*]\n\t\\item To the best of our knowledge, CoRR is the first work which realizes the cooperative training between retriever and ranker in recommender systems. Thanks to the mutual reinforcement between the two modules, the overall recommendation quality can be substantially enhanced.\n\t\\item A couple of strategies are introduced for optimized conduct of CoRR: 1) develop the scalable and adaptive sampling strategy, enabling efficient sampling from the retriever; 2) propose an asymptotic-unbiased approximation of KL divergence, effectively compensating the bias induced by sampled items.\n\n\t\\item We perform comprehensive experimental studies on four benchmark datasets, whose results verify CoRR's advantage against the existing deep recommenders, and its standalone effectiveness to the retriever and ranker.\n\\end{itemize}\n\n\n\n\n\\section{Related Work}\nThis paper studied the cooperation between the ranker and retriever in deep recommender systems. Therefore, we first review some important deep learning techniques for recommendation, and then present negative sampling and knowledge distillation.\n\\subsection{Deep Learning for Recommendation}\nThe earliest successful DL-based methods lie in modeling side information, such as textual data~\\cite{huang2013learning,wang2015collaborative}, visual data~\\cite{he2016vbpr}, knowledge graph~\\cite{zhang2016collaborative} via deep learning techniques. However, behavior data is shallowly modeled based on inner product or Euclidean distance between user vector and item vector. Such deep models, which can be generalized to a two-tower similarity network, have an advantage in conveniently using the SOTA ANN algorithms. These methods have been extended by modeling feature interaction in an explicit or implicit way, whose representative works include Deep\\&Wide~\\cite{cheng2016wide}, NFM~\\cite{he2017neuralfm}, DeepFM~\\cite{guo2017deepfm}, XDeepFM~\\cite{lian2018xdeepfm} and Deep\\&Cross~\\cite{wang2017deep}. Since behavior data is sequential in nature, therefore, it is intuitive to apply sequential deep learning methods for the sequential recommendation, where the representative works include SASRec~\\cite{kang2018self} based on transformer, GRU4Rec based on GRU~\\cite{hidasi2015session}, CASER based on CNN~\\cite{tang2018personalized}. To refine user preferences, DIN~\\cite{zhou2018deep} and DIEN~\\cite{zhou2019deep} further incorporate complex feature interaction between recommended items and interaction history via the attention mechanism. Though recommendation performance can be improved in this way, recommendation efficiency is remarkably affected since the off-the-shelf ANN algorithms can not be applied anymore.\n\n\\subsection{Negative Sampling in RecSys}\nMany methods sample negative items from static distributions, such as uniform distribution~\\cite{rendle2009bpr,he2017neural} and popularity-based distribution~\\cite{rendle2014improving}. To adapt to recommendation models, many advanced sampling methods have been proposed. For example, AOBPR~\\cite{rendle2014improving} transforms adaptive sampling into searching the item at a randomly-drawn rank. CML~\\cite{hsieh2017colla} and WARP~\\cite{weston2010large} draw negative samples for each positive by first drawing candidate items from uniform distribution and then selecting more highly-scored items than positive minus one as negative. Dynamic negative sampling (DNS)~\\cite{zhang2013optimizing} picks a set of negative samples from the uniform distribution and then choose the most highly-scored item as negative. Self-adversarial negative sampling~\\cite{sun2019rotate} draws negative samples from the uniform distribution but weight them with softmax-normalized scores. Kernel-based sampling~\\cite{blanc2018adaptive} picks samples proportionally to a quadratic kernel in a divide and conquer way. Locality Sensitive Hashing (LSH) over randomly perturbed databases enables sublinear time sampling~\\cite{mussmann2016learning} and LSH itself can generate correlated and unnormalized samples~\\cite{spring2017new}. Quantization-based sampling~\\cite{chen2021fast} decomposes the softmax-normalized probability via product quantization into the product of probabilities over clusters, such that sampling an item is in sublinear time.\nIRGAN~\\cite{wang2017irgan} distinguishes positive items from the randomly picked negatives according to the generator, but sampling is time-consuming. Though the generator (sampler) is simultaneously updated along with the discriminator, it is only trained by policy gradient (PG) to promote highly-probable items according to the discriminator. It does not only lack the supervision from the data but also suffers from PG-induced low training efficiency.\n\n\\subsection{Knowledge Distillation in RecSys}\nKD~\\cite{hinton2015distilling} provides a powerful model-agnostic framework for compressing a teacher model into a student model, by learning to imitate predictions from the teacher. Therefore, KD has been applied in RecSys for compressing deep recommenders. A pioneer work is Ranking Distillation (RD)~\\cite{tang2018ranking}, where the top-$k$ recommendation from the teacher model is considered as position-weighted pseudo positive. The subsequent distillation methods improve the use of the top-$k$ recommendations, such as drawing a sample from the top-k items via ranking-based distribution~\\cite{lee2019collaborative}, its mixture with uniform distribution~\\cite{kang2020rrd} and rank discrepancy-aware distribution~\\cite{kweon2021bidirectional}. In addition teacher's prediction, the latent knowledge in the teacher can be distilled into the student via hint regression~\\cite{kang2020rrd} and topology distillation~\\cite{kang2021topology}. In contrast to them, our method does not depend on the top-k results of the retriever at all. \n\n\n\n\\section{Cooperative Retriever and Ranker}\nIn this section, we first overview the training of the CoRR on a training dataset $\\mathcal{D}=\\{(c,k)\\}$, which consists of pairs of a context $c$ and a positive item $k$. The context indicates all information except items, such as user, interaction history, time and location. Following that, we elaborate on how to train the ranker with hard negative and how to train the retriever by knowledge distillation in CoRR. \n\\subsection{Overview}\nFor the sake of scalable recommendation, deep recommender relies on the collaboration of two models: the retrieval model (retriever) and the ranking model (ranker). The retriever targets on selecting a small set of potentially positive items from the whole items with high efficiency. Typically, the retriever is represented by $M_\\theta(i,c)=\\text{sim}\\big(E_{\\theta_1}(i),E_{\\theta_2}(c)\\big)$, the similarity (e.g. Euclidean distance, inner product and cosine similarity) between item embedding $E_{\\theta_1}(i)$ and context embedding $E_{\\theta_2}(c)$. Therefore, we can use off-the-shelf ANNs, such as FAISS and SCANN, for retrieving similar items as candidates in sublinear time. The ranker aims to identify the best items from the retrieval results with high precision, and usually represented by expressive yet time-consuming networks $R_\\phi(i,c)$ taking an item $i$ and a context $c$ as inputs. \n\nDue to time-consuming computation in the ranker, in most cases, traditional training methods learn to discriminate positive from randomly picked negatives. In spite of simplicity, the training of the ranker is independent of the retriever. In some cases, the ranker is trained by discriminating positives from the retrieval results. In spite of establishing a connection with the retriever, the ranker easily suffers from the false negative issue, since potential positives are definitely included in the retrieval results. Moreover, due to the lack of methodological support, this negative sampling strategy may introduce a large bias to gradient, being easily stuck into local optimum. To address the problem, we optimize the ranker with respect to sampled log-softmax, which is an asymptotic-unbiased estimation of log-softmax based on importance sampling. The sampled log-softmax relies on samples efficiently drawn from a proposal distribution. To connect the ranker with the retriever, we construct the proposal distribution from the retriever based on importance resampling. \n\nIn addition to supervision from data, the training of the retriever can be also guided by the ranker based on knowledge distillation since the ranker is more precise in recommendation. Regarding supervision from recommendation data, the sampled log-softmax objective is also exploited for optimizing the retriever based on the same proposal. Regarding knowledge distillation from the ranker, most existing methods distill knowledge in top-k results from the ranker, but the top-k recommendation results are time-consuming to compute for the ranker. In this paper, we distill the ranking order information from the ranker's predictions, by directly matching normalized predictions between the ranker and the retriever based on the KL divergence. To reduce the time cost of optimization, we propose an asymptotic-unbiased estimation -- sampled KL divergence.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{corr_framework.pdf}\n\t\\vspace{-0.5cm}\n\t\\caption{The framework of CoRR. The context $c$ indicates all information except items. $Y(\\cdot)$ indicates the static proposal and $Q_\\mathcal{C}(\\cdot|c)$ indicates resampling probability over the sampled itemset $\\mathcal{C}$. The dotted line means the gradient is stopped.} \n\t\\label{fig:corrframework}\n\\end{figure}\n\nThe retriever and ranker are simultaneously trained with a unified workflow, where both models can be mutually reinforced. Along with the training course, the ranker becomes increasingly precise, which in return provides more informative supervision signals for the retriever; meanwhile, with the improvement of the retriever, generating samples will become increasingly harder, contributing to a higher discriminative capability of the ranker. The overall framework is illustrated in Figure~\\ref{fig:corrframework}. The overall procedure can be referred to in Algorithm~\\ref{alg:corr}.\n\n\\subsection{Training Ranker with Retriever}\n\\subsubsection{Loss Function}\nIn this paper, we optimize the ranker w.r.t sampled log-softmax, which is an asymptotic-unbiased estimation of log-softmax. The use of log-softmax is motivated by its close relationship with the logarithm of Mean Reciprocal Rank (MRR) and the excellent recommendation performance~\\cite{liang2018variational}. Assuming the positive item is $k$ at a context $c$, the log-softmax objective w.r.t $k$ is formulated as follows:\n\\begin{displaymath}\n\\begin{aligned}\n \\ell(k,c) & = R_\\phi(k,c) - \\log \\Big(\\sum_i \\exp\\big(R_\\phi(i,c)\\big)\\Big)\\\\\n & = - \\log \\Big(\\sum_i \\exp\\big(R_\\phi(i,c) - R_\\phi(k,c)\\big)\\Big) \\\\\n &\\le -\\log \\Big(\\sum_i \\mathbb{I}_{\\ge 0}\\big( R_\\phi(i,c) - R_\\phi(k,c) \\big) \\Big)\\\\\n &=\\log \\Big(\\frac{1}{\\text{rank}_R(k,c)}\\Big) = \\log \\text{MRR}\\\\\n\\end{aligned}\n\\end{displaymath}\nwhere the inequality holds due to $\\exp(\\cdot) \\ge \\mathbb{I}_{\\ge 0}(\\cdot)$. \n\\begin{algorithm}[t]\n\t\\KwIn{Training Data $\\mathcal{D}$}\n\t\\KwOut{Retriever $M_\\theta$ and Ranker $R_\\phi$ }\n\t\\Repeat{Convergence or reaching maxepoch}\n\t{\n\t\t\\ForEach{$(c,k) \\in \\mathcal{D}$}\n\t\t{\n\t\t\t$\\mathcal{C} \\gets $ Draw $n$ samples according to $Y(\\cdot)$\\;\n\t\t\t$M_\\theta(\\cdot,c) \\gets $ Forward-Inference($M_\\theta$, $\\mathcal{C}\\cup \\{k\\}$) \\;\n\t\t\t$Q_{\\mathcal{C}\\cup \\{k\\}}(\\cdot|c) \\gets \\text{softmax}\\Big(M_\\theta(\\cdot,c)\/T - \\log Y(\\cdot)\\Big)$\\;\n\t\t\t$\\mathcal{S} \\gets$ Draw $L$ samples according to $Q_{\\mathcal{C}\\cup \\{k\\}}(\\cdot|c)$\\;\n\t\t\tUpdate $M_\\theta$ w.r.t. $\\tilde{\\ell}_\\theta(c,k)+D_{KL}(P_\\mathcal{S}(\\cdot|c)\\parallel Q_\\mathcal{S}(\\cdot|c))$\\;\n\t\t\tUpdate $R_\\phi$ w.r.t. $\\tilde{\\ell}_\\phi(c,k)$\n\t\t}\n\t}\n\t\\caption{Cooperative Retriever and Ranker}\\label{alg:corr}\n\\end{algorithm}\n\nHowever, it is computationally challenging to optimize the log-softmax, since the gradient computation scale linearly with the number of items, i.e.,\n\\begin{displaymath}\n \\nabla \\ell_\\phi(k,c) = \\nabla R_\\phi(k,c) - \\mathbb{E}_{i\\sim P(\\cdot|c)} [ \\nabla R_\\phi(i,c)]\n\\end{displaymath}\nwhere $P(\\cdot|c)$ represents categorical probability distribution over the whole items $\\mathcal{I}$ with the parameter $\\text{softmax}(R_\\phi(\\cdot,c))$, i.e., $P(j|c)=\\frac{\\exp(R_\\phi(j,c)}{\\sum_{i\\in \\mathcal{I}}\\exp(R_\\phi(i,c))}$. To improve the efficiency, importance sampling was used for approximation by drawing $L$ samples $\\mathcal{S}=\\{o_1, \\cdots, o_L\\}$ from the proposal distribution $Q_0(\\cdot|c)$, which is based on the retriever $M_\\theta$. That is,\n\\begin{displaymath}\n \\nabla \\ell_\\phi(k,c) \\approx \\nabla R_\\phi(k,c) - \\sum_{i\\in \\mathcal{S}\\cup \\{k\\}} w(i,c) \\nabla R_\\phi(i,c)\n\\end{displaymath}\nwhere $w(i,c)=\\frac{\\exp\\big(R_\\phi(i,c)-\\log Q_0(i|c)\\big)}{\\sum_{j\\in \\mathcal{S}\\cup \\{k\\}} \\exp\\big(R_\\phi(j,c) - \\log Q_0(j|c)\\big)}$. It is easy to verify that this gradient equals that of the following sampled log-softmax.\n\\begin{displaymath}\n\t\\ell_\\phi(k,c)\\approx \\tilde{\\ell}_\\phi(k,c) =\\log \\frac{\\exp\\big(R_\\phi(k,c) - \\log {Q}_0(k|c)\\big)}{\\sum_{i\\in \\mathcal{S}\\cup \\{k\\}} \\exp\\big(R_\\phi(i,c) - \\log {Q}_0(i|c)\\big)}\n\t\\label{eq:sample_logsoftmax}\n\\end{displaymath}\nwhere ${Q}_0(\\cdot|c)$ can be unnormalized. Note that both positive item $k$ and sampled itemset $\\mathcal{S}$ are used in these formulations. This is because it is necessary to guarantee negativeness of the sampled log-softmax like log-softmax. \n\nLet's consider to set $Q_0(j|c) = \\frac{\\exp(M_\\theta(j,c))}{\\sum_{i \\in \\mathcal{I}} \\exp (M_\\theta(i,c))}$. The sampled log-softmax can be reformulated as $$\\tilde{\\ell}_\\phi(k,c)=\\log \\frac{\\exp\\big(R_\\phi(k,c) - M_\\theta(k,c)\\big)}{\\sum_{i\\in \\mathcal{S}\\cup \\{k\\}} \\exp\\big(R_\\phi(i,c) - M_\\theta(i,c)\\big)}.$$\nIn spite of concision, it is time-consuming to draw samples from $Q_0(j|c)$ as aforementioned. In the next part, we will elaborate on how to accelerate item sampling according to the retriever.\n\n\\subsubsection{Sampling from the Retriever} \\label{sec:resample}\nThe top-k predictions from the retriever absolutely include both hard negatives and potential positives. Existing empirical study shows the only use of the top-k results is sub-optimal for training the model. Therefore, we suggest to draw samples from $Q(i|c) = \\frac{\\exp\\big(M_\\theta(i,c) \/T\\big)}{\\sum_{j\\in \\mathcal{I}} \\exp\\big(M_\\theta(j,c)\/T\\big)}$. The parameter $T$ controls the balance between hardness and randomness of negative samples. When $T\\rightarrow \\infty$, sampling approaches completely random; when $T\\rightarrow 0$, it becomes fully deterministic, being equivalent to argmax. However, as aforementioned, it is time-consuming to draw samples from $Q(i|c)$ while sampling efficiency is important for training the model. \n\nIn this part, we propose a scalable and adaptive sampler for approximately sampling from $Q(i|c)$, which consists of two steps:\n\\begin{itemize}[leftmargin=*]\n\t\\item The first step is to draw a comparatively large pool (of size $n$) of candidate samples from a static categorical distribution $Y(\\cdot)$, such as uniform and popularity-based distribution.\n\t\\item The second step is to draw a small size of ($L